Market Completion and Robust Utility Maximization

Marke Compleion and Robus Uiliy Maximizaion DISSERTATION zur Erlangung des akademischen Grades docor rerum nauralium (Dr. rer. na.) im Fach Mahemaik eingereich an der Mahemaisch-Naurwissenschaflichen Fakulä II Humbold-Universiä zu Berlin von Dipl. mah. oec. Mahias Müller geboren am 12.12.1972 in Dresden Präsiden der Humbold-Universiä zu Berlin: Prof. Dr. Jürgen Mlynek Dekan der Mahemaisch-Naurwissenschaflichen Fakulä II: Prof. Dr. Uwe Küchler Guacher: 1. Prof. Dr. Peer Imkeller 2. Prof. Dr. Alexander Schied 3. Prof. Dr. Said Hamadène Tag der mündlichen Prüfung: 26. Mai 25

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Absrac In his hesis we sudy wo problems of financial mahemaics ha are closely relaed. The firs par proposes a mehod o find prices and hedging sraegies for risky claims exposed o a risk facor ha is no hedgeable on a financial marke. In he second par we calculae he maximal uiliy and opimal rading sraegies on incomplee markes using Backward Sochasic Differenial Equaions. We consider agens wih incomes exposed o a non hedgeable exernal source of risk who complee he marke by creaing eiher a bond or by signing conracs. Anoher possibiliy is a risk bond issued by an insurance company. The sources of risk we hink of may be insurance, weaher or climae risk. Sock prices are seen as exogenuosly given. We calculae prices for he addiional securiies such ha supply is equal o demand, he marke clears parially. The preferences of he agens are described by expeced uiliy. In Chaper 2 hrough Chaper 4 he agens use exponenial uiliy funcions, he model is placed in a Brownian filraion. In order o find he equilibrium price, we use Backward Sochasic Differenial Equaions. Chaper 5 provides a one period model where he agens use uiliy funcions saisfying he Inada condiion. The second par of his hesis considers he robus uiliy maximizaion problem of a small agen on a incomplee financial marke. The model is placed in a Brownian filraion. Eiher he probabiliy measure or drif and volailiy of he sock price process are uncerain. The rading sraegies are consrained o closed convex ses. We apply a maringale argumen and solve a saddle poin problem. The soluion of a Backward Sochasic Differenial Equaion describes he maximizing rading sraegy as well as he probabiliy measure ha is used in he evaluaion of he robus uiliy. We consider he exponenial, he power and he logarihmic uiliy funcions. For he exponenial uiliy funcion we calculae uiliy indifference prices of no perfecly hedgeable claims. Finally, we apply hose echniques o he maximizaion of he expeced uiliy wih respec o a single probabiliy measure. We apply a maringale argumen and solve maximizaion problems insead of saddle poin problems. This allows us o consider closed, in general non convex consrains on he values of rading sraegies. Keywords: marke compleion, incomplee financial marke, uiliy maximizaion, backward sochasic differenial equaions

Zusammenfassung In dieser Arbei sudieren wir zwei Probleme der Finanzmahemaik, die eng zusammenhängen. Der erse Teil beschreib eine Mehode, Auszahlungen zu beweren, die einem auf dem Finanzmark nich absicherbaren Risiken ausgesez sind. Im zweien Teil berechnen wir den maximalen Nuzen und opimale Handelssraegien auf unvollsändigen Märken mi Hilfe von sochasischen Rückwärsgleichungen. Wir berachen Händler, deren Einkommen einer exernen Risikoquelle ausgesez sind. Diese vervollsändigen den Mark, indem sie enweder einen Bond schaffen oder gegenseiig Verräge schließen. Eine andere Möglichkei is eine Anleihe, die von einer Versicherung herausgegeben wird. Die Risikoquellen, die wir in Berach ziehen, können Versicherungs-, Weer oder Klimarisiko sein. Akienpreise sind exogen gegeben. Wir berechnen Preise für die zusäzlichen Anlagen so dass Angebo und Nachfrage dafür gleich sind. Wir haben parielle Markräumung. Die Präferenzen der Händler sind durch erwareen Nuzen gegeben. In Kapiel 2 bis Kapiel 4 haben die Händler exponenielle Nuzenfunkionen. Um den Gleichgewichspreis zu finden, wenden wir sochasische Rückwärsgleichungen an. In Kapiel 5 beschreiben wir ein Einperiodenmodell, wobei die Händler Nuzenfunkionen verwenden, die die Inada-Bedingungen erfüllen. Der zweie Teil dieser Arbei beschäfig sich mi dem robusen Nuzenmaximierungsproblem eines kleinen Händlers auf einem unvollsändigen Finanzmark. Enweder das Wahrscheinlichkeismaß oder die Koeffizienen des Akienmarkes sind ungewiss. Die Handelssraegien sind auf abgeschlossene konvexe Mengen beschränk.wir wenden ein Maringalargumen an und lösen Saelpunkprobleme. Die Lösung der Rückwärsgleichung beschreib die nuzenmaximierende Handelssraegie und das Wahrscheinlichkeismaß, das in der Auswerung des robusen Nuzens benuz wird. Für die exponenielle Nuzenfunkion berechnen wir Nuzenindifferenzpreise für nich absicherbare Auszahlungen. Ausserdem wenden wir diese Techniken auf die Maximierung des erwareen Nuzens bezüglich eines Wahrscheinlichkeismaßes an. Wir nuzen ein Maringalargumen und lösen Maximierungsprobleme anselle von Saelpunkproblemen. Dies erlaub uns, abgeschlossene, im allgemeinen nich konvexe zulässige Mengen für die Handelssraegien zu berachen. Schlagwörer: Markvervollsändigung, unvollsändige Finanzmärke, Nuzenmaximierung, sochasische Rückwärsgleichungen

Conens 1 Inroducion 1 1.1 Par I Marke compleion.................... 1 1.2 Par II Robus uiliy maximizaion............... 5 I Marke compleion, hedging exernal risk facors 15 2 Equilibrium wih risk securiy 17 2.1 The sock, exernal risk...................... 18 2.2 Prices for he risk securiy, rading............... 2 2.3 Uiliy maximizaion....................... 22 2.4 Equilibrium wih parial marke clearing............ 28 3 Marke compleion wih conracs 37 3.1 Socks, prices of risk ransfer................... 38 3.2 Uiliy maximizaion....................... 4 3.3 Equilibrium wih conracs.................... 41 4 A risk bond 45 4.1 The risk bond........................... 46 4.2 Parial marke clearing...................... 53 4.3 Risk bond compleing he marke................ 57 5 General uiliies 65 5.1 Income, preferences, he marke................. 66 5.2 Equilibrium wih parial marke clearing............ 7 6 An incomplee marke 81 v

II Uiliy maximizaion 87 7 Robus uiliy maximizaion 89 7.1 Sock marke and robus uiliy................. 9 7.2 Robus exponenial uiliy maximizaion............ 93 7.3 Power Uiliy........................... 16 8 Uncerain sock price dynamics 113 8.1 Exponenial uiliy........................ 113 8.2 Power uiliy............................ 118 8.3 Logarihmic uiliy........................ 12 9 Uiliy maximizaion 123 9.1 The exponenial uiliy...................... 124 9.2 Power uiliy........................... 131 9.3 Logarihmic Uiliy........................ 137 A BMO maringales 141 vi

Chaper 1 Inroducion 1.1 Par I Marke compleion Pricing and hedging of opions on socks is well undersood. The famous Black Scholes formula gives he price of a call opion. Using maringale mehods relying on he reqiremen ha he opion does no creae an opporuniy of arbirage, price processes and he hedging sraegy of opions wrien on socks in a complee marke can be calculaed. In recen years, new ypes of financial producs have appeared. Insurances aim a ransferring insurance risk o financial markes. This is done by securiizaion, a securiy, e.g. a bond, is creaed ha depends on a non financial risk facor. Those securiies are ofen called CAT bonds. The bes known example are earhquake bonds for California. No insurance company is willing o ake a large par of he earhquake risk because he loss poenial is oo high. Insead, earhquake bonds are sold o large banks or hedge funds. If an earhquake occurs he invesors are no repayed. They even loose he principal of he bond. There are also CAT bonds covering he risk of hurricanes. Many examples are given in he aricle Economic aspecs of securiizaion of risk by Cox e al., (CFP). An example of a securiy on weaher risk is he Heaing Degree Day (HDD) swap. This paper is raded a he Chicago Merkanile Exchange. The payoff of a HDD swap depends on he emperaure during a heaing period. If he emperaure is higher han usual, he buyer has o pay o he seller. If he emperaure is lower, he seller pays. The swap is ideal for energy producers. They can hedge volume risk, he volume of energy sold depends on he emperaure. The Winherur insurance issued a bond ha ransfers he risk of hailsorms. Srucure and pricing of his bond are described in Schmock, (Sch99). 1

2 CHAPTER 1. INTRODUCTION Anoher example is given by he risk a reinsurance company faces due o big accumulaive losses for example in farming or fishing caused by he mos well known shor erm climae even of he El Niño Souhern Oscillaion (ENSO). All hose securiies have in common ha heir payoff depends on non financial, i.e. exernal risk facors. Those risks canno be hedged on a financial marke. How should a claim be priced ha depends on exernal risk? How does he price process of a securiy on an exenal risk facor evolve? Here we skech some pricing mehods described in he lieraure of financial mahemaics. One echnique o consruc prices and hedging sraegies in incomplee markes comes from a uiliy indifference argumen. The rader uses he rading sraegy ha maximizes he expeced uiliy of he erminal wealh aained wih he rading sraegy minus he claim he has o pay a a cerain ime. The uiliy indifference price is given by he adjusmen of he iniial capial such ha he maximal uiliy is he same as wih he no adjused iniial capial wihou he liabiliy. This means he rader is indifferen beween eiher geing he price and acceping he obligaion o pay or doing nohing. The uiliy indifference argumen also yields a hedging sraegy. For exponenial uiliy funcions, uiliy indifference prices are calculaed in Becherer, (Bec1) and Delbaen e el. (Del3). A very closely relaed pricing principle is he resul of an infiniesimal indifference argumen. The price of he nonhedgeable claim is chosen such ha he rader is indifferen beween eiher acceping an infiniesimal small par of he claim or doing nohing. Davis (Dav1) used his argumen o price a emperaure bond. Boh ypes of uiliy indifference argumens ake eiher he poin of view of a buyer or a seller. The preferences of only one rader are aken ino accoun. The quadraic hedging approach sees he price from he perspecive of he buyer and he seller simulanuosly. The expecaion of he square of he difference beween he erminal value of a rading sraegy and he claim is aken. This quaniy is minimized over all rading sraegies. One can compare his funcional over differen iniial capials. The iniial capial such ha his funcional is minimized is he price for he claim. Since gains and losses boh are punished, he price can be seen as a compromise beween a buyer and a seller. A survey can be found in he aricle of Schweizer, (Sch1). Møller (Møl1) uses his approach in order o price insurance conracs. A fundamenally differen approach are equilibrium prices. Karazas e al. (KL9) consider agens who obain a random income. The model is placed in a Brownian filraion. The agens consruc securiies in zero ne supply such ha hey have a complee marke. They rade hose securiies in order

1.1. PART I MARKET COMPLETION 3 o find he rading sraegy ha maximizes heir uiliy from consumpion. Then he prices of he securiies and he ineres rae are chosen such ha all rading sraegies add up o zero. This equilibrium is called Arrow Debreu equilibrium. Barrieu (Bar2) considers he problem of securiy design. An insurance company inends o ransfer some of is insured risk o an invesor. The securiy is consruced such ha he uiliy of he insurance is maximal under he consrain ha he invesor buys i. This means he srucure and he price of he securiy is chosen such ha he uiliy of he invesor does no grow smaller if he buys he securiy. However, he role of he invesor is passive. Insiuional invesors who are aiming a maximizing heir profi migh no be conen wih his siuaion. The invesor should also have he possibiliy o maximize his uiliy. A survey aricle abou securiy design is Duffie, Rahi (DR95). We aim a finding mehods ha allow pricing and hedging of claims ha depend on boh financial and exernal risk. The echniques for incomplee markes we have seen so far migh lead o resuls ha are no very useful for us because he sock marke and he exernal risk facor are independen or no closely relaed. Hedging he exernal risk on he sock marke alone is no enough. The Arrow Debreu equilibrium on he oher hand sees all securiies as equal. All agens wih heir risk exposure have o be modeled and he price for every securiy is he resul of he equilibrium of supply and demand. This approach is no perfecly suiable for our problem. The size of he sock marke and he marke for securiies on exernal risk is very differen. Furhermore, our goal is o explain prices for he exernal risk whereas sock prices are he resul of rading a he sock markes ha we consider as exogenuosly given. So he firs ask in his hesis is he choice of an appropriae economic model. We propose an equilibrium wih parial marke clearing. Our model considers a group of agens wih incomes affeced by boh financial and exernal, non financial risk. Since he exernal risk is no radeable on he sock marke, he agens ineresed in rading his risk creae a marke for i. The agens may rade his risk among hemselves. In our model, hey complee he marke. This is done eiher by creaing an addiional securiy (risk bond) or by signing muual conracs. Given he sock price and a price of exernal risk, he agens choose he claim ha maximizes heir expeced uiliy among all claims hey can afford. This is done by rading wih he sock and eiher by buying and selling he risk securiy or by conracs. In order o achieve our equilibrium wih parial marke clearing, he price for he risk bonds and conracs on claims conaining exernal risk is adjused

4 CHAPTER 1. INTRODUCTION such ha supply and demand are equal. The difference o he usual equilibrium is ha we don change he sock price. The marke clears only parially, here is no clearing condiion on he rading sraegies wih he sock. The reason is ha our agens are considered as small rader on he sock marke. This means, heir demand is small compared o he overall volume of he sock marke. The agens canno change he sock price and hey are assumed o find oher raders o buy from or sell o who migh no belong o he group of agens considered here. Thus sock prices are exogenuosly given and we don require marke clearing for rading wih he sock wihin our group of agens. In Chaper 2 hrough Chaper 4, our agens use he exponenial uiliy funcion wih an individual coefficien of relaive risk aversion. We place ourselves in a Brownian framework. Equilibrium prices are obained by he soluion of a Backward Sochasic Differenial Equaion (BSDE). In Chaper 2, he marke is compleed by a securiy in zero ne supply ha is raded coninuously during he whole rading ime. Since he exernal risk is described by a one dimensional Brownian moion, one addiional securiy is enough o complee he marke. We find a condiion on drif and volailiy of he price process of he risk securiy such ha he marke clears parially. Chaper 3 considers he case of a more complicaed exernal risk described by a finie dimensional Brownian moion. In ha case he invesors sign muual conracs. The price of such a conrac is calculaed using a probabiliy measure ha is equivalen o he reference measure. Such a measure is called pricing measure. Since he price of financial risk canno be changed, a pricing measure has o be choosen from he se of equivalen maringale measures for he sock. The equilibrium is aeined by adjusing he pricing measure. In Chaper 4 an insurance company sells a risk bond in order o ransfer some of is insured risk o he agens who are willing o rade i. We use he erm risk bond because his securiy is no in zero ne supply. The insurance company is ineresed in selling a claim o he agens on he marke. A feedback of he ineres rae payed by he insurance from he price of exernal risk on he marke as well as a dependence on he exernal risk facor are possible. Parial marke clearing means here ha he demand for he bond is equal o he supply provided by he insurance. In conras o Chaper 2, he erminal value, i.e. he payou of he bond, is specified. A candidae of a price process is given by he successive condiional expecaions of he erminal value wih respec o a maringale measure of he sock. We provide a crierion ha characerizes he compleeness of he marke under he equilibrium price as well as a simple example for a risk bond compleing he marke. Chaper 5 considers an absrac one period model where he probabiliy

1.2. PART II ROBUST UTILITY MAXIMIZATION 5 space (Ω, F, P ) is placed in a Borel space. The uiliy funcions of he agens are allowed o be oher han exponenial. The incomes are modeled as random variables. An absrac sock marke is represened by a sub σ algebra G. All random variables measurable wih respec o G are radeable. On he oher hand, he pricing measure is already fixed on G. The agens complee he marke using conracs. Parial marke clearing is defined as in Chaper 3. Chaper 6 finally considers an equilibrium model in an incomplee marke. In conras o he previous chapers, he raders do no complee he marke. They are only willing o rade claims ha are measurable wih respec o a σ algebra T, whereas he incomes migh depend on a larger σ algebra. An inerpreaion for his fac is ha he agens only rade claims ha depend on observable facors in order o exclude moral hazard. In his chaper we use he usual equilibrium idea wihou he addiional sock marke. Chaper 2 and Chaper 3 are published in Hu, Imkeller and Müller (HIM4a). In he paper (CIM4), he pricing mehod presened here is applied o a simple model of climae risk, a paricularly ineresing exernal risk source. Numerical mehods are developed based on he well known correspondence beween non-linear BSDE and viscosiy soluions of quasi-linear PDE o simulae opimal wealh and sraegies of individual agens paricipaing in he marke. We focus on wo or hree agens exposed o he climae phenomenon of ENSO. 1.2 Par II Robus uiliy maximizaion An invesor on a financial marke is ineresed in having an opimal wealh a a fixed ime T. The invesor may represen a company ha has o repor o is shareholders a ha ime. Which crierion describes opimaliy? This depends on he preferences of he invesor. We use wo conceps of preferences on random claims in his hesis: he expeced uiliy wih respec o a fixed probabiliy and on he oher hand he robus uiliy. The laer is he infimum of he expeced uiliies of a random claim over a whole se of probabiliy measures. In his hesis we calculae he opimal self financing rading sraegy in an incomplee marke for boh ypes of preferences. Self financing means ha he invesor does no ake money ou or invess new money wihin he rading inerval. He invess some iniial capial. The wealh of he invesor changes only due o gains or losses by rading wih he sock. We consider he exponenial, he power and he logarihmic uiliy funcions. In he case

6 CHAPTER 1. INTRODUCTION of he exponenial uiliy, he invesor may hedge a liabiliy ha he has o pay ou a he end of he rading ime. Here we describe and compare he robus and he usual uiliy maximizaion. We follow closely Secion 2.5 in he book of F ollmer and Schied (FS2) in our presenaion. A random variable ha represens he erminal wealh of a rading sraegy is inerpreed as a funcion which associaes a real number o each scenario, i. e. a measurable funcion X T on some measure space (Ω, F). Denoe wih X he se of all claims considered. A preference can be seen as a binary relaion ha is asymmeric and negaively ransiive (see Definiion 2.1 in (FS2)). L. J. Savage (Sav54) inroduced a se of axioms which guaranees ha he preference relaion can be represened in he form U(X T ) = E Q [u(x T )] = u(x T (ω))q(dω), X T X (1.1) wih a probabiliy measure Q on (Ω, F) and a funcion u : R R. Of course, if U(XT 1) > U(X2 T ) for X1 T, X2 T X, hen X1 T is preferred. The probabiliy Q is deermined by he preference relaion and can differ from an objecive probabiliy measure. Thus, a real world measure migh be disored owards a more pessimisic or opimisic view. Usually, invesors prefer higher claims and are risk averse. This leads o a growing and concave funcion u ha is called uiliy funcion. However, some very inuiive preferences canno be wrien in a Savage represenaion. Invesors are no only averse agains risk bu also agains uncerainy. A very insrucive example for uncerainy is he Ellsberg paradox (see e.g. Example 2.81 in (FS2)). A player is faced wih he following problem: here are wo urns, each conaining 1 balls which are eiher red or black. The player knows ha in he firs urn here are 51 red and 49 black balls. The proporion of red and black balls in he second urn is unknown. Suppose ha he player ges 1 $ if he draws a red ball and $ for a black ball. The player may choose beween wo random claims, one wih a known and one wih a compleely unknown disribuion. The ypical decision is o draw from he firs urn. On he oher hand, if he player ges 1 $ for a black ball and nohing for a red ball, he usually also draws from he firs urn. If he player draws from he firs urn, he is exposed o risk. A probabiliy measure is fixed ha describes he model. The second urn is differen. The player has no informaion. I is impossible o find an objecive probabiliy measure for his urn. Such a siuaion is called uncerainy. Choosing he firs urn even if he probabiliy o win is less han, 5 is due o uncerainy aversion. The choices of he player define a preference relaion. Describing his relaion wih a Savage represenaion would mean ha we have o find

1.2. PART II ROBUST UTILITY MAXIMIZATION 7 one subjecive probabiliy measure for he second urn such ha in boh cases drawing from he firs urn yields a higher expeced uiliy. This is impossible. Insead of aking only a single measure Q, he robus Savage represenaion considers a whole se Q of probabiliy measures on (Ω, F). The represenaion is U(X T ) = inf Q Q E Q[u(X T )], X T X. (1.2) The invesor sees a whole se of probabilisic views as reasonable and akes a wors case approach in evaluaing he expeced uiliy of a given claim. The preference relaion in he Ellsberg paradox can be represened in his form. Le p r be he lowes probabiliy o draw a red ball in he firs urn for which he player chooses he firs urn in boh games. The se Q consiss of all probabiliy measures ha agree wih he informaion abou he firs urn and assigns he probabiliy for a red ball in he second urn beween p r and 1 p r. Anoher ype of uncerainy appears if coefficiens of a sock price process are no exacly known. Drif and volailiy migh be he resul of a saisic esimae ha yields only a confidence inerval. The robus uiliy of he erminal wealh of a rading sraegy is calculaed in he following way: compare he expeced uiliies for all possible processes of coefficiens. The infimum is he robus uiliy. The expecaion is aken wih respec o a reference probabiliy measure. Schied (Sch4b) considers he robus uiliy maximizaion problem on a complee marke. The price process of he socks is assumed o be a semimaringale wih respec o a probabiliy P. Compleeness means ha here exiss a unique probabiliy P P under which S is a local maringale. The invesor has an iniial capial bu no erminal liabiliy. Schied proves a dualiy resul under he assumpion ha a so called leas favorable measure Q P exiss. The leas favorable measure wih respec o P is defined as he probabiliy Q in Q ha saisfies [ ] [ ] dp dp Q x = inf dq Q x for all x >. Q Q dq If his leas favorable measure exiss, (Sch4b) shows ha for every growing, sricly concave uiliy funcion u : (, ) R, he robus uiliy maximizaion is equivalen o he uiliy maximizaion wih respec o Q. Schied gives examples and characerizaions of he leas favorable measure. The model in (Sch4b) ha is he mos ineresing for his hesis is he following: he sock prices are driven by a m dimensional Brownian moion W under a reference

8 CHAPTER 1. INTRODUCTION probabiliy measure: ds i = S i ( d j=1 σ i,j dw j + b i d), i = 1,..., m. The invesor is uncerain abou he drif b: any drif is possible ha is adaped o he filraion generaed by W and saisfies b C, where C is a nonrandom ime dependen bounded closed subse of R m. Then he se Q of probabiliy measures in he robus Savage represenaion are all probabiliy measures such ha S has a drif wih his properies. The volailiy marix is deerminisic and has full rank. Le α be he elemen in C ha minimizes he norm σ 1 b. If boh α and σ are coninuous, Proposiion 3.2 in(sch4b) saes ha he leas favorable measure is he one under which he drif is equal o α. Of course, our mehod gives he same resul under he assumpions of (Sch4b) for he uiliy funcions we consider. We find a simple resul in a case where he leas favorable measure does no exis. Le he marke be complee. We use an exponenial uiliy funcion. The invesor has a erminal liabiliy F, he uncerainy lies in he probabiliy measures, he drif is known. Then he opimal rading sraegy consiss of wo pars: he hedging sraegy for he sum of F and an addiional explicily given random variable, and he uiliy maximizing rading sraegy under he measure in Q under which he drif of he sock price is minimal (see Theorem 58 on page 15). Gundel (Gun3) provides a dualiy resul for robus uiliy maximizaion in complee and incomplee markes using reverse f projecions. She provides a dualiy resul in he following problem: maximize inf E Q[u(X)] over all X wih Q Q sup E P [X] x P P for a convex se P of equivalen local maringale measures for he sock price process. We consider wo ypes of uncerainy. For he firs one we use an explicily described se of probabiliy measures Q in he robus savage represenaion defined in (1.2). In he second approach, he coefficiens of he sock price process are uncerain. Our model is placed in he filraion generaed by an m dimensional Brownian moion wih respec o a probabiliy measure P. The densiies of he probabiliy measures in Q wih respec o P are sochasic exponenials of sochasic inegrands wih respec o he Brownian moion. The inegrands are resriced o ime dependen random predicable closed convex ses C (ω) of R m, [, T ]. Predicabiliy for se valued processes is explained in

1.2. PART II ROBUST UTILITY MAXIMIZATION 9 Delbaen, (Del3) page 5, or in our hesis in Remark 46 on page 92. All ses C (ω), ω Ω, [, T ] have o lie in a bounded ball around he origin. Our seup covers some muliplicaively sable (m-sable) ses of probabiliy measures in he sense of Definiion 1.2 in Delbaen, (Del3). Muliplicaively sable means ha we ake he densiy of a probabiliy measure in Q up o a sopping ime. Then we coninue wih he densiy of anoher probabiliy measure in Q ha is equivalen o he reference measure. The probabiliy measure wih he densiy composed in his way has also o belong o Q. Theorem 1.4 in Delbaen (Del3) applied o a Brownian filraion saes ha m sable ses of densiies have he same srucure as our se Q. However, we use he addiional assumpion ha he consrains on he inegrands have o be in a bounded ball around he origin. The sock price process in our model is he soluion of a sochasic differenial equaion driven by a Brownian moion. In Chaper 8 he uncerainy lies in he drif and volailiy of he sock price. The invesor has o ake ino accoun all sock price processes where he drif and volailiy process ake values wihin a convex se during he whole rading ime. In fac, he robus uiliy maximizaion problem in Chaper 7 and Chaper 8 can be seen as a saddle poin problem. The saddle poin consiss of he opimal rading sraegy and on he oher hand on a probabiliy measure or drif of he sock price. We find he saddle poin using a maringale argumen. This leads o a Backward Sochasic Differenial Equaion (BSDE). The soluion of he BSDE enables us o consruc he opimal rading sraegy as well as he probabiliy measure or he drif. The powerful ool of BSDE has been inroduced o sochasic conrol heory by Bismu (Bis76). Is mahemaical reamen in erms of sochasic analysis was iniiaed by Pardoux and Peng (PP9), and is paricular significance for he field of uiliy maximizaion in financial sochasics clarified in El Karoui, Peng and Quenez (EKPQ97). In (Pen9), Peng proves a maximum principle for sochasic conrol problems ha is based on BSDE. The mehod we use o calculae he saddle poin is a generalizaion of he approach used in Hu, Imkeller, Müller, (HIM4b). In his paper, we solved he problem of maximizing he expeced uily wih respec o a single probabiliy measure. El Karoui and Hamadène (EKH3) relaes he soluion of a saddle poin of an expecaion of an exponenial cos funcional o a BSDE. Our saddle poin problem doe no saisfy heir boundedness assumpions on he cos funcional. Quenez (Que4) considers he robus uiliy maximizaion if he sock price is given by a semimaringale. Using dualiy mehods she proves exisence of a saddle poin. For a Brownian filraion and a logarihmic resp. a

1 CHAPTER 1. INTRODUCTION power uiliy funcion she finds Backward Sochasic Differenial Equaions ha describe he opimal rading sraegy as well as he probabiliy measure used in he evaluaion of he robus uiliy. However, he coefficiens of he sock price process have o be consan for he power uiliy. We use a direc approach ha does no rely on dualiy mehods. Peng (Pen9) proves a maximum principle for sochasic conrol problems. In Chaper 9 we consider he uiliy maximizaion wih respec o one single probabiliy measure for he exponenial, power and logarihmic uiliy funcions. In he secion 9.1 abou he exponenial uiliy, he invesor may have a erminal liabiliy. We summarize he resuls of Hu, Imkeller and Müller (HIM4b), where he mehod we use has been developed. In conras o he chapers abou he robus uiliy maximizaion, we simply solve a maximizaion problem insead of a saddle poin problem. So he consrains o he values of he rading sraegy are assumed o be closed, bu in general no convex. This direc approach allows us o find he maximizing rading sraegy wihou dualiy argumens. In a relaed paper, El Karoui and Rouge (EKR) compue he value funcion and he opimal sraegy for exponenial uiliy by means of BSDE, assuming more resricively ha he sraegies be confined o a convex cone. Sekine (Sek2) relies on a dualiy resul obained by Cvianic and Karazas (CK92), also describing consrains hrough convex cones. He sudies he maximizaion problem for he exponenial and power uiliy funcions, and uses an aainabiliy condiion which solves he primal and dual problems, finally wriing his condiion as a BSDE. In conras o hese papers, we do no use dualiy, and direcly characerize he soluion of he primal problem. This allows us o pass from convex o closed consrains. Uiliy maximizaion is one of he mos frequen problems in financial mahemaics and has been considered by numerous auhors. Here are some of he milesones viewed from our perspecive of maximizaion under consrains using he ools of BSDEs. For a complee marke, uiliy maximizaion has been considered in Karazas e al. (KL87). Cvianic and Karazas (CK92) prove exisence and uniqueness of he soluion for he uiliy maximizaion problem in a Brownian filraion consraining sraegies o convex ses. There are numerous papers considering general semimaringales as sock price processes. Delbaen e al. (DGR + 2) give a dualiy resul beween he opimal sraegy for he maximizaion of he exponenial uiliy and he maringale measure minimizing he relaive enropy wih respec o he real world measure P. This dualiy can be used o characerize he uiliy indifference price for an opion. Also relying upon dualiy heory, Kramkov and Schachermayer (KS99) and Cvianic e al. (CSW1) give a fairly complee soluion of he uiliy opimizaion problem on incomplee markes for a class

1.2. PART II ROBUST UTILITY MAXIMIZATION 11 of general uiliy funcions no conaining he exponenial one. See also he review paper by Schachermayer (Sch2) for a more complee accoun and furher references. Par II of his hesis is organized as follows: In Chaper 7 we solve he robus uiliy maximizing problem for he exponenial and power uiliies. The uncerainy lies in he choice of probabiliy measures. Chaper 8 explains he uiliy maximizaion for an uncerain drif for exponenial, power uiliies and logarihmic uiliy. Chaper 9 gives he soluion for he uiliy maximizaion problem where he expecaion is aken wih a single probabiliy measure. In his case, we allow nonconvex consrains on he rading sraegies. In all hree chapers, he agen may have a erminal liabiliy if he uses he exponenial uiliy funcion. In his case, we calculae he uiliy indifference price of he liabiliy.

12 CHAPTER 1. INTRODUCTION Noaions We shall use he following noaions. Le Q be a probabiliy measure on F, k N, p 1. Then L p (Q) or L p (Ω, F, Q) sands for he se of equivalence classes of Q a.s. equal F T measurable random variables which are p inegrable wih respec o Q. L (Ω, F, Q) denoes all random variables ha are measurable wih respec o F whereas L (Ω, F, Q) is he se of random variables ha are bounded Q a.s. H k (Q, R d ) denoes he se of all R d valued sochasic processes ϑ ha are predicable and such ha E Q [ ϑ k d] <. Here and in he sequel E Q denoes he expecaion wih respec o Q. We wrie λ for he Lebesgue measure on [, T ] or R. H (Q, R d ) is he se of all predicable R d valued processes ha are l Q a.e. bounded on [, T ] Ω. For a coninuous semimaringale M wih quadraic variaion M he sochasic exponenial E(M) (for an adaped coninuous sochasic process M) is given by E(M) = exp(m 1 2 M ), [, T ]. Le C R n be closed and x R n. The disance dis C (x) is dis C (x) = min x y, y C where denoes he Euklidian norm. The projecion of x on C is he se Π C (x) ha saisfies Π C (x) = {y C x y = min x a }. (1.3) a C If C is convex, Π C (x) consiss of one elemen.

Acknowledgemens Many people have conribued o he compleion of his hesis. Foremos, my warmes hanks go o my advisor Peer Imkeller for posing me his non-sandard problem of finncial mahemaics. He generously shared his excellen mahemaical knowledge wih me and lead me o a beer undersanding of mahemaics. I m also indeped o Ying Hu who invied me o a visi in Rennes and provided much insigh in he heory of Backward Sochasic Differenial Equaions. I also hank Marin Schweizer and Alexander Schied for fruiful discussions and commens. Especially hanks o Ulrich Hors for his advise on microeconomic quesions. Also hanks o all members of he financial mahemaics and sochasics groups of he TU Berlin and he HU Berlin. My family and my friends always suppored me, his hesis would no have been possible wihou heir encouragemen. Thank you, Urnaa, for your love. Financial suppor by he Deusche Forschungsgemeinschaf via Graduierenkolleg ( Sochasic Processes and Probabilisic Analysis ) and via DFG Forschungszenrum ( Maheon ) is graefully acknowledged. 13

14

Par I Marke compleion, hedging exernal risk facors 15

Chaper 2 Equilibrium wih risk securiy In his chaper we calculae an equilibrium wih parial marke clearing in a model where he randomness comes from a wo dimensional Brownian moion wih respec o a probabiliy measure P. One componen of he Brownian moion drives a sock price process X S wih a quoien of drif and volailiy θ S. The oher componen describes he exernal risk. Our mehod works also if he sock depends on boh componens of he Brownian moion. Every agen wihin a finie group obains incomes depending on boh ypes of risk. In order o hedge he exernal risk, hey creae a risk securiy ha complees he marke. Given a candidae of he price process, he agens rade wih boh sock and risk securiy in order o maximize he expeced uiliy of he wealh a he end of he rading period. The agens use exponenial uiliy funcions. In order o obain parial marke clearing, we adjus he drif and he volailiy of he risk securiy X E such ha he rading sraegies for his asse add up o zero. We consider a whole se of quoiens θ E of drif and volailiy for X E. For every θ = (θ S, θ E ) we find a unique probabiliy measure Q θ equivalen o P such ha (X S, X E ) is a Q θ maringale. Since he agens maximize he uiliy of he wealh a he erminal ime, we may ransform our equilibrium condiion on he sraegies ino a condiion on he wealh: he sum of he incomes minus he preferred erminal wealh is a payoff ha is replicable a he sock marke. This difference is simply he sum of he rading sraegies wih he sock. The problem is simplified because we don need o calculae wih he only implicily known opimal sraegies anymore. We apply uiliy maximizaion echniques for complee markes using maringale and BSDE mehods. Maringale mehods are reaed in (KL87), (CH89) and (Pli86). The compleeness of he marke leads o a budge condiion: every payoff ha is no more expensive han he income of an 17

18 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY agen can be replicaed. The price of a payoff is calculaed as is expecaion under he maringale measure Q θ. Using he Legendre ransform, he payoff maximizing he expeced uiliy wihin he budge se is calculaed. For he exponenial uiliy funcion, his payoff depends explicily on θ S and θ E. This explici srucure of he uiliy maximizing erminal wealh of he agens allows us o wrie down a Backward Sochasic Differenial Equaion ha characerizes he quoien of drif and volailiy θ E of he equilibrium price. This chaper is organized as follows: in secion 2.1 we explain our sock marke, he exernal risk facor and he incomes of he agens. Secion 2.2 defines he se of price processes for he risk securiy. Addiionally, admissible rading sraegies for boh he sock and he risk securiy are defined. Secion 2.3 recalls he soluion of he uiliy maximizaion problem in a complee marke. Finally, in Secion 2.4 we define our equilibrium wih parial marke clearing and consruc he price process of he risk securiy ha aains parial marke clearing. 2.1 The sock, exernal risk The mahemaical frame is given by a probabiliy space (Ω, F, P ) carrying a wo dimensional Brownian moion W = (W 1, W 2 ) indexed by he ime inerval [, T ], where T > is a deerminisic ime horizon. Noe here ha sochasic processes indexed by [, T ] will be wrien X = (X ) [,T ]. The filraion F = (F ) [,T ] is he compleion of he naural filraion of W. Le us now explain he firs version of our model in more formal deails. The sock marke is represened by an exogenous F adaped index or sock price process X S indexed by he rading inerval [, T ]. The dynamics of his price process evolves according o he sochasic inegral equaion X S = X S + X S s (b S s ds + σ S s dw 1 s ), [, T ], where X S is a posiive consan, so ha we have ( X S = X S E ) (b S s ds + σs S dws 1 ). (2.1) Throughou he paper we shall work wih he following assumpion concerning he drif b S and volailiy σ S of he sock price process X S :

2.1. THE STOCK, EXTERNAL RISK 19 Assumpion 1 b S H (P, R), σ S H (P, R), here is ε > such ha σ S > ε. Observe ha due o his assumpion he process θ S := bs σ S (2.2) is also conained in H (P, R) and P [X S > for all [, T ]] = 1. Our analysis relies on he fac ha he inegral equaion describing he sock price is driven by only one componen of he Brownian moion. If his is no he case, we have o consruc a new Brownian moion ha saisfies his condiion. Observe ha he coefficiens b S and σ S may depend on he whole filraion F. The following remark considers only he componens of he Brownian moion in he inegral in (2.1). Remark 2 Le he sock price be described by X S = X S + X S s (b S s ds + σ S,1 s dw 1 s + σ S,2 s dw 2 s ), [, T ], where b S, σ S,1 and σ S,2 saisfy Assumpion 1. Then define and W 1 W 2 = σs,1 W 1 + σ S,2 W 2 (σ S,1 ) 2 + (σ S,2 ) 2, [, T ], = σs,2 W 1 + σ S,1 W 2 (σ S,1 ) 2 + (σ S,2 ) 2, [, T ]. Wih he well known characerizaion heorem of Lévy we see ha ( W 1, W 2 ) is a Brownian moion. Furhermore, he inegral equaion for he sock price process S is driven only by W 1. The exernal risk componen eners our model hrough an F adaped sochasic process K, indexed by he rading inerval as well. As an example, one migh hink of a climae process, such as he emperaure process in he Easern Souh Pacific which gives rise o he climae phenomenon of ENSO which largely affecs he naional economies of he neighboring saes. See (CIM4), where he effecs of his phenomenon and risk ransfer sraegies

2 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY based on he conceps of which are developed in his hesis are capured by numerical simulaions. Agens on he marke are symbolized by he elemens a of a finie se I. They can use a bank accoun wih ineres rae zero. Every agen a I is supposed o be endowed wih an iniial capial v a. A he end of he rading inerval a ime T he receives a sochasic income H a which describes he profis ha his agen or he company he represens obains from his usual business. The income H a is supposed o be a real valued bounded F T measurable random variable, funcion of he processes X S and K, i.e. H a = g a (X S, K). A ypical example covered by hese assumpions is he following. Think of wo agens, say a company c and a bank b. c could for example possess an income H c = g c (K) purely dependen on he exerior risk. The bank has an income H b = g b (X S ) which only depends on he sock marke. c wans o hedge flucuaions caused by he exernal facor and signs a conrac wih b o ransfer par of his risk. b s ineres in he conrac could be based on he wish o diversify is porfolio. For concree numerically invesigaed oy examples in he conex of ENSO risks see (CIM4). 2.2 Prices for he risk securiy, rading In his secion we describe he se of price processes we consider for he risk securiy. Then we define rading sraegies using boh he sock and he risk securiy and he wealh process gained by rading. In order o complee he marke, we wan o consruc a second securiy hrough which exernal risk can be raded wih price process X E of a form given by he following sochasic inegral equaion X E = X E + X E s (b E s ds + σ E s dw 2 s ), [, T ], (2.3) wih coefficien processes b E and σ E H 2 (P, R), and such ha for some ε > we have σ E > ε. Le θ E := be σ. E (2.4) The processes θ S, θ E are called marke price of risk of he sock and he insurance securiy. Every marke price of risk θ E of he second securiy is supposed o belong o he following se: { V = θ E H 2 (P, R) θs E dws 2 is a (P, F) BMO maringale }. (2.5)

2.2. PRICES FOR THE RISK SECURITY, TRADING 21 The definiion of BMO maringales as well as imporan resuls are explained in he appendix. We will use he fac ha sochasic exponenials of BMO maringales are uniformly inegrable maringales. The marke price of risk vecor θ ime paramerizes a class of probabiliy measures Q θ for which he price processes (X S, X E ) are maringales. More formally, denoe ( X S X := X E ), θ := ( θ S θ E ) and σ := ( σ S σ E ). (2.6) The marix valued process σ is inverible for all [, T ] P a.s. Wih θ E V and θ S according o Assumpion 1 i is seen by using (A.2) (Appendix) ha he process ( θ sdw s ) [,T ] is a P BMO maringale. This propery in urn guaranees ha he change of measure obained by drifing W by θ induces an equivalen probabiliy. Lemma 3 Suppose ha θ = (θ S, θ E ) wih θ S saisfying Assumpion 1 and θ E V. Then he process Z θ := E( θ dw ) defines he densiy process of an equivalen change of probabiliy. Proof The process Z θ is he sochasic exponenial of a BMO maringale. By Theorem 2.3 in (Kaz94) i is a uniformly inegrable (P, F) maringale. According o Lemma 3 we may define he measure Q θ wih Radon Nikodym densiy wih respec o P given by ( dq θ ) dp = Zθ T = E θ dw T ( = exp θ dw 1 ) θ 2 d. 2 (2.7) This provides he unique probabiliy for which he price process X = (X S, X E ) given by (2.1) and (2.3) is a maringale. Hence he choice of a paricular insurance asse compleing he marke leads o a class of equivalen maringale measures for he price dynamics paramerized by he price of risk processes. By he well known Lévy characerizaion W θ = W + θ sds is a Q θ Brownian moion. The marke being equipped wih his srucure, each agen a I will maximize he erminal wealh obained from his porfolio in he securiies (X S, X E ) and his random risky income subjec o he exerior risk H a, according o his individual preferences. Thereby he will be allowed o follow rading sraegies o be specified in he following. A rading sraegy is given by a 2 dimensional F predicable process π = (π ) T such ha

22 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY π σ 2 d < P a.s., hence ( π 1,s, π 2,s )dx Xs S Xs E s is well defined. This noaion of a rading sraegy describes he number of currency unis invesed in each securiy. The wealh process V = V (π) = V (c, π) of a rading sraegy π wih iniial capial c is given by ( π1,s V = c +, π ) ( ) 2,s X S d s Xs S Xs E Xs E, [, T ]. The number of shares of securiy i is π i,, i = S, E. For he ease of noaion, X i we shall wrie in he sequel dx dxs for he vecor incremen (, dxe ). Trading X X S X E sraegies are self financing. This means ha hose pars of he wealh no invesed ino X S or X E are kep in he bond. Gains or losses are only caused by rading wih he securiies. The wealh process can equivalenly be wrien as V (c, π) = c + π s σ s (dw s + θ s ds) = c + π s σ s dw θ s, [, T ]. (2.8) A se Φ of sraegies is called free of arbirage if here exiss no rading sraegy π Φ such ha V (π) =, V T (π) and P [V T (π) > ] >. We have o resric he se of rading sraegies by defining he se of admissible sraegies in order o exclude opporuniies of arbirage. Definiion 4 (Admissible Sraegies) The se of admissible rading sraegies A is given by he collecion of he 2 dimensional predicable processes π wih π σ 2 d < Q θ a.s. such ha he wealh process V (c, π) is a (Q θ, F) supermaringale. The se of admissible sraegies A is free of arbirage. In fac, we ge from V (, π) = and V T (, π) ha V T (, π) = Q θ and hus P a.s. Examples are sraegies π wih iniial capial v such ha V (v, π) is bounded from below uniformly on [, T ] Ω. In his case, V (v, π) is a local Q θ maringale bounded from below, hence a Q θ supermaringale. 2.3 Uiliy maximizaion Fixing a paricular marke price of risk θ E V, in his secion we describe he individual behavior of an agen a I. In paricular, he impac of he choice of θ E deermining he price process X E of he insurance asse

2.3. UTILITY MAXIMIZATION 23 on his erminal wealh and rading sraegy is clarified. Le us emphasize a his poin ha he inroducion of X E complees he marke wih price process X having componens X S and X E. We use well known resuls abou uiliy maximizing rading sraegies and he associaed erminal wealh in a complee marke. They can be found e.g. in (KL87) for he maximizaion of an expeced uiliy and in (Ame99) for he opimizaion of he condiional expeced uiliy wih respec o a non rivial sigma algebra. Every agen a I has iniial capial v a a his disposal. A he erminal ime T he receives a random income possibly depending on exernal risk and described by an F T measurable bounded random variable H a. The invesor wans o hedge flucuaions in his income H a or diversify his porfolio. His preferences are described by he expeced uiliy using he uiliy funcion u a (x) = exp( α a x) x R, wih an individual risk aversion coefficien α a >. The agens ac as price akers. The individual uiliy maximizaion problem for he raders acing on he whole ime inerval [, T ] hen akes he following mahemaical form. Each one of hem wans o find a rading sraegy π a A which aains Problem 5 (Individual uiliy maximizaion, sar a ) J a (v, a H a, X S, X E ) = sup E[ exp( α a (V T (v, a π) + H a )] π A = sup E π A [ ( ))] exp ( α a v a dx s + π s + H a. X s Since x exp( αx) is bounded from above, he expecaions appearing in Problem 5 are well defined. I will be more convenien o reformulae our uiliy maximizaion problem using he maringale measure Q θ wih Brownian moion W θ of our price process X = (X S, X E ). In paricular, we aim for an alernaive descripion of he budge se, described above as he se of final claims aained by admissible rading sraegies, in erms of he maringale measure. This will urn ou o be imporan in secion 3 where we generalize our model o more complex siuaions: maringale measures will correspond o pricing rules here. A he end of he rading period, every agen has a claim of ξ = V T (v a, π) + H a based on his iniial capial, his invesmens in X and exernal risk exposure. On he one hand, V (v a, π) being a Q θ supermaringale for each admissible rading sraegy π his claim has o saisfy he inequaliy E θ (ξ) v a + E θ (H a ). If i is even a Q θ maringale, equaliy holds. On he oher hand, he marke being complee, every claim

24 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY of his ype can be replicaed by appealing o he maringale represenaion heorem wih respec o he Brownian moion W θ under Q θ. More precisely, H a being bounded, for any ξ L 1 (Q θ ) we may find an F predicable process φ saisfying φ s 2 ds < Q θ a.s. and So we may se ξ H a = E θ [ξ H a ] + = v a + φ s σ 1 dx s s X s = V T (v, a φ σ 1 ). φ s dw θ s π = φ σ 1 (2.9) o obain an admissible sraegy. Here σ is defined by (2.6). To summarize he resul of our argumens in a slighly differen manner: a random variable ξ L 1 (Q θ, F T ) is he sum of he erminal value of he wealh process of an admissible rading sraegy π wih iniial capial v and a erminal income H a if and only if E θ [ξ] = v + E θ [H a ]. This implies ha our problem (5) boils down o he following maximizaion problem over random variables given by he claims. We collec claims ξ composed of final wealhs of admissible sraegies and final incomes H a in he budge se B(v, H a, θ S, θ E ) := {ξ L 1 (Q θ, F T ) : E θ [ξ] v + E θ [H a ]}, (2.1) and hen have o find he random variable ξ a (θ S, θ E ) ha aains J a (v a, H a, θ S, θ E ) := sup E[ exp( α a ξ)]. (2.11) ξ B(v a,ha,θs,θ E ) The soluion is obained by well known mehods via an applicaion of he Fenchel Legendre ransform o he concave funcion x exp( α a x). Theorem 6 Le H a be a bounded F T measurable random variable, v a. Define ξ a (θ S, θ E ) := ξ a (v a, H a, θ S, θ E ) = 1 α a log( 1 α a λ a Z θ T ) where λ a is he unique real number such ha E θ [ 1 α a log( 1 α a λ a Z θ T )] = v a + E θ [H a ]. Then ξ a (θ S, θ E ) is he soluion of he uiliy maximizaion problem (2.11) for agen a I.

2.3. UTILITY MAXIMIZATION 25 Proof The main body of he proof is given by Theorem 2.3.2 of (KL9), saed for uiliy funcions saisfying he Inada condiions, i.e. U ( ) =, U (+) =, and under he hypohesis ha he quadraic variaion of θ sdw s is bounded. In our seing, his process is a BMO maringale for which he quadraic variaion is no necessarily bounded. Therefore we have o show ha for every a I, v R here exiss λ a > saisfying E θ [ 1 α a log( 1 α a λ a Z θ T )] = v. (2.12) A sufficien condiion for his is ha he relaive enropy of Q θ wih respec o P is finie. We recall ha for probabiliy measures Q, R on F he relaive enropy of Q wih respec o R is defined by { E H(Q R) = Q [log dq ], if Q R, dr, if no. Therefore we may finish he proof of he Theorem wih an applicaion of he following Lemma, saed in a more general seing. In fac, i implies ha for θ of he ype we have chosen he relaive enropy H(Q θ P ) is finie. Lemma 7 Le θ = (θ S, θ E ), and suppose ha θ S saisfies Assumpion 1 and θ E V. Then E θ [log Z θ T F τ] is finie P a.s. for every sopping ime τ T. Proof By Theorem 3.3 in (Kaz94), he process M = θ sdws θ, T, is a Q θ BMO maringale. Therefore here exiss a consan c ha does no depend on τ such ha [ 1 T ] E θ θ s 2 ds 2 F τ c. The equaion τ θ s dw s 1 θ s 2 ds = θ s dws θ + 1 θ s 2 ds τ 2 τ τ 2 τ yields E θ [log Z θ T F τ ] = E θ [ 1 2 τ ] θ s 2 ds F τ <.

26 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY So far we deermined he individual uiliy maximizing invesmen sraegy of an agen on our marke, compleed by he insurance asse X E wih parameer θ E for he marke price of exernal risk fixed, who sars rading a ime. We now show ha he migh as well sar acing a a sopping ime τ ha akes is values in [, T ] wihou having o modify his opimal invesmen sraegy. For his purpose, le us recall he resuls of (Ame99) for he maximizaion of a condiional expecaion and apply hem o our exponenial uiliy funcion. Le τ T denoe a sopping ime. We wan o solve he following condiioned maximizaion problem: Problem 8 (Individual uiliy maximizaion, sar a τ) Jτ a (vτ a, H a, θ S, θ E ) = sup E[ exp( α a (V T (vτ a, π) + H a )) F τ ] π A = sup E π A [ ( )) ] exp ( α a vτ a dx s + π s + H a F τ. τ X s Hereby he iniial capial v a τ is an F τ measurable random variable, he wealh process of an admissible rading sraegy a Q θ supermaringale. Exending he argumens made above o reformulae he opimizaion problem in erms of maximizaion over a budge se, and in paricular using Doob s opional sopping heorem, we find ha he problem may be recas in he following way. Define he budge se B(τ, v a τ, H a, θ S, θ E ) using he condiional expecaion wih respec o F τ by B(τ, v τ, H a, θ S, θ E ) := {ξ L 1 (Q θ, F T ) : E θ [ξ F τ ] v τ +E θ [H a F τ ] P a.s.} (2.13) (see (Ame99) Proposiion 4.3). Then we have o solve a maximizaion problem concerning random variables which represen he agens individual claims: J a τ (v a τ, H a, θ S, θ E ) = sup E[ exp( α a ξ) F τ ]. (2.14) ξ B(τ,vτ a,ha,θs,θ E ) The exponenial uiliy funcion does no saisfy he hypohesis made in (Ame99). Bu i is easy o apply he same mehod in our case. In fac, again an applicaion of he Fenchel Legendre ransform will yield he resul wih he usual argumens. Theorem 9 Le H a be a bounded F T measurable random variable, v a τ an F τ measurable random variable. Define ξ a,τ (θ S, θ E ) := ξ a,τ (v a, H a, θ S, θ E ) = 1 α a log( 1 α a Λ a Z θ τ ),

2.3. UTILITY MAXIMIZATION 27 where Λ a is an F τ measurable random variable which saisfies 1 α a log Λ a = v a τ + E θ [H a F τ ] + 1 α a log 1 α a + 1 α a E θ [log Z θ T F τ ]. Then ξ a,τ (θ S, θ E ) is he soluion of he uiliy maximizaion problem (8) for agen a I. Proof Our reasoning via Theorem 2.3.2 of (KL9) his ime leads us o he problem of finding an F τ measurable random variable which saisfies 1 α a log Λ a = v a τ + E θ [H a F τ ] + 1 α a log 1 α a + 1 α a E θ [log Z θ T F τ ]. This again boils down o a finie relaive enropy condiion already covered by Lemma 7. Le us summarize our findings of his secion for ease of laer reference by giving an explici formula for he uiliy maximizing wealh a ime T of agen a I if he uses his opimal sraegy from a sopping ime τ T on wih a Q θ inegrable F τ measurable iniial capial vτ a. We recall ha he parameer θ deermines uniquely he second securiy X E on our marke which is a possible candidae for making he exernal risk radable. The formula we obain from Theorem 9 by employing he explici srucure of he densiy Zτ θ reads ξ a,τ (θ S, θ E ) = 1 α a log + 1 2α a τ ( Λa α a ) + 1αa ( θ S 2 + θ E 2 )d. τ (θ S dw 1 + θ E dw 2 ) (2.15) To emphasize is explici dependence on he price of exernal risk, we furher wrie π a (θ E ) for he uiliy maximizing rading sraegy aaining he claim ξ a (θ S, θ E ) H a = V T (v a, π a (θ E )) = v a + π a (θ E ) s dx s X s. (2.16) The opimal rading sraegy saisfies he principle of dynamic programming: if a ime = an agen a chooses he opimal sraegy π a (θ E ) which provides he wealh V τ (v a, π a (θ E )) a a sopping ime τ, he has o follow he same sraegy if he sars acing a ime τ wih iniial capial V τ (v a, π a (θ E )).

28 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY 2.4 Equilibrium wih parial marke clearing Le us now inroduce our concep of equilibrium wih parial marke clearing for he marke on which he exernal risk due o he risk process K is raded. Le us briefly recall he model componens implemened so far. Every agen a I obains an iniial capial v a and a ime T a random risky income H a ha, besides he economic developmen described by he exogenous sock price process X S, depends on he exernal risk process K. A second (insurance) securiy X E is creaed o make individual risks immanen in he incomes H a and caused by K radable. I depends on he process parameer θ E which describes a possible price of exernal risk in X E. Given such a sysem of pricing risk every agen rades wih X S and X E and calculaes he rading sraegy π a (θ E ) ha maximizes expeced exponenial uiliy wih individual risk aversion α a of he sum of his erminal wealh from rading and he income H a. In order o reach a parial marke clearing, we have o find a marke price of exernal risk process θ E V for which a any ime a marke clearing condiion for he second securiy is saisfied, i.e. a I π 2,(θ E ) =. This equilibrium is called parial since no marke clearing for he sock X S is required. Definiion 1 (equilibrium wih parial marke clearing) Le he iniial capials v a R, he erminal incomes H a, a I, and he sock price process X S be given. A equilibrium wih parial marke clearing consiss of a marke price of exernal risk process θ E V for he second securiy and rading sraegies π a (θ E ), a I, which saisfy he following condiions: 1. for any a I he rading sraegy π a (θ E ) is he soluion of he uiliy maximizaion problem 5 for he sock price process X S and he price process of he second securiy associaed wih marke price of risk θ E, 2. he second componen π2(θ a E ), a I, saisfies he parial marke clearing condiion π2(θ a E ) = P λ a.e. a I The condiion ha he marke clears parially pus a naural consrain on he se of processes of marke price of risk for he second securiy. We shall now invesigae he impac of his consrain. I will compleely deermine he srucure of θ E and herefore also a unique maringale measure Q θ obained via (2.7) for θ = (θ S, θ E ). So we shall have o compue θ E from he condiion ha he marke be in equilibrium wih respec o X E = σe s (dws 2 +θs E ds). Recall ha Assumpion 1 guaranees θ S H (P, R). In

2.4. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 29 he following Lemma he overall effec of our equilibrium condiion emerges. Plainly, if we ake he sum of he erminal incomes and erminal wealh obained by all agens from rading on he securiy marke composed of X S and X E, he condiion of parial marke clearing jus eliminaes he conribuion of X E. Lemma 11 Le θ = (θ S, θ E ) be such ha θ S saisfies Assumpion 1, and θ E V. The marke is in an equilibrium wih parial marke clearing if and[ only if here exis an F predicable real valued sochasic process φ wih ( ) ] 1 E θ T (φ s) 2 2 ds < such ha he opimal claims (ξ a (θ S, θ E )) a I and incomes (H a ) a I saisfy he equaion (ξ a (θ S, θ E ) H a ) = c + a I φ s (dw 1 s + θ S s ds) (2.17) wih some consan c R. Hence π = (π 1, ) wih π 1 = φ(σ S ) 1 = a I πa 1, possesses he properies of an admissible rading sraegy. Proof Firs we apply he represenaion propery (2.9) o he erminal wealh ξ a (θ S, θ E ) H a of each individual agen a I wih iniial capial v, a hen sum over all a I. Using lineariy of he sochasic inegral and recalling (2.8) we hus obain (ξ a (θ S, θ E ) H a ) a I = a I = a I + v a + v a + ( a I ( a I ( a I π a 1,) dxs X S + ( a I π a 1,) σ S (dw 1 + θ S d) π a 2,) σ E (dw 2 + θ E d). π a 2,) dxe X E (2.18) To prove he only if par, wrie now π i = a I πa i, i = 1, 2. Since he marke clears parially, we have π 2 =. Hence he desired equaion (2.17) follows. For he if par, suppose ha a I (ξa (θ S, θ E ) H a ) can be wrien as in (2.17). By comparison wih (2.18) and uniqueness of inegrands in sochasic inegral represenaions we obain π 1 = φ and π σ S 2 =. This esablishes he equivalence. Finally, π = (π 1, ) is admissible, because a I (ξa (θ S, θ E ) H a ) L 1 (Q θ ), and he process π 1,dX S is even a Q θ maringale.

3 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY We now come o he main goal of his secion, he consrucion of θ E for which our equilibrium consrain is saisfied. A he same ime, his will jusify he exisence of an equilibrium wih parial marke clearing. We use he characerizaion of he uiliy maximizing payoffs in our equilibrium described in Lemma 11 and he explici formula (2.15). This will enable us o describe θ E and φ (or π) in erms of he soluion of a BSDE. To abbreviae, we wrie ᾱ = ( 1 ) 1, H = H a + 1 θs S 2 ds. (2.19) α a 2ᾱ a I a I We combine he wo alernaive descripions of a I (ξa (θ S, θ E ) H a ) provided by Lemma 11 and he equaion (ξ a (θ S, θ E ) H a ) (2.2) a I = c 1 + 1 α (θ S dw 1 + θ E dw 2 ) + 1 2α ( θ S 2 + θ E 2 )d a I H a which follows from (2.15) wih a consan c 1 no specified furher a his poin, o obain a condiion deermining θ E in he form of a BSDE. To keep o he habis of he lieraure on BSDE, se z S = θ S ᾱφ, z E = θ E. In his noaion he comparison of (2.17) and (2.2) yields he equaion h = ᾱ H (z S dw 1 + z E dw 2 ) 1 2 ze 2 d θ S z S d. (2.21) Due o Assumpion 1, H is bounded. By exending (2.21) from ime o any ime [, T ] we obain a BSDE whose soluion uniquely deermines z E = θ E. I defines backward in ime a predicable sochasic process (h ) [,T ] H (R, P ) wih erminal value h T = ᾱ H and an inegrand (z = (z S, z E )) [,T ] H 2 (R 2, P ). The following Theorem provides an equilibrium soluion by seing θ E := z E which is obained from known resuls on non-linear BSDE. Theorem 12 The backwards sochasic differenial equaion (BSDE) h = ᾱ H (z S s dw 1 s + z E s dw 2 s ) θ S s z S s ds 1 2 ze s 2 ds, (2.22)

2.4. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 31 [, T ], possesses a unique soluion given by he riple of processes (h, (z S, z E )) H (P, R) H 2 (P, R 2 ). The choice θ E := z E provides an equilibrium wih parial marke clearing for he marke. Proof H is F T measurable and bounded. The process θ S is F predicable and uniformly bounded in (ω, ). By Theorem 2.3 and Theorem 2.6 in (Kob), equaion (2.22) has a unique soluion (h, (z S, z E )) H (P, R) H 2 (P, R 2 ). Le hen θ E := z E and φ := 1 ᾱ (θs z S ). Then, hanks o Lemma 11 we ge a equilibrium wih parial marke clearing, provided we can prove ha z E V. This is done in Lemma 13 below. Given θ E, for he coefficiens b E and σ E we are free o choose for example b E = θ E, σ E = 1. Lemma 13 Le z E be he hird componen of he soluion (h, (z S, z E )) of (2.22). Then he process M = ze s dws 2 is a P BMO maringale. Proof Wihou loss of generaliy, we may suppose ᾱ H nonnegaive. To see his, recall ha ᾱ H is bounded from below by a consan S. We may hen solve he BSDE (2.22) for H = ᾱ H S insead. By uniqueness is soluion (k, (y 1, y 2 )) saisfies k = h S, y 1 = z S, y 2 = z E. If H, he comparison heorem (Theorem 2.6 (Kob)) gives h. For every sopping ime τ T, Iô s formula yields [ ] E H 2 h 2 τ (2h s θs S zs S + zs S 2 )ds F τ τ [ ] [ ] = E (h s + 1) zs E 2 ds F τ E zs E 2 ds F τ. τ To find also an upper bound for he lef hand side in he inequaliy above we noe 2h s θ S s z S s z S s 2 = θ S s 2 h 2 s (θ S s h s + z S s ) 2. Le S 1 denoe an upper bound for H 2 and S 2 an upper bound for θs S 2 h 2 s. Then we ge for every sopping ime τ T [ ] S 1 + T S 2 E zs E 2 ds F τ Therefore M is a P BMO maringale. τ = E [ M T M τ F τ ]. τ

32 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY Here we give an example where our equilibrium price of he exernal risk does no depend on he financial marke. This is he case if he income of he agen is he sum of a payoff ha depends only on financial risk and a payoff ha depends on he exernal risk. Then our BSDE (2.22) decomposes ino wo BSDEs ha can be solved seperaely. Example 14 Le he drif of he sock price θ S be adaped o he filraion F 1 = (F 1 ), he P augmenaion of he filraion generaed by W 1. Le F 2 = (F 2 ) denoe he P augmenaion of he filraion generaed by W 2. We assume ha he sum of he incomes H = a I H a can be decomposed in wo pars: H = H 1 + H 2, where H 1 is measurable wih respec o FT 1, H 2 is FT 2 measurable and boh random variables are bounded. Then we can decompose our BSDE (2.22) ino a BSDE wih respec o W 1 wihin F 1 and a BSDE wih respec o W 2 in he filraion F 2. Here is he firs BSDE: Y 1 = (H 1 + 1 2ᾱ and he second one: Y 2 = H 2 θ S s 2 ds) z E s dw 2 s z S s dw 1 s 1 2 ze s 2 ds. θ S s z S s, Each BSDE can be solved separaely wihin is filraion F 1 and F 2. The inegrands z S and z E are equal o he inegrands of he soluion of (2.22). Furhermore, he proces Y in he soluion of (2.22) saisfies Y = Y 1 +Y 2. The economic inerpreaion is simple: he income H 1 is hedged on he financial marke. The income H 2 is disribued among he agens using he usual equilibrium approach: he marke price of exernal risk θ E is deermined by he fac ha supply and demand for he ransfer of exernal risk is equal. In paricular, under he assumpions in his example, θ E = z E does no depend on he marke price of financial risk θ E and he par of he income H 1 ha is radeable on he financial marke. In he following Theorem we shall show ha he choice θ E = z E made above provides he unique equilibrium price of exernal risk under he assumpions valid for he coefficien processes. Theorem 15 Suppose θ E = b E /σ E is such ha we have an equilibrium wih parial marke clearing. Then z E = θ E is he hird componen of he unique soluion process (h, (z S, z E )) of (2.22).

2.4. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 33 Proof We firs apply Girsanov s Theorem o eliminae he known drif θ S from our consideraions. More formally, consider he probabiliy measure Q given by he densiy d Q ( dp = E (θ S, )dw ). Le W = W + (θs s, )ds be he corresponding Brownian moion under Q. Now define z S = θ S ᾱφ, z E = θ E and z = (z S, z E ) r. Since z E guaranees ha we have an equilibrium wih parial marke clearing, as for (2.21) we deduce wih a consan c c = ᾱ H = ᾱ H (z S dw 1 + z E dw 2 z d W ) 1 2 ze 2 d. Hence we may furher define he process h by h = c + wih he alernaive descripion h = ᾱ H z s d W s z s d W s + 1 2 1 T 2 ze 2 d θ S z S d(2.23) (z E s ) 2 ds, 1 2 ze s 2 ds, [, T ]. (2.24) This yields ha (h, (z S, z E )) solves (2.22). I remains o verify according o Theorem 2.6 in (Kob) ha (z S, z E ) H 2 (P, R 2 ), h is uniformly bounded. Le us firs argue for he square inegrabiliy of (z S, z E ). By he definiion of our equilibrium, we have θ E H 2 (P, R). θ S being bounded, i remains o argue for P -square-inegrabiliy of φ, where φ is given by (2.17). By Burkholder-Davis-Gundy s inequaliy, we have a I (ξa (θ S, θ E ) H a ) L p ( Q) for p 1, and his random variable can be represened as a sochasic inegral wih he inegrand (φ, ) wih respec o he Brownian moion W. Hence, E Q([ (φ s ) 2 ds] p 2 ) <,

34 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY for p 1. Therefore, due o Hölder s inequaliy and E P ([ we also obain (φ s ) 2 ds] p 2 ) = E Q([ E P ([ (φ s ) 2 ds] p 2 E( (θ S, )d W s )) (φ s ) 2 ds] p 2 ) < for all p 1. To prove he boundedness of h, we perform sill anoher equivalen change of measure. Le ˆQ be given by d ˆQ dp = E( Then by virue of (2.24) we ge (θ S, 1 2 ze )dw ). h = E ˆQ[ᾱ H F ], [, T ]. Therefore h has a uniformly bounded version wih he same bounds as ᾱ H. We conclude his secion by showing ha he unique equilibrium consruced persiss if he individual uiliy maximizaion problems of he agens on he marke sar a some sopping ime τ. Remark 16 The marke price of risk θ E ha aains parial marke clearing saisfies a dynamic programming principle. Indeed, le θ E be he unique marke price of risk process in V calculaed for he individual uiliy maximizaion saring a ime =. Le τ T be a sopping ime and le he agens solve he condiioned maximizaion problem 8 beginning a ime τ wih erminal incomes H a. Then he equilibrium is given by θ E as well. For he consrucion of an equilibrium wih parial marke clearing for rading afer τ we proceed in he same way as in he case of he maximizaion of a condiioned expeced uiliy. The definiion of a parial equilibrium remains as in Definiion 1. The saring poin is Lemma 11 adaped o he sigma algebra F τ, where we have o replace he consan c by an F τ measurable bounded random variable c τ. Comparing he explici soluion of he uiliy maximizaion wih respec o a candidae for an equilibrium marke price of risk process θ E o (2.17) yields he following BSDE wih z = ( z S, z E ) h = ᾱ ( 1 H a z s dw s 2 ze s 2 + θ S z s S 1 ) 2 θs s 2 ds, [τ, T ]. a A

2.4. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 35 By uniqueness of he soluion of he BSDE, we derive h = h + 1 2 θs s 2 ds and for he inegrands ( z E, z S ) = (z E, z S ). As for he uiliy maximizaion beginning a = we obain θ E = z E and φ = 1 ᾱ (θs z S ). The marke price of risk process θ E V ha aains he parial clearing is unique. The proof of Theorem 15 remains valid if we replace he consan c in (2.23) wih an F τ measurable bounded random variable.

36 CHAPTER 2. EQUILIBRIUM WITH RISK SECURITY

Chaper 3 Marke compleion wih conracs In his secion, we shall describe an alernaive approach o he problem of ransferring exernal risks by rading on a financial marke in parial equilibrium. This approach is concepually more flexible and herefore beer appropriae for dealing wih risk exposures oo complicaed o be radable by jus one securiy. The ingrediens of he model are basically he same. There is a sock marke wih a sock evolving according o an exogenous price process X S. As in Chaper 2, we consider finiely many agens a I each one of which is endowed wih an iniial capial v a and a random income H a payed ou a he erminal ime T. H a depends on he economic developmen described by X S and a process K represening exernal risk which canno be hedged by rading on he sock marke. In his secion we do no consruc a second securiy o be raded ogeher wih X S. Insead, he agens have he possibiliy o sign muual or mulilaeral conracs in order o exchange random payoffs in addiion o rading wih he sock. Le us firs explain wha corresponds o marke compleion in his version of he model. The agens random payoffs are priced using one and he same pricing rule for he enire marke. The value of a payoff ha is replicable by a rading sraegy mus be equal o he iniial capial of he rader. Therefore, a pricing rule ha is consisen wih he sock price is linear on he replicable payoffs. We only consider pricing rules which are linear on all payoffs. I is well known ha pricing rules ha are coninuous linear funcionals on an L p (P )-space for some p > 1 and preserve consans can be described as expecaions of a probabiliy measure absoluely coninuous wih respec o P. Under he addiional assumpion ha a nonrivial posiive payoff has a posiive price, hese probabiliy measures urn ou o be equivalen o P. A pricing rule meeing all hese claims and being consisen wih he sock price 37

38 CHAPTER 3. MARKET COMPLETION WITH CONTRACTS is herefore given by he expecaion under a probabiliy measure equivalen o P for which X S is a maringale. We call hose measures pricing measures. Given a paricular pricing measure Q, every agen possesses a budge se which mus conain hose random payoffs ha are cheaper han he sum of his iniial capial and he value of his income H a. The preferences of an agen a are described by he expeced exponenial uiliy wih individual risk aversion α a. Now every agen maximizes his uiliy by choosing he bes priced payoff in his budge se under Q. He hen has o replicae he difference beween his payoff and his income H a by rading wih he sock, which is possible since he sock price process is a maringale under Q, and signing conracs wih oher agens. And here is how we inerpre he equilibrium wih parial marke clearing in his seing. Fix again a pricing measure Q for a momen. The random claim of each agen a may be decomposed ino a par which is hedgeable under Q purely wih X S, and an addiional par C a which depends on Q and describes he remaining compound risk of his conracs wih oher agens. So we have o look for an equilibrium pricing measure Q for which he oal compound risk a I Ca vanishes. In oher erms, he difference of offers and demands of payoffs by he differen agens creaes a claim hey are able o hedge on he financial marke alone. We use a version of he explici formula (2.15) for he uiliy maximizing payoff and he parial marke clearing condiion o characerize he densiy of he pricing measure ha aains he equilibrium in erms of he soluion of a BSDE as before. 3.1 Socks, prices of risk ransfer This ime we work on a d dimensional model wih a Brownian moion W = (W 1,..., W d ). The P compleion of he filraion generaed by W is denoed by F = (F ) [,T ]. As in (2.1) he sock price process X is given by he sochasic equaion X S = X S + X S s (b S s ds + σ S s dw 1 s ), [, T ]. (3.1) The basic facs abou our model remain unchanged wih respec o he previous secions. The coefficiens b S and σ S saisfy Assumpion 1 and herefore θ S := b S /σ S is F predicable and uniformly bounded. If he inegral in (3.1) depends on more han one componen of he Brownian moion W, hen we have o consruc a new Brownian moion such ha his inegral is driven by only one componen. This is explained in Remark 2.

3.1. STOCKS, PRICES OF RISK TRANSFER 39 The process K ha describes he exernal risk is F adaped. For a I he income H a ha agen a receives a ime T is again a real-valued bounded F T measurable random variable of he form H a = g a (X S, K). Every agen a is endowed a ime = wih an iniial capial v a, and maximizes his expeced uiliy wih respec o he exponenial uiliy funcion u a (x) = exp( α a x), x R, wih an individual risk aversion coefficien α a >. According o he inroducory remarks we nex specify he sysem of prices admied for pricing he claims of agens on our marke. We aim a considering pricing measures which do no change prices for X S. Hence we le P e be he collecion of all probabiliy measures Q on F T which are equivalen o P and such ha X S is a Q maringale. Remark 17 The price of a claim ξ under Q P e is described by he expecaion E Q [ξ] (3.2) which makes sense for all coningen claims such ha his expecaion is well defined, e.g. for ξ bounded from below. The se of equivalen maringale measures P e parameerizes all linear pricing rules ha are coninuous in an L p (P ) space for p > 1, sricly posiive on L +(P )\{} and consisen wih he sock price process X S. These pricing sysems do no allow arbirage. P e can be described and hus parameerized explicily. I consiss of all probabiliy measures Q θ possessing densiy processes wih respec o P of he following form dq θ dp = Z θ = E F ( (θ S s, θ E s )dw s ), [, T ], (3.3) wih a predicable R d 1 valued process θ E such ha he sochasic exponenial is a uniformly inegrable maringale. We denoe θ = (θ S, θ E ). The process θ E plays he same par as in secion 2.4. Using his paramerized se, he sraegies agens are allowed o use can be formulaed in he following way. Definiion 18 (admissible rading sraegy, wealh process) An admissible rading sraegy wih iniial capial v is a sochasic process

4 CHAPTER 3. MARKET COMPLETION WITH CONTRACTS π wih σs s π s 2 ds < P a.s. and such ha here exiss a probabiliy measure Q θ P e such ha he wealh process is a Q θ supermaringale. V (v, π) = v + π s dx S s X S s, [, T ], The se of admissible rading sraegies is free of arbirage. A sraegy π wih a wealh process V (v, π) ha is bounded from below is admissible. 3.2 Uiliy maximizaion For he purpose of uiliy maximizaion wih respec o our exponenial uiliy funcions he se P e has o be furher resriced o he se P f of equivalen maringale measures wih finie relaive enropy wih respec o P (see secion 2.3). Le Q θ P f for θ = (θ S, θ E ) be given. The condiion under which agens maximize heir expeced uiliy is given by a budge consrain. An individual agen a can choose among all claims ha are no more expensive han he sum of his iniial capial v a and he price of his income E θ (H a ) = E Qθ (H a ). The se of hese claims is called he budge se for agen a, formally given by B a := B(v a, H a, Q θ ) = {D L 1 (Q θ, F T ) : E θ [D] v a + E θ [H a ]}. Every agen a chooses in his budge se he claim ξ a (Q θ ) ha maximizes his expeced uiliy, i.e. he soluion of he following maximizaion problem J a (v a, H a, Q θ ) = sup E[ exp( α a D)]. (3.4) D B(v a,ha,qθ ) According o he well known heory of uiliy maximizaion via Fenchel Legendre ransforms, he soluion is given by he following Theorem. Here we pu I a (y) = ((U a ) ) 1 (y) = 1 α a log y α a, for Q θ P f. Noe ha aking Q θ from his se replaces an appeal o Lemma 7 in he proof. Theorem 19 Le H a be a bounded F T measurable random variable, v a. Define ξ a (Q θ ) = I(λ a Z θ T ) = 1 α a log( 1 α a λ a Z θ T ), (3.5) where λ a is he unique real number such ha E θ [I a (λ a Z θ T )] = v a + E θ [H a ]. Then ξ a (Q θ ) is he soluion of he uiliy maximizaion problem (3.4) for agen a I.

3.3. EQUILIBRIUM WITH CONTRACTS 41 3.3 Equilibrium wih conracs Le us now describe more formally wha we mean by an equilibrium wih parial marke clearing. We wan o consruc a sochasic process θ E and wih θ = (θ S, θ E ) via (3.3) a measure Q = Q θ P f under which he overall difference beween demands and offers of agens claims is replicable on he financial marke, i.e. can be hedged wih he securiy X S. In differen erms, we look for a price measure Q such ha a I (ξa (Q ) H a ) can be represened as a sochasic inegral wih respec o he sock price process X wih an inegrand given by an admissible rading sraegy. Under Q θ, agen a knows he claim ξ a (Q θ ) which maximizes his expeced uiliy. He covers he difference ξ a (Q θ ) H a beween his preferred payoff and his income by wo componens: he erminal wealh of a rading sraegy π a (Q θ ), and he payoff C a (Q θ ) from he muual conracs wih he oher paricipans in he marke. Formally, ξ a (Q θ ) H a = C a (Q θ ) + v a + π a (Q θ ) s dx S s X S s We now define he equilibrium measure Q by claiming ha C a (Q ) =. a I Definiion 2 (equilibrium wih parial marke clearing) Le (H a ) a I be a family of bounded F T measurable incomes, (v) a a I a family of iniial capials of he agens, X S he exogenous sock price process according o (3.1), (U a ) a I a family of exponenial uiliy funcions wih risk aversion coefficiens (α a ) a I, and (ξ a (Q θ )) a I he family of uiliy maximizing claims according o (3.5) for Q P f. A probabiliy measure Q P f aains he equilibrium wih parial marke clearing if here exiss an admissible rading sraegy π such ha we have a I (ξ a (Q ) H a ) = a I v a + πs dxs S. Xs S In view of he preceding remarks, o obain he admissible rading sraegy π of Definiion 2 we have o sum all he individual sraegies π a (Q ) of agens a over a I. Given he equilibrium measure, he exisence of π is equivalen o he exisence of an F predicable real valued sochasic process φ saisfying a I (ξ a (Q ) H a ) = a I v a + φ (dw 1 + θ S d)..

42 CHAPTER 3. MARKET COMPLETION WITH CONTRACTS The process φ and he admissible rading sraegy π are relaed by he equaion π = φ σ. S To consruc Q, we jus have o find an appropriae process θ E appearing in he exponenial of an equivalen measure change and ake Q = Q (θs,θ E ). Bu his jus means ha we can proceed as in secion 2.4 and use he echnology of BSDE. The process θ E will jus be he higher dimensional version of he process θ E consruced here. Since we are in a d dimensional model here, we shall give a few deails of he analogous consrucion. Le z 1 = θ S H = a I H a + 1 2ᾱ θ S 2 d, ᾱφ, z i = θ E i 1, i = 2,... d. We obain he following BSDE h = ᾱ H (z 1,s,..., z d,s )dw s θ S s z 1,s ds 1 2 d i=2 (z i,s ) 2 ds, [, T ]. The process θ S is uniformly bounded by Assumpion 1 and H is also bounded. In his seing he following exisence resul for an equilibrium wih parial marke clearing holds. Theorem 21 The Backward Sochasic Differenial Equaion (BSDE) h = ᾱ H (z 1,s,..., z d,s )dw s θs S z 1,s ds 1 d (z i,s ) 2 ds, (3.6) 2 [, T ], possesses a unique soluion given by he riple of processes (h, (z S, z E )) H (P, R) H 2 (P, R d ). The choice θ E = (z 2,..., z d ) and Q defined via (3.3) wih (θ S, θ E ) gives a pricing measure for which an equilibrium wih parial marke clearing is aained. Proof Due o Theorem 2.3 and Theorem 2.6 in (Kob), (3.6) possesses a unique soluion (h, z) H (P, R) H 2 (P, R d ). Now se i=2 θ E = (z 2,, z d ). (3.7) As in Lemma 13 i follows ha (θs s, θs E )dw s is a P BMO maringale. The sochasic exponenial E( (θs S, θs E )dw s ) is a uniformly inegrable maringale and he Radon Nikodym densiy of a probabiliy measure Q P e wih respec o P. As in Lemma 7, we ge H(Q P ) < and by (2.15) and (2.12) he maximal uiliy for every agen is finie. By virue of φ = 1 ᾱ (θs z 1 ), ( φ s(dws 1 + θs S ds)) [,T ] ) is a Q maringale. Hence, Q defines via (3.3) a pricing measure ha aains he parial marke clearing.

3.3. EQUILIBRIUM WITH CONTRACTS 43 For he corresponding uniqueness resul, we need he echnical condiion ha he sochasic inegral process associaed wih (θ S, θ E ) belongs o BMO. Theorem 22 Le Q (θs,θ E ) P f aain he equilibrium wih parial marke clearing and suppose ha ( (θs s, θs E )dw s ) [,T ] is a P BMO maringale. Then we have θ E = (z 2,..., z d ) and φ = 1 ᾱ (θs z 1 ) where z = (z 1,..., z d ) is given by he soluion of (3.6). Proof The proof of his saemen is quie similar o he one of Theorem 15. We conclude his secion by noing ha as in secion 2.4, θ E saisfies a dynamic programming principle. Remark 23 Le he probabiliy measure Q be given hrough (3.7) and (3.3). If he agens solve he condiioned opimizaion problem 8 for a sopping ime τ T wih he same incomes (H a ), hen Q aains also an equilibrium wih parial marke clearing. The argumens needed o prove his are as for Remark 16.

44 CHAPTER 3. MARKET COMPLETION WITH CONTRACTS

Chaper 4 A risk bond In his chaper we consruc he price process of a bond ha is issued by an insurance company. The insurance ranfers some of is insured risk o he financial marke. In conras o Chaper 2, he erminal value of he bond is specified, in fac, i is chosen by he insurance. Also, he risk bond is no in zero ne supply. The insurance is ineresed in selling i compleely o he agens presen a he marke. The agens wih heir incomes and preferences are modeled as in Chaper 2. They receive a he end of he rading ime an income ha depends on financial and exernal risk. Using he sock and he risk bond, he agens maximize he expeced uiliy of heir risky income and he erminal wealh of he rading sraegy. They apply he exponenial uiliy funcion. The soluions of he uiliy maximizaion problems deermine he demand of he risk bond and he sock. The insurance sells he bond a he beginning of he rading ime. During he rading period, he agens rade he risk bond among hemselves. A he erminal ime, he insurance pays ou he bond. This payou consiss of wo pars. The payou a he end of he rading ime is described by a random variable ha may depend on exernal and financial risk. During he rading period, he insurance may coninuously pay ou an ineres wih a rae ha depends on he exernal risk and also on he marke price for he exernal risk. Thus, a feedback of he opinion of marke abou he exernal risk o he srucure as well as he volume of he risk bond is possible. Since he ineres rae for he bank accoun is equal o zero, here is no difference, if he ineres is payed as a lump sum a he end of he rading period. Ulrich Hors poined ou he imporance of an ineres rae ha depends on he exernal risk as well as on he marke. The mehods developed in Chaper 2 can be easily adaped o his siuaion. The equilibrium condiion in his chaper is sraighforward: he price 45

46 CHAPTER 4. A RISK BOND of he risk bond has o be adjused such ha he rading sraegies add up o one, since we assume ha he insurance has issued exacly one share. Afer selling he bond, he insurance does no rade anymore. Thus, our equilibrium gives exacly he price such ha he bond is compleely sold. As in Chaper 2 and Chaper 3, he agens considered in our model are assumed o be small raders a he sock marke. Thus, he sock price is exogenuosly given and here is no marke clearing for rading wih he sock required wihin our group of agens. Wha are possible price processes for he risk bond? One consrain is imposed by he absence of arbirage condiion. Since he sock price process is already fixed, we can only choose among he maringale measures for he sock price process. A price process of he risk bond is hen he successive condiional expecaion of is erminal value. We consruc via a BSDE he densiy of a maringale measure for he sock price process and hen a risk bond price process such ha marke clears for he bond. A problem is o show ha our bond complees he marke. Using Malliavin calculus we give an absrac crierion as well as an example. The fac ha he propery of compleness depends on he equilibrium price makes he problem very difficul, since his price is only implicily described by he soluion of a BSDE. 4.1 The risk bond In his secion we describe he srucure and he se of price processes ha we consider for he bond. Furhermore, we prove a crierion ha characerizes marke compleion. Finally, for every possible price process of he risk bond, he uiliy maximizing payoffs of he agens are calculaed. The sock marke consiss of a bank accoun wih ineres rae zero. The sock defined in (2.1) saisfies Assumpion 1. Prices for payoffs replicable wih he sock are already fixed: he price is he iniial capial needed o replicae he payoff wih a rading sraegy. This iniial capial is equal o he expecaion of he payoff under a maringale measure for he sock. We consider a subse of maringale measures, where every elemen Q η, η V, is defined by (3.3) in Chaper 2. The se V of possible marke prices of exernal risk is defined in (2.5). In our approach, we fix he erminal value of he risk bond. A candidae of a price process for he risk bond is calculaed by aking successive condiional expecaions. The price for a payoff F under Q η, i.e. he iniial capial ha is needed o replicae F wih boh he sock and he risk bond, is equal o E η [F ].

4.1. THE RISK BOND 47 The payou of he risk bond consiss of wo pars. random erminal paymen H I, H I = g I (K, X S ) The firs one is a where g I may depend on he whole pah of K and X S. We assume ha H I is bounded, bu no necessarily posiive. The second par accumulaes wih a rae r(, θ, ρ (K ), η ) ha depends on several facors. θ is he marke price of risk for all coningen claims replicable wih he sock. This process replaces θ E ha is used in Chaper 2. The insurer may wan o adjus his paymen according o he evoluion of he exernal risk facor. He uses a predicable process (ρ (K )) [,T ] ha describes he risk caused by he exernal facor K from his poin of view. The insurance migh require more capial if he exernal risk is seen as more dangerous. In his case, r(, ρ, ) is negaive. On he oher hand, if he exernal risk evolves in less dangerous way, he insurance migh pay a higher ineres. For hose ρ, r(, ρ, ) is posiive. The mos imporan poin is he possibiliy o le he payou depend on he marke via he marke price of exernal risk η. Here, η replaces θ E. All facors are conneced by a deerminisic funcion r : [, T ] (R d ) 3 R ha is a priori chosen by he insurer. So he insurer pays a ime T H I + r(, θ, ρ, η )d. The payou r(, θ, ρ, η )d migh be inerpreed as an ineres. Since he ineres rae for he bank accoun is zero, i does no maer wheher he ineres r for he risk bond is payed coninuously or as a lump sum a ime T. We assume ha he whole sum is payed a ime T. The price ha he insurer ges for his bond under a pricing measure Q η, η V, is ] h I (η) = E [H η I + r(, θ, ρ, η )d, (4.1) where Q η is defined analoguosly o Q θ in (3.3). How should he feedback of he marke price of he exernal risk on r be chosen by he insurer? The choice reflecs a supply of he insurer depending on he marke. One possibiliy is he following: under a favorable pricing sysem, he insurer is willing o provide a large volume of he bond. Conrary, if he price is no favorable for he insurer, he volume of he bond migh be much less. This can be modeled by an ineres rae r ha is decreasing in η. Consider he following BSDE: Y η = H I z s dw s + (r(s, θ s, ρ s, η s ) z s η s )ds.

48 CHAPTER 4. A RISK BOND Of course, E η [H I + r(, θ, ρ, η )d] = Y η. Applying he comparison heorem for BSDE under appropriae assumpions on r, we see ha E η [H I ] E ˆη [H I ] if η ˆη P λ- a.e. We inend o use he resuls in Kobylanski (Kob) in order o compue an equilibrium pricing densiy η and hus an equilibrium pricing measure Q η. So our assumpions include Assumpion (H1) and (H2) in (Kob). Assumpion 24 Le he erminal payoff H I be bounded and F T measurable. Le r : [, T ] (R m ) 3 R ogeher wih he predicable processes θ and ρ saisfy P λ a.s. for all z R m 1. r(, θ, ρ, z) c z 2 for a consan c < 1 2 and 2. r is differeniable in z and saisfies for a consan c 2 and a coninuous funcion k : [, T ] R:. r z (r(, θ, ρ, z) k() + c 1 z The nex sep is o specify he se of price processes for he risk bond. Here we describe he case where he cumulaive ineres is payed a ime T. Price processes for coninuously payed ineres are given in Remark 25. We consider only price processes such ha here exiss a probabily measure equivalen o P ha sees he sock price and he price of he risk bond as maringales. In fac, we fix a maringale measure for he sock price Q η, where η V, and V is defined in (2.5). Then he risk bond price process B η is defined by B η = E η [H I + = E η [H I ] + r(s, θ s, ρ s, η s )ds F ] (4.2) κ s (η)(dw 1 s + θ s ds) + υ s (η)(dw 2 s + η s ds), where (κ(η), υ(η)) is he inegrand in he represenaion of ] E [H η I + r(s, θ s, ρ, η )ds F (4.3) as a sochasic inegral wih respec o he Q η Brownian moion ( W η W 1 = + θ sds W 2 + η sds ), [, T ].

4.1. THE RISK BOND 49 Since here are wo risky securiies, we have o define a wo dimensional rading sraegy ( ) π S π = π B, [, T ]. The number of shares of sock owned is denoed wih π S, whereas π B sands for he numbers of shares of he risk bond. Le v denoe he iniial capial. The wealh process of a rading sraegy for T is V ( π) = v + = v + + π S u dx S u + π B u db η u π S u (X S u σ S u + κ u (η))(dw 1 u + θ u du) π B u υ u (η)(dw 2 u + η u du) Remark 25 The wealh process V ( π) is he same if he ineres is payed ou coninuously during [, T ]. A price process B η for he bond wih a erminal payou H I and insananeous ineres paymen r(, θ, ρ, η ) is given by he successive condiional expecaion of he payou ha is no ye payed: ] B η = E [H η I + r(s, θ s, ρ s, η s )ds F = v + κ s (η)(dw 1 u + θ u du) + r(s, θ s, ρ s, η s )ds. υ u (η)(dw 2 u + η u )du The inegrands κ and υ are he same as in (4.2). Since in he ime inerval [, ] he ineres r(s, θ s, ρ s, η s )ds is already payed, his payou is no included in he condiioned expecaion. However, his ineres is par of he wealh process, he holder of π u B shares of he bond is eniled o ge he paymen rae π u B r(u, θ u, ρ u, η u ): V ( π) = v + π S u dx S u + Thus, he ineres cancels ou: V ( π) = v + π B u d B η u + π S u X S u σ S u (dw 1 u + θ u du) + π B u r(u, θ u, ρ s η u )du. π B u υ u (η)(dw 2 u + η u du). So he invesor ge for boh ypes of bonds he same wealh process if he uses he same rading sraegy.

5 CHAPTER 4. A RISK BOND In he sequel we assume ha r(s, θ s, ρ s, η s )ds is payed a ime T. The price process we use is B η defined in (4.2). The se of admissible rading sraegies depends on he price process of he risk bond, i.e. on η. Similar o (4), hose rading sraegies are called admissible ha lead o a wealh process ha is a supermaringale under Q η : A(η) = { π : V ( π) is Q η supermaringale}. An agen indexed wih a I ries o find he opimal rading sraegy: [ ( ( X sup E exp ( α a v a S + π u d u π A(η) Bu η he iniial capial of he agens being denoed by v a. )))], (4.4) We inend o use echniques relying on a complee marke. So he nex proposiion gives a sufficien crierion on a risk bond process B η o complee he marke. A simple example of a risk bond compleing he marke is saed in Example 34 on page 61. Proposiion 26 (Complee marke) Le he erminal value H I and he ineres rae r of he bond be according o Assumpion 24. Then a bond price process B η complees he marke if and only if υ (η), P l a.e. (4.5) Then o every F L 1 (Q η, F), here exis a unique rading sraegy ( π S, π B ) A(η) such ha he wealh process is a Q η maringale ha saisfies F = E η [F ] + ( π S, π B )d ( X S B η Proof. The proof consiss of several seps. Firs we apply he maringale represenaion heorem under P (see e.g. (RY91) Th.V.3.5) o M = E[Z η T F F ], where Z η is defined analogously o Z θ in (2.7), and obain an inegrand α H 1 (P, R 2 ) saisfying Iô s formula yields d(m (Z η ) 1 ). M = E[Z η T F ] + α dw, [, T ]. ) = [M (Z η ) 1 (θ, η ) r + (Z η ) 1 α ](dw + (θ, η ) r d).

4.1. THE RISK BOND 51 Thus, he inegrand saisfies β = M (Z η ) 1 (θ, η ) r + (Z η ) 1 α, [, T ], F = E η [F ] + β s dw η s, and M(Z η ) 1 is a Q η maringale. Le (4.5) be saisfied. Then we may se and π B = β 2, υ (η) π S = β 1, π B κ (η). σ S X S This sraegy ( π S, π B ) is admissible since he wealh process is a Q η maringale. In order o see uniqueness, assume (4.5). We apply a well known argumen. Le π and π be wo admissible rading sraegies aaining F such ha heir wealh processes are Q η maringales. Define δ = (δ S, δ B ) by Then δ s = π s π s, s [, T ]. (δ S u σ S u X S u, δ B u υ u (η))dw η s is a Q η maringale wih erminal value zero. Thus he quadraic variaion saisfies (δ S u σ S u X S u ) 2 + (δ B u υ u (η)) 2 ds =, [, T ]. This means, δ = wihin he equivalence class generaed by he vecor space H 2. I remains o show he necessiy of (4.5). This par of he proof uses he Kunia Waanabe decomposiion: for every (Q η, F) maringale M wih M T L 2 (Ω) < here exiss a unique inegrand φ and a unique maringale N H 2 (Q η ) srongly orhogonal o (X S, B η ) such ha M = φ u d(x S u, B η u) r + N. So, if υ = on some se A Ω [, T ] wih P λ[a], we can find a predicable process φ saisfying φ = P λ a.e. on A c and F = M T = φ sdw η s L 2 (Ω), M T in L 2 (Ω). Uniqueness in he Kunia Waanabe decomposiion yields ha F can no be represened as a sochasic inegral wih respec o (X S, B η ) where he inegral is a B η maringale.

52 CHAPTER 4. A RISK BOND Here we describe he preferred payoffs of our agens. They maximize heir uiliy wih respec o an exponenial uiliy funcion. Proposiion 26 leads o a budge consrain: having fixed a risk bond price process B η, a coningen claim ξ is he sum of he income H a and he erminal wealh of a replicable rading sraegy π, if ξ is in he budge se: B a (η) = { ξ L 1 (F T, Q η ) E η [ξ] E η [H a ] + v a }. Observe ha we have he same budge ses as in (2.1) in Chaper 2. The maximizaion problem concerning he se of admissible rading sraegies is equivalen o a maximizaion problem considering he aainable claims in he budge se: sup E [ exp ( α a ξ)]. ξ B a(η) According o Theorem 6, he uiliy maximizing erminal wealh ξ a (η) for a risk bond price process B η, η V, is hen ξ a (η) = c a + 1 θ u dwu 1 + 1 α a α a where c a is a consan choosen such ha E η [ξ a (η)] = E η [H a + η u dw 2 u + 1 2α a r(, θ, ρ, η )d + v a (θ 2 u + η 2 u) du, (4.6) The soluion ( π S,a (η), π B,a (η)) of (4.4) is called opimal rading sraegy for he bond price B η, η V. Since a sraegy wih a wealh process ha is a supermaringale can be opimal, ( π S,a (η), π B,a (η)) is he sraegy ha aains ξ a (η) H a wih a wealh process ha is a Q η maringale. Thus, ( π S,a (η), π B,a (η)) saisfies ξ a (η) H a = v a + = v a + + π u S,a (η)dxu S + ]. π u B,a (η)dbu η π u S,a (η)xu S σu S (dwu 1 + θ u du) π u B,a (η)υ u (η)(dwu 2 + η u )du.

4.2. PARTIAL MARKET CLEARING 53 4.2 Parial marke clearing In his secion we formulae our equilibrium wih parial marke clearing in presence of a risk bond and prove is exisence. Parial marke clearing simply says ha he sraegies using he bond have o sum up o one. Definiion 27 (equilibrium wih risk bond) Le (H a ) a I be a family of bounded F T measurable incomes, H I he bounded F T measurable erminal value of he risk bond and (r(, θ, ρ, η )) [,T ] he ineres rae according o Assumpion 24. Le X S denoe he exogenous sock price process according o (2.1), (u a ) a I a family of exponenial uiliy funcions wih risk aversion coefficiens (α a ) a I, and ( π S,a, π B,a )(η) he uiliy maximizing rading sraegies for he bond price B η, η V. A bond price process B η wih η V ogeher wih he uiliy maximizing rading sraegies ( π S,a, π B,a )(η ) is an equilibrium wih parial marke clearing, if π B,a (η )(ω) = 1 for P λ a.e.(ω, ). (4.7) a I In order o find he marke price of risk saisfying condiion (4.7) we sae an equivalen condiion on he sum of he individual uiliy maximizing erminal wealhs: here exiss an inegrand φ H 1 (Q η, R) saisfying ξ a (η ) = h I (η ) + H I + a I + a I v a + r(, θ, ρ, η )d + a I H a (4.8) φ (dw 1 + θ d). This condion on he marke price of exernal risk η has he following meaning: on he lef hand side we have he sum of he payoffs preferred by he invesors. On he righ hand side are price of he bond h I (η ) (defined in 4.1), he payoff of he risk bond, he sum of he incomes of he invesors and φ he erminal wealh of he cumulaive rading sraegy wih he sock. σ S X S The equaion says ha he price for he risk bond is chosen such ha he invesors buy i compleely. The budge consrain for each invesor yields ha he cumulaive price a ime of he risk bond is equal o he deerminisic value h I (η ). The invesors redisribue heir risky incomes among hemselves. Addiionally hey use he sock in order o hedge financial risks. The marke price of risk a he sock exchange θ is exogenuously given. Recall ha here is no marke clearing required for he sock wihin he group I of invesors. In he nex propsiion we sae he equivalence of (4.7), a

54 CHAPTER 4. A RISK BOND condiion on he rading sraegies, and (4.8), a condiion on he erminal values. Proposiion 28 Le η V. The opimal rading sraegies for he bond π B,a (η) saisfy condiion (4.7) if and only if he sum of he uiliy maximizing erminal wealh saisfies condiion (4.8). Proof. We proceed as in he proof of Lemma 11, where he equilibrium condiion for a bond in zero ne supply is relaed o a condiion on he erminal wealh of he rading sraegies. Firs, we show he only if par. Le (4.7) be saisfied. Since B η T = H I + r(u, θ u, ρ u, η u )du, we obain wih he lineariy of he sochasic inegral a A ξ a (η) a I H a = a I v a + = h I (η) + H I + + a I v a + a I π S,a u (η)dx S u + a I Thus, if (4.7) is saisfied, hen we obain wih φ = a A π S,a (η)x S σ S r(u, θ u, ρ u, η u )du π S,a u (η)dx S u. π B,a (η)dbu η a I also (4.8). Since all ξ a (η) are inegrable wih respec o Q η and he wealh process of an opimal rading sraegy is a Q η maringale, φ H 1 (Q η, R) is saisfied. Now le (4.8) be saisfied. Le π a (η) = ( π S,a sraegy ha replicaes ξ a (η) H a obained by Proposiion 26. The sum of he coningen claims hedged by he agens is equal o ( x a + π S,a (η)dx S + a A ξ a (η) H a = a A (η), π B,a (η)) r be he hedging ) π B,a (η)db η. Recall ha he uiliy maximizing rading sraegies generae wealh processes ha are Q η maringales. The only admissible rading sraegy aaining he erminal value B η T of he bond wih a maringale wealh process is (π S, π B ) = (, 1), [, T ]. Thus H I + r(, θ, ρ, η )d h I (η) = 1dB η.

4.2. PARTIAL MARKET CLEARING 55 Furhermore, since (ξ a (η) H a v) a = a A φ (dw 1 + θ d) + 1dB η, he coningen claim on he lef hand side is replicable by an admissible rading sraegy ( π S, π B )(η) wih π B (η) = 1 and π S (η) = φ. On he X SσS oher hand, (ξ a (η) H a v) a = π S,a (η)dx S + π B,a db η. a I a A The lineariy of he sochasic inegral and he uniqueness of he inegrand yield ha ( a I πs,a (η), 1) is he sum of he individual uiliy maximizing rading sraegies. Our main resul is a BSDE ha characerizes he equilibrium marke price of he exernal risk facor η. The BSDE can be obained by combining (4.8) and he explici srucure of he uiliy maximizing erminal wealh ξ a, a I of he invesors. Before saing he BSDE, we do some preparaions. Le ᾱ saisfy 1 ᾱ = a I 1 α a. Since all invesors use exponenial uiliy funcions wih differen coefficiens of risk aversion, he preferred payoffs differ only by a deerminisic facor and a deerminisic consan. So he sum has a simple srucure: ξ a (η) = c + 1 ᾱ a I (θ, η ) r d ( W 1 W 2 ) + 1 2ᾱ (θ 2 + η 2 )d, (4.9) where c = a I c a wih he consans c a, a I, from (4.6). Denoe ( ) H = ᾱ H a + H I h I (η ) + 1 θ 2 2 sds. a I Now we plug (4.9) ino (4.8) and rearrange he erms. Thus, η has o be chosen such ha H = ᾱc + (θ ᾱφ )dw 1 + η dw 2 + + 1 2 (η ) 2 + θ 2 ᾱφ θ ᾱr(, θ, ρ, η ) d.

56 CHAPTER 4. A RISK BOND A change of variables leads o he noaion used in BSDEs: z S = θ ᾱφ, z B = η. Now we are able o wrie down he BSDE ha characerizes he equilibrium marke price of risk η : Y = H (z S s, z B s ) d ( W 1 s W 2 s ) [ 1 2 ((zb s ) 2 + θ s z S s r(s, θ s, ρ s, z B s ) ] ds (4.1) In he following Theorem we show ha he choice η := z B yields a parial equilibrium. Theorem 29 There exiss a marke price of risk process η V ha leads o an equilibrium wih parial marke clearing. η can be consruced using he soluion (Y, (z S, z B )) of he BSDE (4.1) by seing η = z B. Proof. By Theorem 2.3 and Theorem 2.6 in (Kob), equaion (4.1) has a unique soluion (Y, (z S, z B )) H (P, R) H 2 (P, R 2 ). In Lemma 3 below we prove ha ( zb s dws 2 ) [,T ] is a BMO maringale, hence z B V. The choice η := z B and φ := 1 (θ ᾱ zs ) yields ha he uiliy maximizing wealhs ξ a (η ) saisfy he equilibrium condiion on he erminal wealh (28). Lemma 11 leads o an equilibrium wih parial marke clearing. Via (4.2) we obain a bond price process B η. In he nex lemma we prove he BMO propery ha we need o define he equilibrium prices. Lemma 3 Le (Y, (z S, z B )) be he soluion of he BSDE (4.1). zb s dws 2 is a P BMO maringale. Then Proof. Le Q be defined by dq = E( (θ dp s, )dw s ). We show ha zb s dws 2 is a Q BMO maringale and apply hen Theorem 3.6 in (Kaz94). This is possible because θ sdws 1 is also a P - BMO maringale. We have o show: here exiss a consan c > such ha for all soppping imes τ T E [ τ (z B s ) 2 ds F τ ] c.

4.3. RISK BOND COMPLETING THE MARKET 57 Under he probabiliy measure Q, he condiional expecaions of he BSDE reads E [H Y τ F τ ] = E [ τ ( 1 2 (zb s ) 2 r(s, θ s, ρ s, z B s )ds F τ ]. According o Assumpion 24 on r and he a priori esimae Corollary 2.2 in (Kob), he lef hand side is bounded by a consan c 1 ha does no depend on τ. Assumpion 24 also yields 1 2 (zb s ) 2 r(s, θ s, ρ s z B s ) ( 1 2 a)(zb s ) 2 b for consans a < 1 and b >. Thus we obain for all sopping imes 2 τ T E [ (zs B ) 2 F τ ] c 1 + b 1 τ a. 2 4.3 Risk bond compleing he marke In his secion we give a crierion and an example for a risk bond ha complees he marke. We use Malliavin calculus. Le us explain he Clark Ocone formula for a d dimensional Brownian moion. This formula gives he inegrand in a sochasic inegral as he condiional expecaion of he Malliavin derivaive of he erminal value ha he inegral aains. In order o undersand he d dimensional Clark Ocone formula, use he parameer space explained in Example 1.1.2 in Nualar (Nua95), his is T = [, T ] {1,..., d}. The measure µ is he produc of he Lebesgue measure and he measure ha gives mass one o each poin 1,..., d. The space of all componenwise square inegrable funcions L 2 (R + ; R d ) is isomorphic o Le W = (W 1,... W d ) be a d dimensional Brownian moion. Then W = W i for = (, i), [, ], i {1,..., d}. For any h H, he random variables d W (h) = h i dw i are a cenered Gaussian family of random variables saisfying E[W (h)w (g)] = h( )g( )dµ( ). i=1 T

58 CHAPTER 4. A RISK BOND Le D = D (,i) [, T ], i {1,..., d}, denoe he Malliavin derivaive in he space D 1,2 of Malliavin differeniable random variables (see Definiion 1.2.1 on page 24 and page 27 in (Nua95)). Then he Clarke Ocone formula for a d dimensional Brownian moion reads for F D 1,2 : F = E[F ] + d i=1 E[ D (,i) F F ]dw i. The proof is analogous o he proof for he one dimensional Clark Ocone formula Proposiion 1.3.5 in (Nua95). The noaion of a d dimensional Malliavin derivaive and Clarke Ocone formula are ools o describe he inegrand υ in (4.2) more exacly. The erminal value of he bond B η T is equal o B η T = H I + r(, θ, ρ, η )d. Le H M denoe he payoff ha is disribued among he agens in he group I in he case of an equilibrium wih parial marke clearing wih he price measure Q η and he bond price process B η : H M = a A H a + H I h I (η ) + r(, θ, ρ, η )d + φ (dw 1 + θ d), where φ is he inegrand in he equilibrium condiion (4.8). The following proposiion gives he inegrand υ depending on H M. Proposiion 31 Le η be he marke price of risk process ha aains an equilibrium wih parial marke clearing according o Theorem 29. Assume ha B η T is bounded and B η T D1,2. Le H M be bounded and H M D 1,2. Then he inegrand υ in (4.2) is equal o υ = E η [D (,2) B η T F ] cov (B η = Z 1 T, D (,2)H M ) (4.11) E[Z T D (,2) B η T F T ] (4.12) [ ( ) ] B η T Z T η + (D (,2) θ s, D (,2) ηs)dw s F [ ] B η T Z T (θsd (,2) θs + ηsd (,2) ηs)ds F. Z 1 E +Z 1 E Here, cov denoes he condiional covariance under Q η wih respec o F.

4.3. RISK BOND COMPLETING THE MARKET 59 Proof. The inegrand (4.12) is already saed in (KO91). Le η be he marke price of risk process ha aains an equilibrium wih parial marke clearing according o Theorem 29. In order o keep he noaion simple, we denoe in his proof Z = Z η, Q = Q η, and W he Q Brownian moion consruced via Girsanov ransform applied o W. According o (4.8), we have for a consan c > Z T = c exp ( ᾱh M ). The proof consiss of wo pars. Firs we apply Iô s formula and he maringale represenaion heorem under he probabiliy measure P and he Brownian moion W. Also under P we apply he Clark Ocone formula. Wih Iô s formula we can ransform he inegrals ino inegrals wih respec o W. Se M = E[Z T B η T F ] = E[Z T B η T ] + ζ dw, [, T ], where ζ H 2 (R 2 ) is he inegrand in he represenaion of M. This supposes only B η T L1 (Q ). Thus Iô s formula yields B η Hence = E [B η T ] + B η = Z 1 E[Z T B η T F ] = Z 1 M. ( dw Zs 1 (ζ s + M s (θ s, ηs)) 1 s + θ s ds dws 2 + ηsds ), [, T ]. υ = Z 1 (ζ 2 + M η ). (4.13) ζ 2 denoes he second componen of ζ. In order o ge more informaion abou υ, we apply he Clarke Ocone formula o Z T B η T o obain ζ more explicily: ζ i = E[D (,i) (Z T B η T ) F ], i = 1, 2. We have o use he produc and chain rules for Malliavin differeniaion. Since H M is bounded, here exiss a consan c such ha c H M and we define { exp( x), x > ᾱc, ẽ(x) = exp( ᾱc ), x ᾱc.

6 CHAPTER 4. A RISK BOND Thus, exp( ᾱh M ) = ẽ(h M ) and ẽ is Lipschiz coninuous. Since H M is assumed o be bounded and in D 1,2, we obain wih Proposiion 1.2.3 in Nualar (Nua95) Since B η T D (,i) Z T = D (,i) ẽ(h M ) = ᾱz T D (,i) H M. is in D1,2 and bounded, we obain D (,i) (Z T B η T ) = Z T D (,i) B η T ᾱbη T Z T D (,i) H M. (4.14) Applying he Clark Ocone formula o Z T = 1 Z s(θ s, ηs)dw s we can wrie η = Z 1 E[D (,2) Z T F ] = Z 1 ᾱe[ Z T D (,2) H M F ]. Combining all hese derivaives we ge from (4.13) υ = Z 1 +Z 1 E[ ᾱz T B η T D (,2)H M + Z T D (,2) B η T F ] } {{ } ζ 2 E[Z T B η T F ] } {{ } M Z 1 E[ᾱZ T D (,2) H M F ] } {{ } η = E [D (,2) B η T F ] ᾱ cov (B η T, D (,2)H M ), [, T ]. (4.15) In order o see (4.12), we use Z T = E( (θ, η )dw ). The Malliavin derivaive of a sochasic inegral is saed in (1.46) on page 38 in (Nua95). This yields (4.12). So far, wo no explicily known parameers appear in Proposiion 31. The firs one is he random variable φ s(dws 1 + θ s ds) wihin H M. The second one is η ha changes he ineres rae of he bond. Here we give an example where he condiions saed in Proposiion 31 involve only he incomes and he erminal value of he bond. We use he idea of Example 14 in Chaper 2. Le F i = (F i ) be he P augmenaion of he filraion generaed by W i, i = 1, 2. Lemma 32 Assume ha he drif of he sock price θ S is adaped o F 1. Furhermore le he sum of he incomes H = a I H a be decomposed ino wo pars: H = H 1 + H 2, where H i is measurable wih respec o FT i, i = 1, 2, and boh random variables are bounded. Furhermore, le he ineres rae r of he bond be equal o zero and H I be FT 2 measurable. Then (4.11) simplifies o υ = E η [D (,2) H I F ] cov (H I, D (,2) (H I + H 2 )). (4.16)

4.3. RISK BOND COMPLETING THE MARKET 61 Proof. Wih he same argumens as in Example 14, we see ha he marke price of he exernal risk η depends only on (H 2 + H I ) and saisfies for a consan c H 2 + H I = c + η sdw 2 s + 1 2 (η s) 2 ds. Furhermore, he adjusmen of he marke porfolio φ s(dw 1 s + θ s ds) is measurable wih respec o F 1 T. Observe ha D (,2)F = for any Malliavin differeniable random variable F ha is F 1 T measurable. In his siuaion one may ask if he represenaion propery of B() enails he represenaion propery of B(η) for every η V. In erms of he inegrands: υ() P λ a.e. yields υ(η) Q η λ a.e. In general, his is no rue. Here is a counerexample for a one dimensional Brownian moion. Example 33 Le η = 1, [, T ]. Le for a < k < T Then H I = 1 2 (W 2 T W 2 k ). D H I = W T 1 T W s 1 k, where 1 k = 1 for k and oherwise. Thus υ () = (W W k )1 >k. So we have a random variable wih an inegral represenaion ha is equal o zero for < k. However, under he equivalen probabiliy measure Q η wih η = 1, we obain an inegrand υ(η) saisfying υ(η) Q η λ a.e. To see his, observe ha D η s =,, s [, T ]. Hence { E υ (η) = η [W T W k F ] = E η [ η k sds F ] = k T, k, E η [W T F ] = W (T ), > k. Here we give a simple example of a marke ha is compleed by a weaher bond. We use he explici formula (4.13). Example 34 (Temperaure bond) The rading inerval consiss of N days, i.e. T = N. The exernal risk facor is he emperaure curve during a heaing period, modelled by a mean revering Ornsein- Uhlenbeck process dk = a(m K )d + dw 2, K =.

62 CHAPTER 4. A RISK BOND The agens represen energy deliverers. During he heaing period hey sell more volume if he emperaure is low. On he oher hand, if he demand on energy volume is oo large, he reailer has o buy more himself and ges no furher benefi. So he income of an agen may have he following form: H a = c a N i=1 ((k K i ) + (k 1 K i ) + ) for emperaure resholds k 1 < k and weighs c a >, a A. In order o keep formulas shor, denoe H s = N ((k K i ) + (k 1 K i ) + ) i=1 and c A = a A c a. Wih he noaion of Lemma 32, we have H = H 2 = c A H s, H 1 =. Wha is he srucure of a risk bond ha complees he marke? We choose he same srucure as in he incomes bu wih he opposie sign. Se he ineres rae equal o zero: r =. and he erminal payoff H I ha is equal o he value of he bond B η T T as H I = B η T = c IH s a ime wih a consan c I o be deermined laer. We may apply Proposiion 26 and Lemma 32. The marke is complee if and only if he inegrand υ in (4.2) saisfies υ P l a.e. (4.16) yields The choice υ = E η [ c I D,2 H s ( ᾱ(c A c I )H s E η [H s F ] 1 ) F ]. c I c A implies ha υ E [ c I D,2 H s F ].

4.3. RISK BOND COMPLETING THE MARKET 63 Proposiion 1.2.3 in Nualar (Nua95) yields D (,2) H = N 1 i exp( a(n i))1 k1 K i k. i=1 Our emperae process K is an Ornsein Uhlenbeck process. The condiional law of K i given F, < i < T, under P is Gaussian, hus equivalen o he Lebesgue measure. Thus P [k 1 K i k F ] >, P a.s., < i < T, Since our equilibrium price measure Q η is equivalen o P, we obain for all [, T ] [ N ] E 1 i exp( a(n i))1 k1 K i k F >, Q a.s. i=1 In every inerval (i 1, i), i = 1,..., N), we have he successive condiioned expecaions of he same random variable. We may esimae he inegrand υ by a piecewise coninuous version of he condiional expecaion. Thus, here exiss a version of υ such ha υ > for P λ- a.e. (ω, ).

64 CHAPTER 4. A RISK BOND

Chaper 5 Equilibrium wih general uiliy funcions In his chaper we consider a larger class of uiliies han he exponenial. The uiliy funcions are defined for posiive wealh and required o saisfy he Inada condiions. However, we have o pay a price. In conras o he exponenial uiliy, i is no possible o characerize he equilibrium price by a BSDE. We calculae our equilibrium wih parial marke clearing in a one period model in a probabiliy space (Ω, F, P ) where (Ω, F) is a Borel space. The concep of he equilibrium wih parial marke clearing in his chaper is he same as in Chapers 2 hrough 4. We have wo sources of risk: financial and exernal risk. We have agens concerned by boh risk facors. Financial risk can be hedged on a sock marke. In order o ransfer exernal risk, he agens complee he marke: hey sign muual conracs. Financial risk is represened by a σ algebra G F. An example illusraes he exernal risk facor: his migh be described by a random variable K. Then F = σ(k) G. The risky incomes of he agens are modeled as F measurable random variables. In he sequel we don use he fac ha he risk facor is described as a real valued random variable. Prices are considered as linear and given by he expecaion under a probabiliy measure equivalen o P. This is explained in Remark 17 on page 39. The probabiliy measures ha we use here in order o calculae prices are called pricing measures. The sock marke is given by G and an exogenously fixed pricing measure Q G on G. Every G measurable Q G inegrable payoff R can be raded on he financial marke. The price is of course equal o E QG [R]. We call hose claims replicable a he financial marke. Reasonable pricing measures are free of arbirage. This means, he pricing measures we consider here agree on G wih Q G, analogously o he maringale 65

66 CHAPTER 5. GENERAL UTILITIES measures for he sock in he previous chapers. Given a pricing measure, he agens choose he payoffs ha maximize heir expeced uiliy among all random variables ha are no more expensive han heir income. In order o do so hey rade on he sock marke and sign muual conracs. The equilibrium wih parial marke clearing is defined as in Chaper 3.2: he difference beween he sum of he incomes and he sum of he preferred incomes has o be replicable on he sock marke. Then he muual conracs add up o zero. We adap he echniques used in he book Föllmer and Schied, (FS2), Chaper 3.4 o our siuaion. In (FS2), an Arrow Debreu equilibrium is consruced. In his classical model, here is no exogenuos financial marke. The agens obain risky incomes and rade hem among hemselves. A pricing measure is consruced such ha he sum of he uiliy maximizing payoffs is equal o he sum of he incomes. The marke clears oally, no only parially. 5.1 Income, preferences, he marke In his secion we describe he income of he agens, prices of random payoffs, he preferences and he marke. We place ourselves in a probabiliy space (Ω, F, P ). Since we aim a consrucing coninuous versions of condiioned expecaions ha depend on a parameer, we assume ha regular condiined disribuions exiss. This is he case if (Ω, F) is a Borel space. The agens in our model are exposed o wo sources of risk. The firs one is economic or financial risk ha can be hedged on a financial marke. This risk is represened by a σ algebra G F. The effec on he income of an agen can e.g. be described by a G measurable random variable S a, a I. The second yp of risk is caused by an exernal facor modeled by a real valued F measurable random variable K. The income H a of an agen depends on boh sources of risk. An example is H a = g a (S a, K), a I, where g a : R 2 R. We use he fac ha all incomes H a, a I, are F measurable non-negaive bounded random variables. The funcions g a and he random variables S a, K are inroduced in order o give an example. The sum of all incomes H = a I H a is called he marke porfolio. The price of a random payoff is calculaed by aking he expecaion under a probabiliy measure equivalen o P. We call a random variable

5.1. INCOME, PREFERENCES, THE MARKET 67 φ L 1 (Ω, F, P ) price densiy. Wih φ, we define a probabiliy measure Q φ by dq φ dp = φ E[φ]. The normalized price of an F measurable random payoff ξ is equal o E[φξ] E[φ] = E φ [ξ], (5.1) where E φ is he expecaion wih respec o Q φ. If we compare he price of wo coningen claims under he same pricing measure, he regularizing denominaor can be ignored. Here we describe he finanial marke in deail. All payoffs conaining only financial risk can be replicaed on he financial marke. This means hey can be sold or bough a a cerain price. In our model, hose risks are given by G measurable random variables. On he oher hand, prices on he financial marke are exogenously fixed. Here, we have a G measurable price densiy φ G. In Assumpion 35 below we sae a condiion on φ G ha we use o calculae our equilibrium price densiy. Assumpion 35 The price densiy φ G is bounded from above and away from zero: here exiss consans < δ < δ 1 such ha < δ φ G δ 1 P a.s. The se of payoffs ha are replicable on he financial marke is R = {R L 1 (Ω, G, Q G ) E[ φ G R ] < }. (5.2) Since he agens are supposed o be small raders, hey can buy or sell any amoun of replicable coningen claims a he price deermined by (5.1) wih he price densiy φ G. There is no marke clearing required on he sock marke. The nex remark compares he se R of replicable payoffs o he sock marke in Chaper 3. Remark 36 Le T > be he end of a rading ime. Le he probabiliy space (Ω, F, P ) carry a d dimensional Brownian moion W = (W 1,..., W d ) and le F = F T, where (F ) [,T ] is he P compleion of he filraion generaed by W. Now le (F 1 ) [,T ] be he compleion of he filraion generaed by W 1. X S denoes he sock price process according o (3.1) in Chaper 3 wih he addiional assumpion ha θ S is predicable wih respec o (F 1 ).

68 CHAPTER 5. GENERAL UTILITIES In conras o Chaper 3, we resric here rading sraegies o be predicable wih respec o (F 1 ). The σ algebra G in (5.2) is equal o FT 1. In Chaper 3, rading sraegies where allowed o use he whole informaion (F ) ha is available o an agen. Le us explain he reason for his resricion. We could se R as he se of random variables such ha here exiss an inegrand π predicable o F ha is an admissible rading sraegy. Then G would be he σ algebra generaed by R: G = σ(r). Unforunaly, no all σ(r) measurable random variables are replicable. Here is a simple example: le W = (W 1, W 2 ). We consider sochasic inegrals wih respec o W 1. An F predicable inegrand would be f(w 2 ) for a bounded funcion f : R R, hus F = f(w 2 )dw 1 R. Now consider he square of F : F 2 = f(ws 2 )dws 1 f(w 2 )dw 1 + f 2 (W 2 )d. Thus, F 2 is measurable wih respec o σ(f ), bu i is i impossible o represen his random variable wih a sochasic inegral wih respec o W 2. In order o avoid opporuniies of arbirage, he agens wihin he group I assign he same price o a replicable payoff as he financial marke does. A price densiy φ is free of arbirage if and only if he normalized price for a replicable payoff is he same under boh price densiies: E[φ G R] E[φ G ] This is saisfied if and only if = E[φR] E[φ] R R (5.3) E[φ G] E[φ] = φ G E[φ G ] P a.s. (5.4) Hence we define he se C of pricing densiies consisen wih he financial marke as C = { φ φ >, E[φ G] = cφ G for a c > P a.s.}. (5.5) The se C has he same meaning as he se of maringale measures for he sock as explained in Remark 17 in Chaper 3. Our equilibrium price will be

5.1. INCOME, PREFERENCES, THE MARKET 69 a price densiy in C. For a given price densiy φ C an agen can choose a payoff in his budge se, i.e. he se of payoffs ha are under φ no more expensive han his income H a : B a (φ) = {ξ L 1 (Ω, F, Q φ ) ξ, E[φξ] E[φH a ] } (5.6) Every agen acs on a complee marke. He chooses he coningen claim in his budge se ha maximizes his expeced uiliy and solves he maximizaion problem ξ a (φ) = argmax ξ Ba(φ) E[u a (ξ)] (5.7) where u a : [, ] R is sricly growing, sricly concave, coninuously differeniable on (, ) and saisfies he Inada condiions lim x u a(x) =, lim u x a(x) =. (5.8) Addiionally we impose ha here exiss a κ > such ha lim sup xu a(x) = κ <. (5.9) x Observe ha he uiliy funcion is only defined on R +, a negaive wealh is no allowed. In he budge se of he agens, here is replicable and nonreplicable income included. However, on he financial marke, he agens can a mos sell he replicable income R a = ess sup{r R R, R H a }. If an agen wans o buy a replicable payoff ha is more expensive han he replicable par of his income R a, he has o buy i from oher agens wihin he group I and o pay wih some of his nonreplicable income. However, he accumulaed purchases of he agens canno be more expensive han he replicable par of heir income a I R a. We make he asumpion ha he group of agens has enough nonrandom income h such ha hey can afford he payoffs ha are he soluion of he uiliy maximizaion problems (5.7) for he budge ses B a (φ) for a class of price densiies ha conains our equilibrium price densiy. This consan h has o be large enough.this depends in a nonrivial way on he uiliy funcions ha will be specified in Remark 39 below. Assumpion 37 The income H a is posiive, bounded and saisfies P [H a ] > for all a A.

7 CHAPTER 5. GENERAL UTILITIES The marke porfolio H saisfies he following condiion: H = h + H for a consan h > ha is specified in (5.26) wihin Remark 39) and a bounded random variable H. There exis consans < ɛ < s such ha < ɛ H s. Here we describe he soluion of he uiliy maximizaion problem. Define I a : (, ) (, ) as he coninuous, sricly decreasing inverse funcion of u a. Thus, lim I a(y) =, y lim I a (y) =. (5.1) y Applying he Legendre Fenchel ransform, we see ha a random variable X a (φ) is he soluion of he uiliy maximizaion problem (5.7) if and only if i saisfies ξ a (φ) = I a (c φ) (5.11) for a consan c such ha E[φI a (c φ)] = E[φH a ]. This maximizer ξ a (φ) is unique. 5.2 Equilibrium wih parial marke clearing Firs we explain he usual idea of an equilibrium where he agens may only rade among hemselves, i.e. wihou financial marke. This means, he agens redisribue he marke porfolio. This is he sum of all incomes H = a H a. An Arrow Debreu equilibrium is a collecion of nonnegaive payoffs ξa, e a I, ogeher wih a pricing densiy φ e ha saisfy: ξa e = H, ξa e = ξ a (φ e ), a I, a I i. e. ξ e a solves he uiliy maximizaion problem of agen a I wih respec o he pricing densiy φ e. The pricing densiy φ e in a usual Arrow Debreu equilbrium does no need o be in C, because here is no exogenously fixed price, hence no arbirage. On he oher hand, supply and demand mus be equal. We use he same concep of an equilibrium wih parial marke clearing as in Chaper 3. Fix a price densiy φ C. The difference beween uiliy maximizing wealh ξ a (φ) and he endowmen H a of an agen is he sum of a

5.2. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 71 replicable payoff R a (φ) R ha he has raded on he financial marke and and a random payoff C a (φ) ha he has bough from oher agens, i.e. ξ a (φ) H a = R a (φ) + C a (φ). A price densiy φ aains an equilibrium wih parial marke clearing, if he rades among he agens add up o zero: C a (φ ) = P a.s. a I Of course he sum of he replicable payoffs purchased by he agens R(φ ) = a I R a (φ ) (5.12) is in general no equal o zero, bu due he budge consrains yield E[φ G R(φ )] =. We call R(φ ) adjusmen of he marke porfolio wih respec o φ. This leads o our definiion of he equilibrium wih parial marke clearing for one period models. Definiion 38 (Equilibrium wih parial marke clearing) Le (H a ) a I be he bounded nonnegaive F measurable incomes, u a, a I, uiliy funcions according o (5.8), (5.9) and (5.1). An equilibrium wih parial marke clearing consiss of a price densiy φ C, he soluions of he uiliy maximizaion problems (5.7) ξ a (φ ) for he agens a I and a replicable payoff R(φ ) R according o (5.12) saisfying ξ a (φ ) = H a + R(φ ). (5.13) a A a I There are wo differences o he usual Arrow Debreu equilibrium: on one par of he marke we don require ha here is no marke clearing, bu on he oher hand he price densiy on his par of he marke is already fixed. Exisence of an Arrow Debreu equilibrium In order o consruc an Arrow Debreu equilibrium, i is useful o adop he view of a represenaive agen. He akes all he income H = a I H a and redisribues i among he agens. Le H be a nonnegaive random variable. An allocaion ζ = (ζ a ) a I of H consiss of nonnegaive random variables ζ a, a I, saisfying ζ a = H. a I

72 CHAPTER 5. GENERAL UTILITIES The se A( H) of allocaions of H describes all possibiliies o disribue H among he agens in he group I: { A( H) = ζ = (ζ a ) a I, ζ a L (Ω, F, P ), ζ a, } ζ a = H. a According o Lemma 3.57 in (FS2), here exiss a unique allocaion ζ λ ha solves (5.14). Here we skech he ideas and echniques used in (FS2) for he consrucion of an Arrow Debreu equilibrium (see Föllmer/Schied (FS2), proof of Lemma 3.57 on page 149 and Theorem 3.55 on page 148 and 153). We use sricer assumpions han Föllmer and Schied (FS2) for he consrucion of our equilibrium. The incomes (in (FS2) called endowmens) H a have o be in L +(Ω, F, P ). This means hey have o be nonnegaive, F measurable and are considered as equal if hey are P a.s. equal. The marke porfolio H = a I H a saisfies E[H] <. A nonnegaive random variable φ is called pricing densiy if E[φH] <. The uiliy funcions u a : [, ) R have o be coninuosly differeniable on (, ) and o saisfy (5.9). In order o keep our noaion simple, we describe he resul of (FS2) under our addiional assumion (5.8). The goal is o find an allocaion of he marke porfolio ζ A(H) such ha all agens are saisfied, i.e. ζ ogeher wih a pricing densiy φ e is an Arrow Debreu equilibrium. The firs sep is o solve weighed uiliy maximizaion problems. Define Λ = {λ = (λ a ) a I [, 1] I a I λ a = 1}. The number λ a describes he imporance ha he represenaive agen assigns o agen a I. For every λ Λ, he marke porfolio H is redisribued in order o solve he following opimizaion problem: sup ζ A(H) λ a E[u a (ζ a )]. (5.14) a I According o Lemma 3.57 on page 149 in (FS2), here exiss a unique allocaion ζ λ of H ha solves (5.14). This allocaion ζ λ is called λ- efficien. Furhermore, Lemma 3.57 (FS2) saes a firs order condiion, i. e. here exiss a price densiy φ λ such ha This firs order condiion yields λ a u a(ζ λ a ) φ λ, wih equaliy on {ζ λ a > }. (5.15) ζ λ a = I a ( φ λ λ a ).

5.2. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 73 Furhermore, ζ λ a maximizes E[u a (ξ)] over all nonnegaive ξ L (Ω, F, P ) saisfying E[φ λ ξ] E[φ λ ζ λ a ]. The coningen claim ζ λ a is he soluion of an individual uiliy maximizaion problem wih respec o he price densiy φ λ and a budge se defined wih ζ λ a insead of H a. This means, φ λ, ζ λ a, a I is an Arrow Debreu equilibrium if for all a I E[φ λ ζ λ a ] = E[φ λ H a ]. Then all budge condiions are me. Oherwise, he weighs λ a have o be adjused. Observe ha I a ( x λ a ) is increasing in λ a. This means, he weigh λ a of an agen ha obains a oo expensive payoff ζa λ has o decrease, he weigh λ a of an agen wih a oo cheap payoff has o increase. To his end, define g(λ) = (g a (λ)) a I by g a (λ) = λ a + 1 E[V ] E[φλ (H a ζ λ a )], λ Λ, where V = κ(1 + H), and κ according o (5.9). Brouwers fixed poin heorem yields a fixed poin λ e Λ saisfying g(λ e ) = λ e. Thus, he payoffs ζa λe, a I, wih he price densiy φ λe consiue our equilibrium. Equilibrium wih parial marke clearing Our problem o find an equilibrium wih parial marke clearing is closely relaed o he consrucion of an Arrow Debreu equilibrium. However, here is a difference. Our equilibrium price densiy φ has o saisfy φ C. On he oher hand, we can adjus he marke porfolio, i.e. add o H a replicable random payoff R saisfying E[φ G R] =. The price densiy obained in he firs order condiion (5.15) for a λ Λ depends on he payoff H disribued among he agens. Our idea is o add a replicable coningen claim R λ R wih E[φ G R λ ] = o H. Then we solve he weighed uiliy maximizaion problem (5.14) over all allocaions of H + R λ. In order o disinguish beween allocaions of he marke porfolio H and he allocaion of an adjused payoff H + R λ, he soluion is denoed ζ λ. R λ is chosen in Lemma 4 such ha he firs order condiion (5.15) applied o ζ λ yields a pricing densiy φ λ C. According o Lemma 3.57 (c) in (FS2), ζ λ a maximizes E[u a (ξ)] over all ξ L (Ω, F, P ) saisfying E[ φ λ aξ] E[ φ λ a ζ λ a.]

74 CHAPTER 5. GENERAL UTILITIES I remains o find a λ Λ and φ λ, R λ λ such ha he componens ζ a, a I, of he allocaion ζ λ of H + R λ ha solve (5.14) for λ saisfy he budge condiion wih equaliy. This is done using a fixed poin argumen: define he funcion g = (g a (λ)) a A as g a (λ) = λ a + 1 E[κ(1 + H + R λ )] E[ φ λ (H a ζ λ a )]. In Lemma 41 we show ha our funcion g saisfies g(λ) Λ and ha g is coninuous. Then Brouwers fixed poin heorem yields a fixed poin λ of g. Thus, he individual budge consrains are saisfied. Then he price densiy φ, he uiliy maximizing payoffs ζ a, a I and he replicable payoff R λ are an equilibrium wih parial marke clearing. This is saed in Theorem 42. Le us firs explain how a pricing densiy gained by he firs order condiion depends on he payoff ha is disribued among he agens. The funcion f : Λ [, ] defined as f(λ, y) := a A I a ( y λ a ) is for fixed y > bounded from above and away from zero in λ Λ, joinly coninuous in all (λ, y) Λ [, ) and sricly decreasing in y. (5.8) yields lim y f(λ, y) = + and lim y f(λ, y) =. This funcion f is already used in (FS2) o find he soluion of (5.14). Define h : Λ [, ] as he unique soluion of f(λ, h(λ, x)) = x. (5.16) The funcion h(λ, x) is sricly decreasing in x and for fixed x > bounded from above and away from zero in λ. Furhermore h(λ, +) = + and h(λ, ) =. In Föllmer/Schied (FS2) page 153 i is shown ha h is coninuous in λ using he coninuiy of f in (λ, y) and he compacness of [, ]. Their argumen shows in fac ha h is joinly coninuous in (λ, x). A pricing densiy φ λ resuling from he firs order condiion applied on a λ efficien allocaion ζ λ of H saisfies f(λ, φ λ ) = a A ( ) φ λ I a = H. λ a On he oher hand, we can apply h o he coefficiens λ = (λ a ) and he marke porfolio H: h(λ, H) = φ λ λ Λ. (5.17)

5.2. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 75 Using (5.17) we consruc he adjusmen of he marke porfolio R λ R such ha he price densiy φ λ = h(λ, H + R λ ) saisfies for a funcion c : Λ R + \{} E[h(λ, H + R λ ) G] = c(λ)φ G P a.s., (5.18) hence φ λ C, hus his price densiy is free of arbirage. Since he uiliy funcions u a, a I are only defined on [, ), he adjused marke porfolio mus be nonnegaive. Furhermore, R λ has price zero on he financial marke. Thus, R λ has o saisfy H + R λ >, (5.19) E[φ G R λ ] =. (5.2) Consrucion of R λ Here we skech he consrucion of R λ and specify he consan h saed in Assumpion 37. Of course, we summarize he resul in Lemma 4 on page 77 and prove i. In order o find R λ saisfying E[φ G R λ ] =, we aim a applying he inermediae value heorem for coninuous funcions. The firs sep is o find a consan c m > and for every λ Λ a R λ m R saisfying R λ m, H + R λ m ε > (5.21) and E[h(λ, H + R λ m) G] = c m φ G P a.s. Then we show ha here exiss a consan c p > and for every λ Λ a R λ p R saisfying R λ p and E[h(λ, H + R λ p) G] = c p φ G P a.s. Since h(λ, ) is sricly decreasing, we have c p < c m. Nex we show ha for every c [c p, c m ] here exiss R λ c saisfying R λ m R λ c R λ p and E[h(λ, H + R λ c ) G] = cφ G P a.s. The key in our proof is ha c E[φ G R λ c ], c [c p, c m ] is coninuous. Since E[φ G Rm] λ and E[φ G Rp] λ, here exiss a c(λ) and a R λ := Rc(λ) λ saisfying (5.18) and (5.2).

76 CHAPTER 5. GENERAL UTILITIES In order o consruc R λ we define random funcions ha are coninuous in (λ, x) and versions of he condiioned expecaion E[h(λ, h + H + x) G]. Recall ha we assume H = h + H for a consan h and a random variable saisfying ɛ < H < s for consans < ɛ < s (see assumpion 37). Since (Ω, F) is assumed o be a Borel space, we max fix a version of he regular condiioned disribuion P [ H dw G](ω) and define Ψ ω (λ, x) = h(λ, h + w + x)p [ H dw G](ω), x h. (5.22) Since h(λ, ) is decreasing and < ɛ < H, here exiss a se N F, P [N] = such ha he funcionals Ψ ω (λ, x), ω Ω\N are uniformly bounded and coninuous in (λ, x). For ω N, we modify our funcionals o Ψ ω (λ, x) = h(λ, ɛ + h + x). In he nex remark we specify he consan h ha we require in Assumpion 37 as well as he consans c m and c p. Remark 39 In Assumpion 37, we saed ha he sum of he incomes of he agens H saisfies H = h + H for a bounded nonnegaive random variable H and a consan h. This consan mus be large enough o allow he consrucion of Rm λ such ha H + Rm λ > ε for all λ Λ. Recall ha δ and δ 1 are he lower and upper bound of φ G. In order o find Rm, λ we aim a adjusing h and c m > such ha Ψ ω (λ, ) c m δ c m δ 1 Ψ ω (λ, h + ɛ). (5.23) Here we esimae he funcionals Ψ ω (λ, x) simulanuosly for all ω Ω wih a deerminisic funcion of x. Since s H ɛ and h(λ, ) is sricly decreasing, we have for x > h h(λ, h + s + x) Ψ ω (λ, x) h(λ, h + ɛ + x) P a.s. λ Λ. (5.24) Thus, (5.23) is saisfied if we find h and c m such ha h(λ, h + ɛ ) c m δ < c m δ 1 h(λ, s ). The following choice fulfills our requiremen: c m = 1 δ 1 min λ Λ h(λ, s ) >. (5.25) and h = max λ Λ f(λ, c mδ ) ɛ. (5.26)

5.2. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 77 In order o prepare he consrucion of R λ p, we specify c p. Since Ψ ω (λ, ) is sricly decreasing and lim x Ψ ω (λ, ) = for all λ Λ and ω Ω, a consan c p saisfying Ψ ω (λ, ) > c p δ 1 is apropriae. Recall ha for fixed λ Λ, h(λ, ) is he inverse funcion of f(λ, ), see (5.16). Now we have o find c p. Thus, (5.24) yields ha c p = 1 δ 1 max λ Λ h(λ, h + s ) (5.27) is sufficien. Now le y [δ, δ 1 ] and λ Λ. Then here exiss an x m [ h +ɛ, ] saisfying h(λ, h +x m ) = c m y and an x p such ha h(λ, h + x p ) = c p y. Since h(λ, ) is decreasing, we have c p c m, hus he inerval [c p, c m ] is well defined. Lemma 4 Suppose ha he marke porfolio H saisfies assumpion (37) wih he consan h saed in (5.26). For every λ Λ here exiss a G- measurable random variable R λ and a coninuous funcion c : Λ R + saisfying for all λ Λ E[h(λ, H + R λ ) G] = c(λ)φ G P a.s., (5.28) E[φ G R λ ] = (5.29) H + R λ ε > P a.s. R λ (ω) can be choosen for all ω Ω as a coninuous funcion of λ. Furhermore, here exiss a consan b such ha h < R λ b for all ω Ω. Proof. Le Y be a G measurable random variable saisfying Y (ω) Ψ ω (λ, h ), ω Ω. Since Ψ ω (λ, ) is sricly decreasing and Ψ ω (λ, x) is G measurable for all x > h, here exiss a unique G measurable random variable R =: Ψ 1 ω (λ, Y (ω)) saisfying Ψ ω (λ, R) = Y (ω). Inequaliy (5.24) yields ha Rm λ = Ψ 1 ω (λ, c m φ G (ω)) saisfies h R λ m, hus, E[φ G R λ m]. Furhermore, le R λ p = Ψ 1 ω (λ, c p φ G )δ 1.

78 CHAPTER 5. GENERAL UTILITIES Applying (5.23), we see ha R λ p saisfies R λ p max λ Λ f(λ, c pδ ) =: b, λ Λ, P a.s. In order o find R λ ha saisfies (5.28) and (5.29) we define for every c [c p, c m ] a G measurable random variable R λ c ha saisfies Ψ ω (λ, R λ c ) = cφ G P a.s. For every ω Ω, he funcion Ψ ω (λ, x) is joinly coninuous in (λ, x). Wih he same argumen as saed in (FS2) on page 153 for he equaion f(λ, y) = x, using he compacness of [, ], we see ha he soluion x of Ψ ω (λ, x) = cy depends joinly coninuously on (λ, c). Thus for every ω Ω, Rc λ (ω) is coninuous in (λ, c). Furhermore, Rc λ is sricly decreasing in c and bounded: for a b >. Thus h < R λ m < R λ c < R λ p b, ω Ω, (λ, c) E[φ G R λ c ] defines a coninuous funcion ha is sricly decreasing (applying dominaed convergence). For every λ Λ here exiss a c(λ) [c m, c p ] saisfying E[φ G R λ c(λ)] =. The soluion c(λ) and R λ (ω) := Rc(λ) λ (ω) for all ω Ω, depend coninuously on λ. In order o prepare he fixed poin argumen we use in he consrucion of our equilibrium, we prove ha he funcion g ha adjuss he weighs of he agens, is coninuous and he image of Λ is conained in Λ. Lemma 41 The funcion g = (g a (λ)) a A defined as g a (λ) = λ a + is coninuous and g(λ) Λ. 1 E[κ(1 + H + R λ )] E[ φ λ (H a ζ λ a )] (5.3)

5.2. EQUILIBRIUM WITH PARTIAL MARKET CLEARING 79 Proof We may apply he heorem of bounded convergence. For all ω Ω, R λ (ω) is coninuous in λ. Recall ha φ λ = h(λ, h + H + R λ ) wih h R λ b and ɛ H s. Thus φ λ (ω) is for all ω Ω uniformly bounded above and away from zero and coninuous in λ. Hence, ( ) φλ ζ a λ = I a. λ a is also coninuous in λ and bounded. The heorem of Lebesgue yields ha g(λ) is coninuous for all λ Λ. As explained on p. 151 in Föllmer/Schied (FS2), κ defined in (5.9) saisfies u a( ζ λ a ) ζ λ a κ(1 + H + R λ ) L 1 (P ). (5.31) The firs order condiion (5.15) yields g a (λ). The sum a A g a(λ) is equal o 1 because (H a ζ a λ ) = R λ, a A his is a G measurable random variable wih So a A g a (λ) = a A E[ φ λ R λ ] = E[φ G R λ ] =. ( ) 1 λ a + E[κ(1 + H + R λ )] E[ φ λ (H a ζ a λ )] = 1. The following heorem summarizes he exisence of a parial equilibrium. Theorem 42 Le he sum of he endowmens H saisfy Assumpion 37, he pricing densiy saisfy Assumpion 35 and he uiliy funcions be given according o (5.8), (5.9) and (5.1). Then here exiss a price densiy φ C ha is consisen wih he price densiy φ G on G such ha he uiliy maximizing coningen claims (ξ a (φ )) a I wih respec o φ saisfy he parial marke clearing condiion ξ a (φ ) = H a + R(φ ) a I a I for a replicable payoff R(φ ) saisfying E[φ G R(φ )] =. Thus, φ, ξ a (φ ), a I and R(φ ) are an equilibrium wih parial marke clearing according o Definiion 38.

8 CHAPTER 5. GENERAL UTILITIES Proof. The funcion g : Λ Λ defined in (5.3) is coninuous and he se Λ is convex and compac. Brouwer s fixed poin heorem yields a λ Λ saisfying g(λ ) = λ. The price densiy φ = φ λ = h(λ, H + R λ ) wih a replicable payoff R λ R consruced in Lemma 4 is consisen wih he financial marke, because for a c(λ ) > we have E[ φ λ G] = E[h(λ, H + R λ ) G] = c(λ )φ G P a.s. Le ζ λ be he soluion of he weighed uiliy maximizaion problem (5.14) for λ and he adjused marke porfolio H + R λ. Of course, ζ a λ = H a + R λ. a I a I Since λ is a fixed poin of g, ζ λ a saisfies λ E[ φ ζλ a ] = E[ φ λ H a ], a I. ζ λ a Thus, he random payoff is in he budge se B a ( φ λ ) of agen a. According o Lemma 3.57 in (FS2), ξ a (φ ) = ζ λ a, a I solves he individual uiliy maximizaion problem of agen a wih respec o φ = φ λ. Thus, φ = φ λ, ξ a (φ λ ) = ζ a, a I and R(φ ) = R λ are an equilibrium wih parial marke clearing.

Chaper 6 An incomplee marke In his secion we urn o an equilibrium in an incomplee marke. Le P be a probabiliy measure on a Borel space (Ω, F). We assume ha he incompleeness is described in a special way: he agens may rade only random payoffs ha are measurable wih respec o some sub σ algebra T F. Chaper 3.4 in he book of Föllmer and Schied (FS2) consrucs an Arrow Debreu equilibrium in a complee marke: in heir seup, all F measurable payoffs are radeable. Why is i reasonable o consider an incomplee marke? The income of an agen may depend on an observable influence like he emperaure and on oher non observable facors. Then he communiy of agens is willing o ransfer risks only depending on he observable facor. The informaion ha is observable is represened by T. Since he agens fear moral hazard, he agens won ake risks caused by non observable facors. Any payoff ha is no T measurable can be decomposed in a radeable and a non radeable par: he radeable par is simply he essenial supremum of all posiive T measurable random variables ha are smaller han he payoff considered. We prove he exisence of an Arrow Debreu equilibrium on T. This consiss of a T measurable price densiy φ and T measurable random payoffs represening he demand of radeable risk ransfer. The supply is given by he radeable par of he income of he agens. In fac, we ransform he uiliy maximizaion problem in an incomplee marke ino a maximizaion problem on a smaller complee marke wih random uiliy funcions. Then he mehods and argumens saed in (FS2) yields he exisence of he equilibrium in our incomplee marke. Denoe he se of radeable payoffs or coningen claims X (T ) = L +(T, P ). 81

82 CHAPTER 6. AN INCOMPLETE MARKET We have a finie se I of raders each endowed wih a F measurable payoff Ĥ a, a I. Which coningen claims can an agen sell on he marke? Since a negaive erminal wealh is no allowed, he can sell he essenial supremum H a of all radeable coningen claims ha are smaller or equal o his income H a = ess sup{x X Ĥa P a.s., X X (T ) }. So he income Ĥa can be decomposed ino he radable componen H a and a nonradable componen H a : Ĥ a = H a + H a (6.1) We assume ha H a is bounded from above: here exiss a consan s such ha H a s, P a.s. The marke porfolio H is he sum of all radeable pars of he incomes of he agens: H = a I H a. In he model wih he complee marke, all he income of he agens is he marke porfolio. This is he income ha can be redisribued. In order o be consisen wih his seup, we define here also he income as marke porfolio ha can be ransferred. As in (FS2) page 144, we assume and P [H a > ] > for all a I E[H] <. (6.2) Le φ denoe a price densiy on T, i.e. a T measurable sricly posiive inegrable random variable saisfying E[φH] <. The budge se B a (φ) of agen a consiss of all radeable (T measurable) random variables ha are no more expensive han his income H a. Furhermore, an agen can buy more han he marke porfolio. B a (φ) = {ξ X (T ) ξ H, E[φξ] E[φH a ]}. The agen aims a maximizing he expeced uiliy of he sum of he non radable par of his income and he coningen claims in his budge se. He uses a uiliy funcion u a : [, ] R, where u a is coninuously differeniable, sricly growing and sricly concave. So he agens wans o find ξ a (φ) = arg max{e[u a ( H a + ξ)] ξ B a (φ)}. (6.3) We use he following assumpions on he uiliy funcions and incomes:

83 Assumpion 43 1. For every agen a I, he uiliy funcion u a, he non radeable income H a and he marke porfolio H saisfy E[u a ( H a + H)] <. (6.4) 2. The non radeable income H a, he marke porfolio H and he derivaive u a of he uiliy funcion u a saisfy [ ( E u a H a + H )] <. (6.5) I 3. The firs derivaive u a of he uiliy funcion u a saisfies lim sup xu a(x) <. (6.6) x So far, we face an incomplee uiliy maximizaion problem. In order o find an equilibrium, we ransform his incomplee uiliy maximizaion ino a problem in a complee marke on he σ algebra T wih random T measurable preferences. The projeciviy of he condiional expecaion is he key. Since E[u a ( H a + ξ)] = E[ E[u a ( H a + ξ) T ] ] for every ξ X (T ), we inerpre E[u a ( H a + ξ) T ] as random preferences. Since our model is placed in a Borel space, we chose a version of he condiional probabiliy P [ H a T ] and wrie Ψ a (ω, x) := u a ( w + x)p [ H a d w T ](ω). There exiss a se N F wih P [N] = such ha for all ω Ω\N he funcions Ψ a (ω, x) are sricly growing, sricly concave and coninuously differeniable wih derivaive Ψ (ω, x) = u a( w + x)p [ H a d w T ](ω), and Ψ(ω, x) = E[u a ( H a + x) T ]. For ω N, we se Ψ a (ω, x) := u a (x). So, (6.3) is equivalen o finding ξ a (φ) = arg max {E[ Ψ a (ξ) ] ξ B a (φ)}. (6.7) Le us now describe he soluion of (6.7). Ψ a is decreasing and Ψ a(x) u a(x), Ψ a(x) u a(x + s ), x >.

84 CHAPTER 6. AN INCOMPLETE MARKET Denoe as in (FS2) page 135 a(ω) := lim x Ψ a(ω, x), b(ω) := lim x Ψ a(ω, x). Define I a + (ω, ) : (a(ω), b(ω)) (, ) as he coninuous, bijecive, sricly decreasing inverse funcion of Ψ a on (a, b). We max exend I + coninuosly o he full half axis by seing I + a (ω, y) := { for y b, + for y a (6.8) Corollary 3.45 in (FS2) saes ha he unique soluion of he uiliy maximizaion problem ξ a (φ) wih a given price densiy φ is ξ a (φ) = I + a (ω, c a φ) H for a consan c a >. Observe ha I + a (ω, c a φ) is T measurable. Now we are able o give he main heorem of his secion. Theorem 44 Le he risky incomes (H a ) a I given as in (6.1) saisfying assumpion 43. Then here exiss an Arrow Debreu equilibrium, i.e. a price densiy φ and an allocaion (ξ a) of he marke porfolio such ha for every a I, ξ a is he uiliy maximizing coningen claim of agen a for he price densiy φ. Proof. In fac, all argumens needed for his proof are already given in he proof of Theorem 3.59 and Theorem 3.55 in (FS2). The firs sep is he weighed uiliy problem { max λ a E[Ψ a (ω, ξ a (ω))] ξ a X (T ), ξ a = } W a a I For every λ Λ, his problem has a unique soluion ξa λ. This is a consequence of he more absrac Remark 3.39. According o Corollary 3.45 in (FS2), he coningen claim ξa λ maximizes E[Ψ a (ξ)] under all coningen claims ξ saisfying ξ ξa λ, E[φ λ ξ] E[φ λ ξa λ ]. Furhermore, (ξa λ ) saisfy a firs order condiion: here exiss a price densiy φ λ saisfying λ a Ψ a(ξ λ a ) φ λ, wih equaliy on {ξ λ a > }. Now define g a (λ) = λ a + 1 E[κ(1 + W )] E[φλ (W a ξ λ a )].

85 If g a (λ) = λ a for all a I, hen E[φ λ ξa λ ] = E[φ λ H a for all a I, and hus (ξ a ) and φ λ are an Arrow Debreu equilibrium. If his is no he case, an agen ges oo few or oo much. His weigh is increased or decreased by g. As in F ollmer / Schied (FS2), we see ha g is a coninuous mapping from Λ o Λ. So Brouwer s fixed poin heorem yields a λ Λ saisfying g(λ ) = λ. Thus, (ξa λ ), φ λ is an Arrow Debreu equilibrium.

86 CHAPTER 6. AN INCOMPLETE MARKET

Par II Uiliy maximizaion 87

Chaper 7 Robus uiliy maximizaion Inroducion In his chaper we consider he problem of finding he rading sraegy ha maximizes he robus uiliy of a small rader in an incomplee marke. The model is placed in a Brownian filraion. Thus, we have o maximize a funcional as defined in (1.2) over he erminal wealh of all possible rading sraegies. The se Q ha includes all probabiliy measures we consider is an m sable se of probabiliies Q in he sens of Delbaen (Del3), Definiion 1.2. We consider he exponenial and he power uiliy funcions. In he case of an exponenial uiliy funcion we are able o solve a more general problem: he invesor has a erminal liabiliy and ries o hedge i. He has sold an opion and is obliged o pay a random sum a he erminal ime. In general i is impossible o replicae every coningen claim in an incomplee marke. The invesor maximizes he robus uiliy of he erminal wealh gained by a rading sraegy minus he liabiliy. The se of rading sraegies a rader may use is resriced. For example, a negaive number of shares is no possible or he invesmen in risky socks is no allowed o exceed a cerain reshold. Every rading sraegy has o ake is values in a convex se ha can be sochasic and ime dependend. The mehod we use is a generalisaion of he approach in Hu, Imkeller and Müller, (HIM4b). In order o find he opimal rading sraegy, we compare he expeced uiliy of all rading sraegies under all probabiliy measures in Q. In fac, we have o solve a max min problem. The goal is now o find a saddle poin. Since he model is placed in a Brownian framework, we may represen he densiy of every equivalen probabiliy measure as he sochasic exponenial of a sochasic inegral wih respec o he Brownian moion. To his end, we consruc a family of sochasic processes R(p, ν) indexed 89

9 CHAPTER 7. ROBUST UTILITY MAXIMIZATION wih all possible rading sraegies p and inegrands ν in he represenaion of he densiies of he probabiliy measures. The erminal value R T (p, ν) is he produc of wo facors: he firs one is he densiy of an equivalen probabiliy measure wih inegrand ν. The second facor is u(x p T ), where X p T is he erminal wealh of he rading sraegy p and u he uiliy funcion we consider. The iniial value R is he same for all (p, ν). The key is he following propery: here exiss a special rading sraegy p and probabiliy measure indexed wih ν such ha R(p, ν ) is a maringale. This special (p, ν ) will urn ou o be our saddle poin, because we consruc R such ha R(p, ν) is a submaringale and R(p, ν ) is a supermaringale for all admissible rading sraegies p and possible measures changes indexed wih ν. This means, R (p, ν ) is he maximal aainable robus uiliy, p is he rading sraegy aaining i and ν describes he probabiliy measure aaining he minimum in he robus uiliy for he erminal wealh of p. We find he processes R(p, ν) by consrucing a quadraic BSDE. The driver of he BSDE depends on saddle values for finie dimensional saddle poin problems. Wih he unique soluion of his BSDE, one can calculae he opimal rading sraegy p and he inegrand ν for he measure Q. For every (ω, ), p and ν solve finie dimensional saddle poins. In a complee marke we ge a fairly explici descripion of he opimal rading sraegy saed in Theorem 58. This is possible in presence of a erminal liabiliy F and random consrains on he inegral in he presenaion of he densiies in Q as sochasic exponenials of inegrals wih respec o he Brownian moion. This is a case where he leas favorable measure does no necessarily exiss. This chaper is organized as follows: Secion 7.1 explains he financial marke and he robus uiliy, in secion 7.2 we consider he exponenial uiliy funcion, and in secion 7.3 he power uiliy. 7.1 Sock marke and robus uiliy A probabiliy space (Ω, F, P ) carrying an m dimensional Brownian moion (W ) [,T ] is given. The filraion F is he compleion of he filraion generaed by W. The financial marke consiss of one bond wih ineres rae zero and d m socks. In case d < m we face an incomplee marke. The price process of sock i evolves according o he equaion ds i S i = b i d + σ i dw, i = 1,..., d, (7.1)

7.1. STOCK MARKET AND ROBUST UTILITY 91 where b i (resp. σ i ) is an R valued (resp. R 1 m valued) predicable uniformly bounded sochasic process. The lines of he d m marix σ are given by he vecor σ i, i = 1,..., d. The volailiy marix σ = (σ i ) i=1,...,d has full rank and we assume ha σσ r is uniformly ellipic, i.e. KI d σσ r εi d, P a.s. for consans K > ε >. The predicable R m valued process θ = σ r (σ σ r ) 1 b, [, T ], (7.2) is hen also uniformly bounded. We will see laer ha only θ eners he soluion of he opimizaion problem. For simpliciy we will call θ drif. There are several possibiliies o define a rading sraegy. One can wrie down he number of shares of each sock held by he invesor, he amoun of money invesed or he par of he wealh. We will choose he noaion ha fis well o our maximizaion problem: for he exponenial uiliy in secion 7.2 we use he amoun of money (see Definiion 47) and for he power uiliy in secion 7.3, we consider rading sraegies ha are wrien as par of he wealh (see Definiion 59). The definiion of a wealh process depends also on he choice of he descripion of a rading sraegy. Robus uiliy The preferences of our invesor on replicable coningen claims are described by robus uiliy as explained on page 7. In conras o he usual expeced uiliy wih respec o a single real world probabiliy, he invesor considers a whole se Q of probabiliy measures. In order o calculae he robus uiliy for a coningen claim, he invesor chooses he measure wihin Q ha minimizes he expeced uiliy of his random variable. The robus uiliy can be seen as a wors case approach. I akes ino accoun, ha he invesor is averse agains risk caused by he random sock price and he uncerainy since he doesn have an objecive probabiliy measure. More informaion abou his opic can be found in he book of Föllmer / Schied (FS2) in Chaper 2.5. So he invesor has o solve a max min problem. Under our assumpions, his leads o a saddle poin problem. The robus uiliy of a payoff F is inf E Q[u(F )] Q Q where u is a convex increasing funcion called uiliy funcion. The densiies of he probabiliy measures in Q wih respec o P are sochasic exponenials of sochasic inegrands wih respec o he Brownian moion. The inegrands are resriced o ime dependend random predicable closed convex subses C (ω) of R m, [, T ], ω Ω. Predicable means here he se {((ω, )), v) v C (ω)} Ω [, T ] R m is P B(R m ) measurable. Here we summarize our assumpions on he se Q:

92 CHAPTER 7. ROBUST UTILITY MAXIMIZATION Assumpion 45 Le he closed convex subses C (ω), [, T ], ω Ω be predicable and conained in a bounded ball around he origin. We assume ha he Radon Nikodym densiy of every Q Q can be wrien as dq dp = E ( ν s dw s ) for a predicable, R m valued process ν where ν C. We denoe he se of inegrands in he represenaion (7.3) wih T (7.3) V = {(ν ) [,T ] predicable ν C P λ a.s.} (7.4) So he se of probabiliy measures Q is paramerized by he se of inegrands V. For he expecaion of a random variable F wih respec o Q ν we also wrie E Q ν[f ] =: E ν [F ]. Our se Q is closely relaed o ses of muliplicaively sable ses of probabiliy measures as defined in Delbaen, (Del3). Remark 46 Le S be a se of probabiliy measures such ha for a reference probabiliy measure Q r S for every Q S he densiy dq saisfies dq dq r dq r L 1 (Q r ). Then he densiy process Z Q = E r [ dq F dq r ] is well defined, where E r denoes he expecaion wih respec o Q r. Furhermore, le he se of hose densiies be closed in L 1 (Q r ). Le Z denoe he densiy process of a Q Q r and Z he densiy process of a Q Q r. For every sopping ime τ T define { Z L =, τ, Zτ Z Z τ, > τ. Assume also ha every nonnegaive F measurable random variable Z saisfying E r [Z ] = 1 defines by dq = Z dq r a probabiliy Q ha is in S. Then S is m sable if every L defined as above is he densiy process of a probabiliy measure Q L S. Denoe wih S e he subse of S ha consiss of measures equivalen o he reference measure Q r. If S is be m sable, Q r S given by exponenials of inegrands wih respec o a coninuous maringale M and Q r S, hen Theorem 1.4 in (Del3) saes ha here exiss a predicable, closed, convex mulivalued mapping (C ) such ha S e is equal o he se of processes Z = E( qdm) where Z is a sricly posiive maringale and q(, ω) C (ω). Our seup is he Brownian filraion generaed by he Brownian moion W. The iniial σ algebra F is rivial. The densiy of any probabiliy measure

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 93 equivalen o P can be wrien as he sochasic exponenial of a sochasic inegral wih respec o W. In order o describe he preferences of our invesor, we use an m sable se of probabiliy measures ha saisfies an addiional assumpion. 7.2 Robus exponenial uiliy maximizaion Suppose an invesor has a liabiliy F a ime T. This random variable F is assumed o be F T measurable and bounded, bu no necessarily posiive. The invesor ries o find a rading sraegy such ha he erminal wealh of he rading sraegy minus he liabiliy F maximizes he robus uiliy. In his secion we consider he uiliy funcion U(x) = exp( αx), x R for a parameer α > ha is called he absolue risk aversion. Here we formally describe rading sraegies as we use for he robus uiliy maximizaion problem wih he exponenial uiliy funcion. A d dimensional F predicable process π = (π ) T is called rading sraegy if π ds is well defined, e.g. π S σ 2 d < P a.s. For 1 i d, he process π i describes he amoun of money invesed in sock i a ime. The number of shares is πi. The wealh process X π of a rading sraegy π wih S i iniial capial x saisfies he equaion X x,π = x + d i=1 π i,u S i,u ds i,u = x + π u σ u (dw u + θ u du), [, T ]. In his noaion π has o be aken as a vecor in R 1 d. Trading sraegies are self financing. Gains or losses are only obained by rading wih he sock. The condiions on he rading sraegies of he following definiion guaranee ha here is no arbirage. In addiion, we allow consrains on he rading sraegies. Formally, hey are supposed o ake heir values in a closed convex se à R1 d, i.e. π (ω) à for λ P a.e. (ω, ) Ω [, T ]. I is also possible o consider random predicable, closed, convex consrains, see Remark 57 for more deails. For echnical reasons we impose some furher inegrabiliy condiions on our rading sraegies. Definiion 47 (Admissible Sraegies wih consrains) Le à be a closed se in R 1 d. The se of admissible rading sraegies à consiss of all d dimensional predicable processes π = (π ) T which saisfy π à for λ P a.e. (ω, ) Ω [, T ], π sσ s dw s is a P BMO maringale, and E(U (XT π )) >.

94 CHAPTER 7. ROBUST UTILITY MAXIMIZATION The definiion and main resuls abou BMO maringales are saed in he appendix. We use BMO maringales because sochasic exponenials of hem are uniformly inegrable maringales. Our ime inerval is resriced. According o (A.2) on page 141, every uniformly bounded rading sraegy π is admissible, bu his is no a necessary condiion. The boundedness of θ and Theorem 3.6 in (Kaz94) imply ha he wealh process X π is a BMO maringale under he equivalen probabiliy measure Q wih Radon-Nikodym densiy dq = E( θdw ). Therefore he se à is dp free of arbirage, i.e. in his se here is no rading sraegy π wih iniial capial X π =, erminal wealh XT π P a.s. and P [Xπ T > ] >. For [, T ], ω Ω define he se A (ω) R m by A (ω) = Ãσ (ω). (7.5) The enries of he marix valued process σ are uniformly bounded. Therefore we ge for λ P a.e. (, ω) and some consan k 1. Remark 48 Wriing min{ a : a A (ω) } k 1. (7.6) p = π σ, [, T ], he se of admissible rading sraegies à is equivalen o a se A of R1 m valued predicable sochasic processes p wih p A if p (ω) A (ω) P λ a.e. and p sdw s is a P BMO maringale. Such a process p A will also be named sraegy, and X x,p denoes is wealh process. Wih his definiion of a rading sraegy we define our maximizaion problem: Problem 49 (Robus uiliy maximizaion) Le F be a bounded F T - measurable random variable. A soluion of he robus uiliy maximizaion problem consiss of an admissible rading sraegy p A and a probabiliy measure Q ν Q (resp. ν V) aaining V (x, F ) := sup inf E ν [ exp( α(x x,p T F ))]. (7.7) ν V p A Under he reference measure P he expecaion in (7.7) reads K(p, ν) = E [ ( exp α(x x,p T F ) + ν s dw s 1 )] ν s 2 ds. (7.8) 2

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 95 So problem 49 consiss in finding a p A and ν V aaining V (x, F ) = sup inf K(p, ν). (7.9) p A ν V Before saing and proving he main heorem we skech he ideas leading o he soluion of he robus uiliy maximizaion problem. We aim o show ha he funcional K has a saddle poin ( p, ν). This saddle poin saisfies K(p, ν) K( p, ν) K( p, ν) p A, ν V, hence ( p, ν) is a soluion of problem 49. We can exploi he exponenial srucure of he maximizaion problem and apply a generalisaion of he maringale argumen developed in Hu, Imkeller, Müller (HIM4b). In order o solve our opimizaion problem 49, we consruc a family of processes R(p, ν) = (R (p, ν)) [,T ] indexed wih he admissible rading sraegies p A and all possible inegrands for he change of measure ν V. Togeher wih R we have o find a rading sraegy p A and a ν V such ha R saisfies for all p A and ν V: R T (p, ν) = exp ( α(x x,p T F ) + ν sdw s 1 2 R (p, ν) = R does no depend on p and ν, R(p, ν) is a P supermaringale for all p A, R( p, ν) is a P submaringale for all ν V, R( p, ν) is a P maringale. ν s 2 ds Wih his family of processes R(p, ν), p A, ν V, we obain E[R T (p, ν)] R ( p, ν) = E[R T ( p, ν)] E[R T ( p, ν)] p A, ν V. Thus ( p, ν) is a saddle poin of K and he soluion of problem 49. We se ( R (p, ν) = exp αx + αy + ν s dw s 1 ) ν s 2 ds 2 where Y is defined by a BSDE wih erminal value F : Y = F Z s dw s ), f(s, θ s, Z s )ds, [, T ]. (7.1) Now we have o consruc he driver f of he BSDE such ha R saisfies he properies described above. We can find f by solving deerminisic saddle

96 CHAPTER 7. ROBUST UTILITY MAXIMIZATION poin problems. The idea is o wrie R(p, ν) as he produc of a maringale and an increasing or decreasing process depending on p and ν: ( ) R (p, ν) = exp( αx + αy )E ( αp s + αz s + ν s )dw s ( ) exp g(p s, ν s, θ s, Z s ) + αf(s, θ s, Z s ) where g : R m R m R m R m R is equal o g(q, v, z, θ ) = 1 2 ( αq + αz + v)2 qθ 1 2 v2 (7.11) ( ( = α2 q z + 1 2 α v + 1 2 )) α θ (7.12) 1 2 (v + θ ) 2 αzθ = αv(z q) + 1 2 α2 q 2 q(α 2 z + θ ) + 1 2 α2 z 2. We wrie he funcion g wih variables q, v, z R m in order o disinguish clearly beween he saddle poin analysis in R m R m and on he ses of processes A V. Here, q akes he place of p and v replaces ν, z sands for Z. In order o obain he desired properies for R(p, ν), we have o choose f such ha here exiss p A and ν V saisfying for all [, T ] P a.s. g(p, ν, Z, θ ) + αf(, θ, Z ) p A g( p, ν, Z, θ ) + αf(, θ, Z ) ν A g( p, ν, Z, θ ) + αf(, θ, Z ) = Then R is he produc of a negaive maringale wih an in / decreasing / consan process, hence a super / sub / maringale. The processes p resp. ν are consrain o be in convex ses A (ω) resp. C (ω) for almos every (ω, ). The firs sep is o prove for fixed (z, θ ) R m R m he exisence of a saddle poin of he funcion g i.e. ( q, v) A C saisfying g( q, v, z, θ ) g( q, v, z, θ ) g(q, v, z, θ ) q A, v C (7.13) (see Lemma 52 below). The value of g on he saddle poin is ḡ(, z, θ ) = g( q, v, z, θ ) = inf sup g(q, r, z, θ ) q A v C ( ( α 2 = sup v C 2 dis2 A z + 1 ) α (v + θ ) 12 ) (v + θ ) 2 αzθ. (7.14)

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 97 Then we choose f(, Z, θ ) = 1 αḡ(, Z, θ ). We aim a applying Theorem 2.3 of Kobylanski (Kob) o prove he exisence of a soluion of he BSDE (7.1) wih his choice of f. In order o do so, we have o show an esimae of he saddle value (Lemma 53 below). Using a soluion (Y, Z) of he BSDE we can find he processes ( p, ν) ha solves he original saddle poin problem: a measurable selecion heorem (Lemma 1 in (Ben7)) yields wo predicable processes p and ν such ha ( p, ν) is a saddle poin of g(q, v, Z, θ ), where q A and v C. Recall he consrucion of R, ( p, ν) is also a saddle poin of he funcional K and he soluion of he uiliy maximizaion problem 49. Theorem 4.3 in El Karoui, Hamadène (EKH3) relaes he saddle poin of a risk-sensiive zero-sum game o a BSDE. Their conrol problem has he following form: sup v V inf u U Eu,v [ { 1 }] exp h(s, x, u s, v s )ds where he expecaion is aken under a probabiliy measure P u,v wih densiy ( P u,v ) dp = E σ 1 (s, x )f(s, x, u s, v s )db s, he conrols u and v are predicable processes on some meric spaces and x saisfies he following SDE: dx = f(, x, u, v )d + σ(, x )db u,v, x = x R d. Assumpion (A4.3) in (EKH3) saes ha h has o be bounded. Our conrol problem does no saisfy his assumpion. Having compleed his overview we sae he main heorem ha gives he soluion of he robus uiliy maximizaion problem in erms of a BSDE. Theorem 5 There exiss a soluion ( p, ν) of he robus uiliy maximizaion problem 49. This soluion is a saddle poin of he funcional K defined in (7.8). For every (ω, ) Ω [, T ], ( p, ν) is a saddle poin of g(,, Z, θ ), i.e. i saisfies g( p, r, Z, θ ) g( p, ν, Z, θ ) g(q, ν, Z, θ ) q A, r C. (7.15) The pair (Y, Z) is he soluion of he BSDE (7.1) and g is defined in (7.11). The value funcion is V (x, F ) = exp( α(x Y ). (7.16)

98 CHAPTER 7. ROBUST UTILITY MAXIMIZATION Applying (7.16), we may calculae he uiliy indifference price x F of F. This is he exra iniial capial ha he invesor needs in order o ge he same maximal uiliy in presence he liabiliy F han wihou F. The invesor is indifferen beween geing x F and acceping he obligaion o pay F and on he oher hand doing nohing. Remark 51 (Uiliy indifference price) Le x > be he iniial capial of an agen. The uiliy indifference price x F of F is defined as he soluion of he equaion V (x + x F, F ) = V (x, ). Le (Y F, Z F ) denoe he soluion of he BSDE (7.1) wih erminal value F, whereas (Y, Z ) is he soluion of (7.1) wih erminal value. According o (7.16), x F = Y F Y. In order o prove Theorem 5, we use some Lemmaa. The proof of he heorem is summarized on page 13. The firs sep is o show he exisence of a saddle poin of he funcion g in R m R m in Lemma 52. An esimae of he value of g a he saddle poin is saed in Lemma 53. This leads o he exisence of a soluion of he BSDE 7.1 proven in Lemma 54. The selecion of an admissible rading sraegy p A and of an inegrand ν V solving he robus uiliy maximizaion problem is proven in Lemma 56 and Lemma 55. We sar proving he differen Lemmaa leading o Theorem 5 while showing he exisence of a saddle poin of he funcion g. Lemma 52 Le (A ) and (C ) be he consrains according o (7.5) and Assumpion 45. For every z R m and (ω, ) Ω [, T ], here exiss a saddle poin ( q, v) of he funcion g defined in (7.11). Proof. Fix z R m. We aim o apply Theorem 37.3 in Rockafellar (Roc7). The funcion g is convex in q and linear in v, hence convex concave. We have o show: he convex funcions g(, v), v ric, have no common direcion of recession neiher does he funcions g(q, ) for q ria. ri denoes he relaive inerior of a convex se (see Secion 6 in (Roc7)). Here we have o describe he definiion of a direcion of recession. If a convex se A is no consrained, here exiss q A and y such ha q + λy A for all λ >. Due o he convexiy of A we have q + λy A for all q A and λ >. Such a y is a direcion of recession, if for every q A lim inf λ g(q + λy, r) < +,

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 99 (see Theorem 8.6 in Rockafellar (Roc7)). If he se of consrains is bounded, hen here exiss no direcion of recession. Since g is quadraic in q (coercive), he funcions g(, v) have no direcion of recession. The funcions g(q, ) are linear, bu hey are only defined on he bounded se C. Hence hey also have no direcion of recession. Theorem 37.3 in Rockafellar (Roc7) saes he exisence of a saddle poin ( q, v) A C wih g( q, v, z, θ ) <. The nex lemma gives an esimae of he saddle value. Lemma 53 Le ( q, v)(, z, θ ) be he saddle poin of g depending on (z, θ ) and on ime depending consrains A, C ha exiss according o Lemma 52 for g defined in (7.11). The saddle value funcion ḡ defined in (7.14) saisfies for a consan c > and ḡ(, z, θ ) c( z 2 + θ 2 + 1) (7.17) ḡ(, z, θ ) 1 2 dis2 (θ, C ) α z θ, z R m. (7.18) Furhermore, he norm of q can be esimaed wih a consan c 1 by ( q sup dis A (z + 1 ) v C α (v + θ )) + z + v + θ c 1 (1 + z + θ ). (7.19) Proof. For every v C we have ( ( α 2 inf q z + 1 q A 2 α v + 1 )) 2 α θ α2 2 where q aains min q A q. Applying (7.6) gives inf q A g(q, v, z, θ ) c( q 2 + v 2 + θ 2 + 1) ( ( q z + 1 α v + 1 2 )) α θ, v C for a c >. The uniform boundedness of he ses C, (ω, ) Ω [, T ], yields (7.17). Equaion (7.18) follows from g(q, v, z, θ ) 1 2 (v + θ ) 2 αzθ q A, v C. In order o show (7.19), we see wih (7.14) ha q saisfies q (z + 1 α ( v + θ )) = dis A (z + 1 α ( v + θ )). Since he ses A saisfy (7.6) and he ses C are uniformly bounded, we obain (7.19).

1 CHAPTER 7. ROBUST UTILITY MAXIMIZATION In order o define he BSDE, we have o show ha our driver f(, θ, z) = 1 αḡ(, θ, z), [, T ], defines a predicable process. This is done in Lemma 56 below. The BSDE ha we use o consruc he opimal sraegy has he following form: Y = F Z s dw s + 1 α ḡ(s, Z s, θ s )ds. (7.2) Now we are able o sae he exisence of a soluion of (7.2). Lemma 54 For a bounded F T measurable erminal value F, he BSDE (7.2) has a unique soluion (Y, Z) H (R) H 2 (R m ). Proof. According o Lemma 56, (ḡ(, θ, )) [,T ] is predicable. In order o ge exisence of a soluion, we apply Theorem 2.3 of Kobylanski (Kob). Due o he boundednesss of θ, (7.6) and (7.17), condiion (H1) in Kobylanski is saisfied, i.e. ḡ(, θ s, z) c z 2 + b for consans b, c >. Now we prove uniqueness of he soluion. Suppose ha (Y 1, Z 1 ) and (Y 2, Z 2 ) are soluions of he BSDE (7.2). They saisfy Y 1 Y 2 = (Z 1 s Z 2 s )dw s (ḡ(s, θ s, Z 1 s ) ḡ(s, θ s, Z 2 s )ds. In he firs sep we esimae he difference in he ds inegral. In order o disinguish beween sochasic processes and elemens of R m, we wrie as on page 96 q for p, v for ν and z for Z. a According o (7.11), we can wrie ḡ(, θ, z) = sup v C ( α 2 2 dis2 A ( z + 1 ) α (v + θ ) 12 ) (v + θ ) 2 αzθ. Fix z 1, z 2 R m. There exiss v 1, v 2 R m ha aain he sup for z 1 resp. z 2. Thus ( ḡ(, θ, z 1 ) ḡ(, θ, z 2 ) ḡ(, v 1, z 1 ) α2 2 dis2 A z 2 + 1 ) α (v 1 + θ ) + + α2 4 (v 1 + θ ) 2 + αz 2 θ

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 11 = α2 2 dis2 A α2 2 dis2 A ( z 1 + 1 ) α (v 1 + θ ) (7.21) ( z 2 + 1 ) α (v 1 + θ ) + αθ (z 1 z 2 ). Using he uniform boundedness of C and he Lipschiz coninuiy of he disance funcion from a closed se we obain wih a consan c > ḡ(, θ, z 1 ) ḡ(, θ, z 2 ) c(1 + z 1 + z 2 + θ ) z 1 z 2. The same inequaliy is valid if we change z 1 and z 2, hus we have an esimae for he absolue value of his difference. Se β = { ḡ(,θ,z 1 ) ḡ(,θ,z2 ), if Z Z 1 Z2 1 Z 2, if Z 1 Z 2 =. Since θ is uniformly bounded, we obain for a consan c 1 > β c 1 (1 + Z 1 + Z 2 ), [, T ]. As shown in Lemma 55, he maringales Zi sdw s, i = 1, 2, are P BMO maringales. So β sdw s is also a P BMO maringale. Thus we may define an equivalen probabiliy measure Q by ( ) dq dp = E β s dw s and a Q Brownian moion W Q = W + β sds. Thus we can wrie he difference of he soluions in he following way: Y 1 Y 2 = Since F = YT 1 = Y T 2 Zs 1 = Zs 2 in H 2. = (Z 1 s Z 2 s )dw s (Z 1 s Z 2 s )dw Q s. β s (Z 1 s Z 2 s )ds 1 we conclude ha Y = Y 2 P λ a.e. on Ω [, T ] and Lemma 55 Le (Y, Z) H (R) H 2 (R m ) be a soluion of he BSDE (7.2). Then he process Z sdw s is a P BMO maringale.

12 CHAPTER 7. ROBUST UTILITY MAXIMIZATION Proof. According o Corollary 2.2 of Kobylanski (Kob), he process Y is uniformly bounded. Le k denoe he upper bound. For every sopping ime τ T Iôs formula applied o he process (Y k) yields [ ] E Z s 2 ds F τ = E[(F k) 2 F τ ] (Y τ k) 2 τ + 2 [ ] α E (Y s k)ḡ(s, Z s, θ s )ds F τ. Using he boundedness of Y and F, he non posiiviy of (Y k), (7.18) and he fac ha θ is uniformly bounded, we obain for consans c 1, c 2, c 3 [ ] [ ] E Z s 2 ds F τ c 1 + c 2 E Z s θ s ds F τ τ τ c 3 + 1 [ ] 2 E Z s 2 ds F τ. The second inequaliy is a consequence of a b 1 2 a2 + 2 b 2 for all a, b R, and he fac ha θ is uniformly bounded. Thus [ ] E Z s 2 ds F τ c τ for all sopping imes τ T and a consan c ha does no depend on τ. In he nex Lemma we prove he exisence of predicable processes ( p, ν) aaining he saddle value ḡ(, Z, θ ) applying a measurable selecion heorem proved in Beneš (Ben7). Lemma 56 Le ḡ(, θ, z) be he saddle value as defined in (7.13) where θ is given in (7.2) and z R m. For every z R m, he process (ḡ(, θ, z)) is predicable. More exacly: ḡ is P B(R d ) B(R) measurable. Moreover, le Z be a predicable process, e.g. he inegrand par of he soluion of (7.2). There exis wo predicable processes p and ν saisfying τ τ g( p, ν, Z, θ ) = ḡ(, Z, θ ) P λ a.s., i.e. for amos every (ω, ) ( p, ν ) is a saddle poin of (7.13).

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 13 Proof. Define 1 for q A (ω), v C (ω), I((ω, ), q, r) = for q A (ω), v / C (ω), + for q / A (ω). Since he graphs of (A ) and (C ) are P B(R d ) measurable, he funcion I is P B(R d ) B( R) measurable, where R d = R d { } {+ }. Furhermore, ḡ(, θ, z) = min q R d max r R d I((ω, ), q, r)g(, q, r, θ, z). So ḡ is P B(R d ) B( R) measurable. Since ḡ is finie, we have also P B(R d ) B(R) measurabiliy. Now we urn o prove exisence of he predicable processes ( p, ν) ha are saddle poins for every (ω, ). In order o do so, we apply a measurable selecion heorem saed in Benes, (Ben7) Lemma 1. I is useful o inerpre g in he following way: (q, r, (ω, )) g(q, r, Z, θ ) is coninuos in (q, r) for every fixed (ω, ) and predicable for every fixed (q, r). The saddle value ḡ(, Z, θ ) is also predicable and finie. Thus, Lemma 1 in (Ben7) yields he resul. Using all hese Lemmaa we prove Theorem 5. Proof of Theorem 5. The exisence of a saddle poin of he funcion g(q, v, z, θ ) is shown in Lemma 52. Lemma 53 saes an esimae of he value of g a he saddle poin in erms of z and θ uniformly for all A, C, (ω, ) Ω [, T ]). This esimae is used o proof exisence of a soluion of he BSDE (7.1) in Lemma 54. Lemma 56 saes ha here exis predicable processes p and ν such ha for all (ω, ) Ω [, T ], ( p, ν ) is a saddle poin of g(q, v, Z, θ ) wih consrains q A and v C. Lemma 55 ogeher wih Lemma 53 and he fac ha θ is bounded yield ha p dw and ν dw are P -BMO maringales, hence p A and ν V. The consrucion of R(p, ν) shows ha ( p, ν) is indeed a saddle poin of he funcional K defined in (7.8). Since K is concave in p and convex in ν, he saddle value of K is unique. Thus ( p, ν) is a soluion of he robus uiliy maximizaion problem 49. However, K is linear in ν for fixed p. So he saddle poin ( p, ν) may no be unique.

14 CHAPTER 7. ROBUST UTILITY MAXIMIZATION The consrucion of he opimal rading sraegy is sill possible under slighly more general condiions. Remark 57 The consrains on rading sraegies can be formulaed more general han in Definiion 47: he consrains on he rading sraegies π may be random and ime dependen. We formulae our assumpion for he noion of rading sraegies according o Remark 48. The se {(q, (ω, )) q A } is assumed o be P B measurable. The ses A, (ω, ) Ω [, T ] are closed, convex and saisfy (7.6). Then Theorem 5 remains valid. We don need he assumpion ha he ses (A ) are generaed by a process of marices applied o a deerminisic se à resp. C. The complee marke In he siuaion of a complee marke we ge a more explici resul. On a complee marke, every random payoff is hedgeable by a rading sraegy. This is he case if he number of socks d is equal o he number of dimensions m of he Brownian moion. The assumpions on he volailiy marix saed below (7.1) yields hen ha σ is regular for all [, T ]. Furhermore, here are no resricions on he values π ha a rading sraegy akes a a paricular ime [, T ]. This means, Definiion 47 describing he se A of admissible rading sraegies remains valid wih à = Rm. Of course, he rading sraegy p in he noaion of Remark 48 also may ake values in R m, hence A = R m for all [, T ]. We find he saddle poin of he funcion g explicily and obain a simple BSDE ha reveals he srucure of he opimal rading sraegy. I is even possible o deermine ν. We have o calculae min max g(q, v, z, θ ). q R m v C There exiss a saddle poin, so we can change min and max. For a fixed v C, he minimum is aained for q(v) = z + 1 α (v + θ ), and we have Hence g(q(v), v, z, θ ) = 1 2 (θ + v) 2 αzθ. v = Π C ( θ ),

7.2. ROBUST EXPONENTIAL UTILITY MAXIMIZATION 15 and ḡ(, z, θ ) = 1 2 dis2 C ( θ ) αzθ. (7.22) In his case he BSDE simplifies o Y = F Z s dw s + 1 2α dis2 C ( θ s ) + Z s θ s ds. In order o solve he BSDE we change he probabiliy measure o Q via d Q ( ) dp = E θ s dw s and obain he Q Brownian Moion W = W + θ sds. The BSDE reads Y = F Z s d W s + T 1 2α dis2 C ( θ s )ds. Then Y is given by successive condiioned expecaions under Q: [ ] Y = Ẽ 1 F + 2α dis2 C ( θ s )ds F. Z is he inegrand saisfying [ Y + Z s d W s = Ẽ F + Now we sae he resul in a heorem. ] 1 2α dis2 C ( θ s )ds F, [, T ]. (7.23) Theorem 58 Le he sock price process S be given according o (7.1), where he number of socks d is equal o he number of dimensions of he Brownian moion m. Le he marke be complee, i.e. he consrains in Definiion 47 of admissible rading saegies given by A = R m for all (ω, ) Ω [, T ]. Furhermore le he se of probabiliy measures Q in he robus preferences be given by Assumpion 45. Then he soluion of he robus uiliy maximizaion problem is given in he following way: he opimal sraegy p is p = Z + 1 α θ + 1 α Π C ( θ ), where Z is he inegrand in (7.23). The inegrand ν for he probabiliy measure Q = Q ν aaining he minimum in (7.7) is ν = Π C ( θ ).

16 CHAPTER 7. ROBUST UTILITY MAXIMIZATION The projecion Π C (x) of an x R m on a closed convex se C is defined in (1.3) on page 12. Le us ake a closer look a he sucure of he opimal rading sraegy in a complee marke. The opimal sraegy consiss of wo pars: 1. he unique hedging sraegy Z for F + prices wih drif θ, and 1 2α dis2 C s ( θ s )ds under sock 2. he opimal sraegy for he maximizaion of he expeced uiliy wih respec o Q ν for F =. Observe ha he drif of he sock price process under Q ν is equal o θ + Π C ( θ ). If 1 2α dis2 C s ( θ s )ds is deerminisic, hen his erm does no affec he hedging sraegy. Schied (Sch4b) finds a so called leas favorable measure under he assumpion ha F =, (C ) is deerminisic and he marke is complee. This is he probabiliy in Q under which he Euclidian norm of he drif of he sock price process is minimized. This probabiliy does no depend on he uiliy funcion. Observe ha he marke price of risk for he sock under he measure Q ν is equal o θ + ν. Of course, under he same assumpions our probabiliy Q ν is he same as in (Sch4b), Proposiion 3.2. The exponenial uiliy funcion allows us o find he same srucure of Q ν even if (C ) is no deerminisic and F. Q ν neiher depends on he erminal liabiliy F nor on he risk aversion α in he exponenial uiliy funcion. 7.3 Power Uiliy Our goal is he characerizaion of he opimal rading sraegy for he robus uiliy maximizaion problem. In conras o he previous secion, we use anoher ype of uiliy funcions. They are called power uiliy and have he following form: U γ (x) = 1 γ xγ, x, γ (, 1). The se of probabiliy measures in our robus uiliy maximizaion problem is he same as defined in Assumpion 45. This ime, he addiional liabiliy is equal o zero, i.e. F =. In his secion, we use a noion of rading sraegy ha is beer adaped o he problem: ρ = ( ρ i ) i=1,...,d denoes he par of he wealh invesed in sock i. The number of shares of sock i is given by ρ i X S i. A d dimensional F predicable process ρ = ( ρ ) T is called rading

7.3. POWER UTILITY 17 sraegy (par of wealh) if he following wealh process is well defined: X ( ρ) = x + d i=1 X s ( ρ) ρ i,s ds i,s = x + S i,s X ( ρ) s ρ s σ s (dw s + θ s ds), (7.24) and he iniial capial x is posiive. The wealh process X ( ρ) can be wrien as: ( ) X ( ρ) = xe ρ s σ s (dw s + θ s ds), [, T ]. The rading sraegies are consrained o ake values in a closed convex se Ā 2 R d. Observe ha he par of wealh invesed in he socks is consrained. This is a difference o he consrains on he number of shares considered in secion 7.2. As before, i is more convenien o inroduce ρ = ρ σ, [, T ]. Accordingly, ρ is consrain o ake is values in A (ω) = Ã2σ (ω) [, T ], ω Ω. The ses A are closed convex subses of R m and saisfy (7.6). Before saing he maximizaion problem, we define he se of admissible rading sraegies. Definiion 59 The se of admissible rading sraegies à consiss of all m- dimensional predicable processes ρ = (ρ ) T saisfying ρ A (ω) P λ a.s and ρ sσ s 2 ds < P a.s. Now we are able o sae he robus uiliy maximizaion problem. Problem 6 Le < γ < 1. The soluion of he robus uiliy maximizaion problem wih uiliy funcion U(x) = x γ consiss of a rading sraegy ρ A and a probabiliy measure Q ν Q wih a ν V aaining V (x) = sup inf ρ A ν V Eν [(X (ρ) T )γ ]. Recall ha Q and V are defined in Assumpion 45 and in (7.4). Our aim is o prove exisence of a saddle poin ( ρ, ν). Of course, his saddle poin is hen he soluion of problem 6. We aim o use he same echniques as in secion 7.2. Define for ρ A, ν V K(ρ, ν) = E ν [(X (ρ) [ ( = E x γ exp γ T )γ ] (ρ s dw s + ρ s θ s 1 2 ρ2 sds) exp( ν s dw s 1 )] 2 ν2 s ds).

18 CHAPTER 7. ROBUST UTILITY MAXIMIZATION We have o o prove ha K has a saddle poin. This means, we have o find a ρ A and a ν V saisfying K(ρ, ν) K( ρ, ν) K( ρ, ν) ρ A, ν V. In order o find he saddle poin, we consruc a family of sochasic processes (R (ρ, ν)) [,T ] indexed wih A V wih he following properies: he iniial value R (ρ, ν) does no depend on ρ and ν and he erminal value R T (ρ, ν) is equal o dqν dp (X(ρ) T )γ, he random variable in he expecaion in K(ρ, η). Furhermore, here exiss ρ A and a ν V such ha R(ρ, ν) is a supermaringale, R( ρ, ν) is a maringale, and R( ρ, ν) is a submaringale. Thus, for all ρ A and ν V E[R T (ρ, ν)] E[R T ( ρ, ν)] = R ( ρ, ν)) E[R T ( ρ, ν)]. Since K(ρ, ν) = E[R T (ρ, ν)], ( ρ, ν) is he soluion of problem 6. We can exploi he exponenial srucure of he problem and se R (ρ, ν) = x γ exp( exp(y + γρ s dw s + γρ s θ s γ 2 ρ2 sds) exp( Z s dw s + f(s, θ s, Z s )ds). The processes (Y, Z) are he soluion of a BSDE in he form Y = Z s dw s f(s, θ s, Z s )ds ν s dw s 1 2 ν2 s ds) where f : [, T ] R m R m has o be consruced such ha R(ρ, ν) has he properies menioned above. Observe ha he erminal value Y T is equal o zero and f does no depend on Y. So he iniial and erminal condiion on R will be saisfied as soon as we find f. Now define he funcion g : R m R m R m R m R as g(r, v, θ, z) = 1 2 (γr + v + z)2 + γrθ γ 2 r2 1 2 v2 (7.25) ( γ(1 γ) = r + 1 ) 2 2 γ 1 (v + θ + z) + ( γ + v + θ + z ) 2 1 2(1 γ) γ 2γ z2 θ z, where r R m akes he place of ρ, v R m he place of ν and z R m sands for Z. In order o keep reading simple, we don replace θ. Wih his

7.3. POWER UTILITY 19 noaion we have ( ) R (ρ, ν) = x γ exp(y )E (γρ s + ν s + Z s )dw s ( ) exp g(ρ s, ν s, θ s, Z s ) + f(s, θ s, Z s )ds. For every θ, z and (ω, ), here exiss a saddle poin ( r, ῡ) of g wih he following consrains: r A, v C, i.e. r A and v C saisfying g(r, v, θ, z) g( r, v, θ, z) := ḡ(, θ, z) g( r, v, θ, z) r A, v C. Observe ha he funcions g(, v, θ, z), v C, and g(r,, θ, z), r A, have no direcions of recession, since g is negaive quadraic in r and v is consrain in he bounded se C. This is even more han assumed in Theorem 37.3 in Rockafellar (Roc7) ha saes ha he saddle poin ( r, v) exiss and ha he saddle value ḡ(, θ, z) is finie. Since he consrains depend on (ω, ), he saddle poin and he saddle value also do. The saddle poin ( r, v) may no be unique since g is linear in v. We have o choose f(, θ, Z ) = ḡ(, θ, Z ). In order o find a soluion of he BSDE we firs have o esimae he saddle value ḡ(, θ, z) from above and below. Le us sar wih he esimae from above. We have ḡ(, θ, z) sup r A g(r, v, θ, z) v C. This supremum is aein for ( r(, v, θ, z) = Π A 1 ) γ 1 (v + θ + z). (7.26) Due o (7.6) and since he ses (C ) are uniformly bounded, we obain for some consan k > ḡ(, θ, z) k(1 + θ 2 + z 2 ) z R m (7.27) and for he Euklidian norm of he firs par of he saddle poin r depending on (, θ, z) r(, θ, z) 2 k(1 + θ 2 + z 2 ). (7.28) For he esimae from below we use he uniform boundedness of he ses C, [, T ], and ge an r A wih r k 1 uniformly for all (ω, ). This yields for a consan k 2 ḡ(, θ, z) g(r, v, θ, z) k 2 z R m. (7.29)

11 CHAPTER 7. ROBUST UTILITY MAXIMIZATION Now we are able o sae he BSDE ha leads o he soluion of he robus uiliy maximizaion problem: Y = Z s dw s + ḡ(s, θ s, Z s )ds (7.3) Lemma 61 The BSDE (7.3) has a unique soluion (Y, Z) H (R) H 2 (R). Furhermore, Z sdw s is a P BMO maringale. Proof. As in he proof of Lemma 56, we see ha ḡ(, θ, ) is predicable. Since θ is uniformly bounded, esimaes (7.27) and (7.29) yield ha ḡ(, θ, z) saisfies condiion H1 in (Kob). So he BSDE (7.3) has a soluion (Y, Z). The key o prove uniqueness of he soluion for he BSDE (7.2) is esimae (7.21). Since here exiss a unique saddle poin of g, we may exchange sup and inf and obain ḡ(, θ, z) = inf v C + { γ 2(1 γ) ) 1 γ (v + θ + z) } ( γ(1 γ) 1 dis 2 A 2 ( v + θ + 1 ) 2 γ z 1 2γ z2 θ z. (7.31) Using he same calculaion as in he proof on Lemma 54 we obain for he dependence on he saddle poin ḡ(, θ, z) on z he following esimae ḡ(, θ, z 1 ) ḡ(, θ, z 2 ) c(1 + θ + z 1 + z 2 ) z 1 z 2. The uniqueness of he soluion (Y, Z) in H (R) H 2 (R m ) follows also as in he proof of Lemma 54. Furhermore, since Y is uniformly bounded and ḡ saisfies (7.29), he inegral Z sdw s is a P BMO maringale. (see Lemma 55). Now everyhing is prepared for he resul of he robus uiliy maximizaion problem wih a power uiliy funcion. Theorem 62 Le (Y, Z) be he soluion of he BSDE (7.3). There exiss a soluion ( ρ, ν) of he robus uiliy maximizaion problem 6. For almos every (ω, ) Ω [, T ] his soluion saisfies g(r, ν, θ, Z ) g( ρ, ν, θ, Z ) = ḡ(, θ, Z ) g( ρ, v, θ, Z ) r A, v C. (7.32) The maximal uiliy V (x) is equal o V (x) = x γ (Y ) γ, x >.

7.3. POWER UTILITY 111 Proof. Le (Y, Z) be he soluion of he BSDE (7.3). Lemma 1 in Beneš yields exisence of wo predicable processes ρ and ν saisfying (7.32) (see he proof of Lemma 56). Since Z sdw s is a P BMO maringale, (7.28) and he boundedness of he ses (C ) yield ha ρ sdw s and ν sdw s are also P BMO maringales. So, ρ A and ν V. Due o he consrucion of R, we see ha ( ρ, ν) (resp. ρ and he probabiliy measure Q ν ) is indeed he soluion of he robus uiliy maximizaion problem 6. Furhermore, V (x) = R. The Complee marke Le us now consider he siuaion of a complee marke, i.e. he number of socks d is equal o he dimension of he Brownian moion m and here are no consrain on he rading sraegies: Ã 2 = R m. Due o he assumpions on σ below (7.1), he marices σ are inverible. Hence A = R m. Since here exiss a saddle poin of g(r, v, θ, z) for every z R m and every (ω, ) wih consrains A, C, we may change sup and inf. So ḡ(, θ, z) = inf sup g(r, v, θ, z) v C r R m = 1 ( γ 2 1 γ dis2 C θ z ) γ The BSDE reads for all [, T ] Y = 1 γ Z s dw s 2 1 γ dis2 C [ 1 2γ Z2 s + θ s Z s 1 2γ z2 θ z. ( θ s 1 ) γ Z s ds (7.33) ] ds. (7.34) Of course, if (C ) is deerminisic, we have he leas favorable measure in he saddle poin. Remark 63 Le he ses C, [, T ] be deerminisic. Then he probabiliy measure Q ν appearing in he saddle poin is he leas favorable measure saed in Schied, (Sch4b) Proposiion 3.2 The second componen of he soluion (Y, Z) of (7.33) saisfies Z =. The inegrand for he measure Q ν is ν = Π C ( θ ),

112 CHAPTER 7. ROBUST UTILITY MAXIMIZATION and he opimal rading sraegy is ρ = 1 γ 1 (Π C ( θ ) + θ ). Observe ha he measure Q ν does no depend on γ. For a maximizaion of he expeced uiliy under Q ν, he drif is equal o θ + ν. So he agen maximizes he uiliy under he probabiliy measure ha minimizes he drif of he sock.

Chaper 8 Uncerain sock price dynamics Inroducion In his chaper we solve he robus uiliy maximizaion problem where he drif (b ) and he volailiy (σ ) of he sock prices are no exacly known. In conras o Chaper 7, he invesor maximizes under a fixed probabiliy measure P. The uncerainy lies in he coefficiens of he sock price process. We assume ha θ = σ r (σ σ r ) 1 b is conained in a closed bounded convex se for all [, T ], bu we don know he exac value. This knowledge of he sock price process migh be he resul of saisics where he coefficiens are esimaed o be in a cerain confidence inerval. In his seup, he robus uiliy of a rading sraegy is defined in he following way: he invesor compares he expeced uiliy of he erminal wealh for every possible θ. Similar o he robus maximizaion wih a se of probabiliy measures, he invesor akes he wors θ ha is possible. He ries o find he rading sraegy ha maximizes his robus uiliy wih uncerain drif. We consider he exponenial, power and logarihmic uiliy funcions. If he invesor uses he exponenial uiliy funcion, he may hedge a liabiliy he has o pay ou a he end of he rading inerval ha is described by an F T measurable bounded random variable F. The uiliy indifference price for F wih respec o he robus uiliy maximizaion under uncerain drif is saed in (8.11). 8.1 Exponenial uiliy In his secion we consider he exponenial uiliy funcion. Le us firs describe he parameers of he sock price process. Since he coefficiens b and σ ener he maximizaion problem only via θ = b σ, we specify only θ 113

114 CHAPTER 8. UNCERTAIN STOCK PRICE DYNAMICS and call his process drif. For every (ω, ) Ω [, T ], θ mus be conained in he se C (ω) defined in as follows: Assumpion 64 Le (C (ω)) denoe a predicable mulivariae mapping of convex ses in he following sense: for every (ω, ) Ω [, T ], C (ω) R m is closed and convex and he graph of (C ) i.e. he se {((ω, )), v) v C (ω)} Ω [, T ] R m is P B(R m ) measurable. Furhermore, he ses C (ω) are conained in a bounded ball around he origin for all (ω, ) Ω [, T ]. So we define Θ = {θ R m valued, predicable, θ C [, T ] P a.s. }. (8.1) In his secion, we use he se of admissible rading sraegies as defined in Definiion 47 and Remark 48 for he exponenial uiliy funcion. The opimizaion problem is he following saddle poin problem: Problem 65 Le F be an F T measurable bounded random variable. A soluion of he robus uiliy maximizaion problem consiss of a rading sraegy p A and a θ Θ aaining V (x, F ) = sup p A inf E θ Θ [ exp ( ( ))] α x + p (θ d + dw ) F. (8.2) We use he same maringale argumen as in secion 7.2 o find he saddle poin in problem 65. The process θ akes now he role of ν. Se ( ) R (p, θ) = exp( αx + αy )E ( αp s + αz s )dw s (8.3) ( ( ) ) 1 exp 2 ( αp s + αz s ) 2 αp s θ s + αf(s, Z s ) ds, where f sill has o be deermined. The pair of processes (Y, Z) is he soluion of a BSDE: Y = F Z s dw s f(s, Z s )ds. (8.4) Now we prove he seps ha lead o he soluion of problem 65. By choosing f we consruc a family of sochasic processes indexed wih A Θ. For every (p, θ), he erminal value R T (p, θ) is equal o he random variable in F ))], where X θ,p is he wealh process for he rading sraegy p A and drif θ Θ. The goal is o find a driver f for he BSDE such ha here exiss a ( p, θ) A Θ saisfying he following condiions: R( p, θ) is a maringale, R(p, θ) is he expecaion in (8.2), i.e. E[R T (p, θ)] = E[ exp(α(x θ,p T

8.1. EXPONENTIAL UTILITY 115 a supermaringale for every p A and R( p, θ) is a submaringale for every θ Θ. Then ( p, θ) is our saddle poin and he soluion of he robus uiliy maximizaion problem 65 because E[R(p, θ)] R = E[R( p, θ)] E[R( p, θ)] p A, θ Θ. In order o find he appropriae funcion f we solve a deerminisic finie dimensional saddle poin problem. Our noaion disinguishes beween sochasic processes and elemens of R m. We wrie q R m for p, u R m for θ and z R m for Z. Define g : R m R m R m R as g(q, u, z) = αuq + α2 2 (q2 2qz + z 2 ) ( q = α2 2 ( z + 1 α u )) 2 αuz 1 2 u2 The funcion f : Ω [, T ] R m is chosen such ha here exiss for every (ω,, z) Ω [, T ] R m a ( q, ū) depending on (ω,, z) saisfying g(q, ū, z) + αf(, z) q A, z R m, (8.5) g( q, ū, z) + αf(, z) = z R m, g( q, u, z) + αf(, z) u C, z R m. So we ake f(, z) = 1 αḡ(, z), [, T ], z Rn, where ḡ(, z) is he saddle value of g saisfying g( q, u, z) g( q, ū, z) = g( q, ū, z) g(q, ū, z), q A, u C. Observe ha he dependence of f on (ω, ) is a resul of he fac ha he consrains A and C depend on (ω, ). There exiss a saddle poin ( q, ū) of g ha depends on (ω, ) and z. The saddle value ḡ(, z) can be esimaed as in Lemma 53. To his end observe ha ( α 2 ḡ(, z) = sup v C 2 dis2 A (z + 1 α u)) αuz 1 ) 2 u2. (8.6) Since all ses C, [, T ], are conained in a bounded ball around he origin and he ses A saisfy (7.6), we obain for consans c 1, c 2 uniformly for (ω, ) Ω [, T ] and all z R m αc 1 z ḡ(, z) c 2 (1 + z 2 ). (8.7)

116 CHAPTER 8. UNCERTAIN STOCK PRICE DYNAMICS We obain an esimae for he norm of he saddle poin. Since he ses C are conained in a ball around he origin, here exiss a consan c > such ha ū c P a.s. For he firs componen of he saddle poin we have q c(1 + z ), u R m. (8.8) Wih his saddle value funcion, we have he driver of he BSDE (8.4) f(, z) = 1 αḡ(, z), [, T ], z Rm. (8.9) Analoguosly o Lemma 56, (f(, )) [,T ] is a predicable process. Theorem 2.3 of Kobylanski (Kob) saes ha a soluion (Y, Z) of he BSDE (8.4) wih driver (8.9) exiss. As in Lemma 54, one can show ha (Y, Z) is he unique soluion of (8.4) in H H 2. Furhermore, he boundedness of F and (8.6) yields ha Z sdw s is a P BMO maringale. This leads o he soluion of he robus uiliy maximizaion problem wih unknown drif. Theorem 66 A soluion ( p, θ) of Problem 65 exiss. Le (Y, Z) be he soluion of he BSDE (8.4) wih driver (8.9). The opimal rading sraegy p and he drif θ saisfy for P λ a. e. (ω, ), g(q, θ, Z ) g( p, θ, Z ) g( p, u, Z ) q A, u C. (8.1) The value of he robus uiliy maximizaion problem 65 is V (x, F ) = exp( αx + αy ). Proof. Firs, here exis predicable processes p, θ saisfying (8.1). This is a consequence of Lemma 1 in Beneš (see Lemma 56). Furhermore, p A: his rading sraegy saisfies p A P λ a.e. The fac ha Z sdw s is a P BMO maringale and (8.8) implies ha p sdw s is also a P BMO maringale. The opimaliy of ( p, θ) follows from (8.5) and (8.3), see he consrucion of he processes R(p, θ). So ( p, θ) is he soluion of problem (65). Of course, we can calculae he uiliy indifference price of F for he robus uiliy maximizaion wih uncerain drif. This is he addiional iniial capial x F ha he invesor needs o ge he same maximal uiliy wih he liabiliy o pay ou F han wihou his liabiliy, see Remark 51. Le (Y F, Z F ) denoe

8.1. EXPONENTIAL UTILITY 117 he soluion of (8.4) wih erminal value F and (Y, Z ) he soluion of (8.4) wih erminal value. The uiliy indifference price is x F = Y F Y. (8.11) The complee marke Le he marke be complee wihou rading consrains, i.e. we have an m dimensional Brownian moion, d = m socks and à = Rm in Definiion 47 of admissible rading sraegies. So we have also A = R m. Then ḡ(, z) = max ( 12 (u + u C αz)2 + 12 ) z2. Using his expression, one can easily find he saddle poin ū = Π C ( αz) and q = z + 1 α Π C( αz). The BSDE has he following form: Y = F Z s dw s ( dis 2 C( αz s ) + 1 2 Z2 s )ds. (8.12) In conras o (7.23), we canno see he soluion of he BSDE (8.12) so easily. However, in he siuaion of Schied, (Sch4b) Proposiion 3.2, we obain he leas favorable measure ha is defined here. Remark 67 Le F = and he ses C deerminisic. Then we can wrie down he soluion of he BSDE and he soluion of he robus uiliy maximizing problem 65 explicily. Since dis 2 C () is also deerminisic, (Y, Z) wih Y = dis 2 C ()ds, Z = is a soluion of he BSDE. The opimal rading sraegy p and he marke price of risk θ aaining he saddle poin have a paricularly simple srucure: θ = Π C (), p = 1 α Π C (). Thus, he Euclidian norm of he drif is minimized. The drif θ is he same as he drif saed in Schied (Sch4b), Proposiion 3.2.

118 CHAPTER 8. UNCERTAIN STOCK PRICE DYNAMICS 8.2 Power uiliy In his secion we solve he robus uiliy maximizaion problem for he power uiliy funcion U(x) = x γ in a seup where he drif and volailiy of he sock are uncerain. We use se à of admissible rading sraegies described in Definiion 59 ogeher wih he se Θ of possible drif processes defined in (8.1). The drif θ saisfies θ C where (C ) is given according o (64). The agen solves he wors case scenario wrien in he following opimizaion problem: Problem 68 The soluion of he robus uiliy maximizaion problem wih uncerain drif for a uiliy funcion U(x) = x γ wih γ (, 1) consiss of an admissible rading sraegy ρ à and a drif θ Θ aaining V (x) = sup inf E ρ A θ Θ [ ( ( x γ exp γ ρ s dw s + γ θ s ρ s 1 ) )] 2 ρ2 s ds. (8.13) Wihin he expecaion is wrien he uiliy of he erminal wealh for a rading sraegy ρ and a sock price process wih drif θ. We aim a finding a saddle poin for his opimizaion problem. Our mehod o solve i is o consruc a family of sochasic processes R(ρ, θ) such ha R T (ρ, θ) is equal o he random variable in he expecaion in (8.13), he iniial value R does no depend on (ρ, θ), here exiss ρ and θ such ha R( ρ, θ) is a maringale, R( ρ, θ) is a submaringale and R(ρ, θ) is a supermaringale for all ρ A and θ Θ. Similarly o (8.3) for he exponenial uiliy funcion, we se ( ) R (ρ, θ) = x γ exp(y )E (γρ s + Z s )dw s ( exp T 1 2 (γρ s + Z s ) 2 ds + γρ s θ s 1 2 γρ2 s + f(s, Z s )ds As in he case of he robus uiliy maximizaion problem for he exponenial uiliy funcion, (Y, Z) is he soluion of he BSDE ). Y = Z s dw s f(s, Z s )ds. (8.14)

8.2. POWER UTILITY 119 We have o choose he driver f such ha R has he desired properies. Define g(r, u, z) = 1 2 (γr + z)2 + γru 1 2 γr2, r, u, z R m = 1 2 γ(γ 1)r2 + γr(u + z) + 1 2 z2 ( γ(1 γ) = r 1 ) 2 γ (u + z) 2 1 γ 2(1 γ) (u + z)2 + 1 2 z2. Here we replaced ρ by r R m, Z by z R m and θ by u R m. Theorem 37.3 in Rockafellar and he boundedness of he consrains (C ) yields a saddle poin, i.e. ( r, ū) depending on z and A, C saisfying g(r, u, z) g( r, ū, z) g( r, u, z) r A, u C, z R m. One can also esimae ḡ: and for consans k 1, k 2 >. We choose ḡ(, z) k(1 + z 2 ) ḡ(, z) k 2 f(, Z ) = ḡ(, Z ). Since his driver f saisfies condiion (H1) in (Kob), he BSDE has a soluion (Y, Z). This soluion is unique in H (R) H 2 (R m ).The process Z sdw s is a P BMO maringale. So we have he soluion of he uiliy maximizaion problem: Theorem 69 Le (Y, Z) be he soluion of he BSDE wih driver ḡ. Then here exiss a couple of predicable processes ( ρ, θ) saisfying ρ A, θ Θ and g(r, θ, Z ) g( ρ, θ, Z ) g( ρ, u, Z ) r A, u C. So ( ρ, θ) is he soluion of he robus uiliy maximizaion problem 6. The maximal robus uiliy is equal o V (x) = (x γ ) exp(y ), x >. The complee marke

12 CHAPTER 8. UNCERTAIN STOCK PRICE DYNAMICS Again, in he case of a complee marke, he BSDE simplifies. Since A = R m, for u C, z R m r = 1 (u + z) 1 γ aains sup r R m g(r, u, z). Thus γ ḡ(, z) = inf u C 2(1 γ) (u + z)2 + 1 2 z2. This infimum is aained by ū = Π C ( z). So r = 1 (Π 1 γ C ( z) + z). The corresponding BSDE is Y = Z s dw s + γ 2(1 γ) (Π C ( Z s ) + Z s ) 2 1 2 Z2 s ds. If he consrains on he drif (C ) are deerminisic, we obain he same drif as Schied, (Sch4b) Proposiion 3.2. Remark 7 If he consrains (C ) are deerminisic, hen Z =, and Y = γ 2(1 γ) (Π C () is he soluion of he BSDE (8.14). The soluion of he robus uiliy maximizaion problem 68 is θ = Π C (), ρ = 1 1 γ Π C (). So he wors case drif is he same as in Remark 67 for he exponenial uiliy funcion. 8.3 Logarihmic uiliy To complee he specrum of imporan uiliy funcions, we consider he uiliy funcion U(x) = log x. We solve he robus uiliy maximizaion problem for an uncerain drif of he sock price process. As in he preceding secion, he invesor has no erminal liabiliy. The noion of rading sraegies is as in Chaper 7.3 for he power uiliy, ρ i, i = 1,..., d, [, T ], describes he amoun of money invesed in sock i a ime. In order o simplify he calculaions, we wrie ρ = ρ σ, where σ is he volailiy marix of he sock price process defined in 7.1. For every (ω, ) Ω [, T ], he rading sraegy ρ akes values in a closed, convex se A (ω) ha is defined in (7.5). The se of admissible rading sraegies A l is given as in Definiion 59 wih an addiional mild inegrabiliy condiion:

8.3. LOGARITHMIC UTILITY 121 Definiion 71 The se of admissible rading sraegies A l consiss of all R d valued predicable processes ρ saisfying E[ ρ s 2 ds] < and ρ A P λ a.s. The se of drif processes considered in he robus uiliy maximizaion Θ is defined in (8.1). Recall from (7.24) ha he wealh process X (ρ,θ) for a rading sraegy ρ A l and drif θ Θ for he sock price process is X (ρ,θ) T So he log of he wealh process is ( [ = x exp ρ s dw s + ρ s θ s 1 ] ) 2 ρ2 s ds. [ log X (ρ,θ) T = x + ρ s dw s + ρ s θ s 1 ] 2 ρ2 s ds. The invesor ries o solve he robus uiliy maximizing problem wih a no exacly known drif: Problem 72 The soluion of he robus uiliy maximizaion problem wih he logarihmic uiliy funcion is given by a rading sraegy ρ A l and a drif θ Θ aaining V (x) = sup inf ρ Ã = sup inf E θ Θ ρ Ã Observe ha we can wrie log X (ρ,θ) T = x + E[log θ Θ Xρ,θ ] [ x + ρ s dw s + ρ s dw s + [ ρ s θ s 1 ] ] 2 ρ2 s ds. [ 1 2 (ρ s θ s ) 2 + 1 ] 2 θ2 s ds. The expeced uiliy of he erminal wealh of a rading sraegy ρ A l and a sock price wih drif θ Θ is [ E ( 1 2 (ρ s θ s ) 2 + 1 ) ] 2 θ2 s ds. Disinguishing sochasic processes and elemens of R m, we wrie r for ρ and u for θ. So we define g(r, u) = ru 1 2 r2, r A, u C.

122 CHAPTER 8. UNCERTAIN STOCK PRICE DYNAMICS In order o solve Problem 72, we have o find for every (ω, ) Ω [, T ] a saddle poin of he funcion g. This saddle poins ( r, ū)(ω, ) exiss. According o Lemma 56, here exis predicable processes ( ρ, θ) ha are for each (ω, ) equal o ( r, ū). Since he ses (C ) are conained in a bounded ball around he origin, he process ρ is also uniformly bounded and admissible. The following heorem summarizes his resul. Theorem 73 There exis wo predicable processes ρ and θ, ρ A l and θ Θ, such ha for every (ω, ) Ω [, T ], ( ρ, θ) is a saddle poin of he funcion g wih consrains A, C. This couple ( ρ, θ) of processes solves he saddle poin problem 72. Now le he marke be complee, i.e. he number of socks is equal o he dimension of he Bromnian moion and we have no rading consrain. This means, he consrains on he rading sraegies saisfy A = R m. Then we see ha 1 ḡ() = min u C 2 u2, and he soluion ( ρ, θ) of Problem 72 is θ = Π C (), ρ = Π C (). This minimal drif corresponds o he resul saed in Schied, (Sch4b), Proposiion 3.2. In conras o he robus uiliy maximizaion wih uncerain drif for he exponenial and he power uiliy we don need he assumpion of (Sch4b), Proposiion 3.2 ha he consrains (C ) are deerminisic. Here we summarize he resuls of he robus uiliy maximizing problems wih uncerain drif for he differen uiliy funcions ha we have considered. Remark 74 Le he consrains for he drif (C ) be deerminisic. Then we have for he uiliy funcions U(x) = exp( αx), α >, for U(x) = x γ, γ (, 1) and for U(x) = log x ha he drif in he soluion of he robus uiliy maximizaion problem is θ = Π C (), [, T ]. So we find ha he drif ha is used in he robus uiliy maximizaion for our uiliy funcions is he same as he drif saed in Proposiion 3.2 in Schied (Sch4b).

Chaper 9 Uiliy maximizaion wih nonconvex consrains This chaper is a summary of he resuls proven in Hu, Imkeller and Müller, (HIM4b). We consider he uiliy maximizaion problem wih respec o one probabiliy measure. An invesor ries o find a rading sraegy ha maximizes he expeced uiliy of his wealh a he end of a finie ime inerval [, T ]. In conras o he previous chaper, he considers only a single probabiliy measure ha he sees as objecive or real world measure. He maximizes a concave funcional, because he is risk averse. However, he does no ake uncerainy ino accoun since he parameers of he sock price process as well as he probabiliy measure he uses are assumed o be known. We consider hree ypes of uiliy funcions: he exponenial, he power and he logarihmic uiliy. In he case of an exponenial uiliy, he invesor may have a erminal liabiliy: his is he obligaion o pay ou a random amoun of money described by a bounded random variable F. Then he expeced uiliy of he erminal wealh of he rading sraegy minus he liabiliy has o be maximized. The model is placed in a Brownian filraion. So we may work wih Backward Sochasic Differenial Equaions (BSDE). The maximal expeced uiliy and an opimal rading sraegy are obained by he soluion of a BSDE. The presence of only one probabiliy measure simplifies our analysis. In order o consruc our BSDE, we have o solve maximizaion problems insead of a saddle poin problems. The assumpions on he consrains of he rading sraegies can be relaxed, for every (ω, ) Ω [, T ] he rading sraegy is resriced o be in a closed se. The assumpion ha he se is convex can be dropped. The mehod ha we apply in order o obain value funcion and opimal sraegy is simple. We propose o consruc a sochasic process R ρ depending 123

124 CHAPTER 9. UTILITY MAXIMIZATION on he invesor s rading sraegy ρ, and such ha is erminal value equals he uiliy of he rader s erminal wealh. As menioned above, o model he consrain, rading sraegies are supposed o ake heir values in a closed se. We don assume ha his se is convex. In our marke, he absence of compleeness is no explicily described by a se of maringale measures equivalen o he hisorical probabiliy. Insead, we choose R ρ such ha ha for every rading sraegy ρ, R ρ is a supermaringale. Moreover, here exiss a leas one paricular rading sraegy ρ such ha R ρ is a maringale. Hereby, he iniial value is supposed no o depend on he sraegy. Evidenly, he sraegy ρ relaed o he maringale has o be he opimal one. Then he value funcion of he opimizaion problem is jus given by he iniial value of R ρ. We obain he paricular conrol process ρ by he soluion of a BSDE. Our direc approach does no use any dualiy, bu characerizes direcly he soluion of our opimizaion problem. As in Chaper 7, we consider hree ypes of uiliy funcions: he exponenial, he power and he logarihmic uiliy. This chaper is organized as follows: in secion 9.1 we solve our uiliy maximzaion problem for an exponenial uiliy in presence of a erminal liabiliy. We give in (9.9) he uiliy indifference price of F. In secion 9.2 we consider he power uiliy funcions and in secion 9.3 he logarihmic uiliy. In boh cases, he invesor has no erminal liabiliy. 9.1 The exponenial uiliy The exponenial uiliy maximizaion allows he presence of a erminal liabiliy ha he invesor wans o hedge. This liabiliy is represened by a bounded F T measurable random variable F : F L (P, F T ). The sock price process is as in (3.2) wih a drif θ described in (7.2). In his subsecion, a rading sraegy is a d dimensional predicable process π ha describes he amoun of he currency invesed in he socks. The wealh process for a rading sraegy π is X π = x + d i=1 π i,u S i,u ds i,u = x + π u σ u (dw u + θ u du), [, T ]. The definiion of admissibiliy formalises he consrains and guaranees ha here is no arbirage. Formally, hey are supposed o ake heir values in a closed se, i.e. π (ω) Ã, wih à R1 d. We emphasize ha à is no assumed o be convex. We also impose a differen inegrabiliy condiion.

9.1. THE EXPONENTIAL UTILITY 125 Definiion 75 (Admissible Sraegies wih consrains) Le à be a closed se in R 1 d. The se of admissible rading sraegies à consiss of all d dimensional predicable processes π = (π ) T which saisfy E[ π σ 2 d] < and π à λ P a.s., as well as {exp( αx π τ ) : τ sopping ime wih values in [, T ]} is a uniformly inegrable family. The invesor faces he following opimizaion problem: sup E [ exp( α(xt π F ))]. (9.1) π à This means, he ries o find he rading sraegy ha maximizes he sum of he erminal wealh of he rading sraegy and he liabiliy. Remark 76 We shall show below ha he sup is aken by a paricular sraegy π which is admissible in he sense of our definiion. Noe ha his process migh no lead o a wealh process which is bounded from below, and herefore no admissible in his sense. For furher deails see Schachermayer (Sch4a) and Meron (Mer71). Remark 77 The condiion of square inegrabiliy in Definiion 75 guaranees ha here is no arbirage. In fac, he square inegrabiliy condiion on π and he boundedness of θ yields ha E[sup T (X π ) 2 ] <. According o Theorem 2.1 in Pardoux, Peng (PP9), (X, π σ ) is he unique soluion of he BSDE X = X T (π s σ s )dw s (π s σ s )θ s ds, wih E[ (Xπ s ) 2 ds] <, E[ (π sσ s ) 2 ds] <. So he iniial capial X π needed o aain XT π is uniquely deermined. In paricular, Theorem 2.2 in El Karoui, Peng, Quenez (EKPQ97) yields if X π = and XT π P a.s., hen XT π = P a.s. Remark 78 In accordance wih he classical lieraure (see Dellacherie, Meyer (DM75)) he uniform inegrabiliy condiion in Definiion 1 coincides wih he noion of class D. Remark 79 If X π is square inegrable and π à λ P a.s., as well as X π is bounded from below on [, T ], i is obvious ha π Ã.

126 CHAPTER 9. UTILITY MAXIMIZATION For [, T ], ω Ω define he se A (ω) R m by A (ω) = Ãσ (ω). (9.2) The enries of he marix valued process σ are uniformly bounded. Therefore we ge min{ a : a A (ω) } k 1 for λ P a.e. (, ω) (9.3) wih a consan k 1. Furhermore, for every (ω, ), he se A (ω) is closed. This is crucial for our analysis. Remark 8 Wriing p = π σ, [, T ], he se of admissible rading sraegies à is equivalen o a se A of R1 m valued predicable sochasic processes p wih p A iff E[ p() 2 d] < and p (ω) A (ω) P a.s., as well as {exp( αx p τ ) : τ sopping ime wih values in [, T ]} is a uniformly inegrable family. Such a process p A will also be named sraegy, and X (p) denoes is wealh process. The maximizaion problem (9.1) is evidenly equivalen o [ ( ( exp α x + p (dw + θ d) F V (x) = sup E p A ))]. (9.4) In order o find he value funcion and an opimal sraegy we consruc a family of sochasic processes R (p) wih he following properies: R (p) T = exp( α(xp T F )) for all p A, R (p) = R is consan for all p A, R (p) is a supermaringale for all p A and here exiss a p A such ha R (p ) is a maringale. The process R (p) and is iniial value R depend of course on he iniial capial x. Given processes possessing hese properies we can compare he expeced uiliies of he sraegies p A and p A by E[ exp( α(x p T F ))] R (x) = E[ exp( α(x p T F ))] = V (x), (9.5)

9.1. THE EXPONENTIAL UTILITY 127 whence p is he desired opimal sraegy. To consruc his family, we se R (p) := exp( α(x (p) Y )), [, T ], p A, where (Y, Z) is a soluion of he BSDE Y = F Z s dw s f(s, Z s )ds, [, T ]. In hese erms we are bound o choose a funcion f for which R (p) is a supermaringale for all p A and here exiss a p A such ha R (p ) is a maringale. This funcion f also depends on he consrain se (A ) where (p ) akes is values (see (9.2)). We ge V (x) = R (p,x) = exp( α(x Y )), for all p A. In order o calculae f, we wrie R as he produc of a (local) maringale M (p) and a (no sricly) decreasing process Ã(p) ha is consan for some p A. For [, T ] define M (p) = exp( α(x Y )) exp ( Comparing R (p) and M (p) à (p) yields à (p) where v is defined as = exp( for [, T ] and z R m. evidenly f has o saisfy and α(p s Z s )dw s 1 2 v(s, p s, Z s )ds), [, T ], v(, p, z) = αp θ + αf(, z) + 1 2 α2 p z 2 v(, p, Z ) ) α 2 (p s Z s ) 2 ds. In order o obain a decreasing process Ã(p) v(, p, Z ) = for all p A for some paricular p A. For [, T ] we have 1 α v(, p, Z ) = α 2 p 2 αp (Z + 1 α θ ) + α 2 Z 2 + f(, Z ) = α 2 p (Z + 1 α θ ) 2 α 2 Z + 1 α θ 2 + α 2 Z2 + f(, Z ) = α 2 p (Z + 1 α θ ) 2 Z θ 1 2α θ 2 + f(, Z ).

128 CHAPTER 9. UTILITY MAXIMIZATION Now se f(, z) = α 2 dis2 A (z + 1 α θ ) + zθ + 1 2α θ 2. For his choice we ge v(, p, z) and for ( p Π A(ω) Z + 1 ) α θ, [, T ], we obain v(, p, Z) =. Here we see why he se à and hence A on which rading sraegies are resriced is assumed o be closed. In order o find he value funcion we have o minimize he disance beween a poin and a se. Furhermore here mus exis some elemen in A realizing he minimal disance. Boh requiremens are saisfied for closed ses. In a convex se he minimizer is unique. This would lead o a unique uiliy maximizing rading sraegy. However, we prove exisence of a possibly non unique rading sraegy solving he maximizaion problem for closed bu no necessarily convex consrains. Theorem 81 The value funcion of he opimizaion problem (9.4) is given by V (x) = exp( α(x Y )), where Y is defined by he unique soluion (Y, Z) H (R) H 2 (R m ) of he BSDE wih Y = F Z s dw s f(s, Z s )ds, [, T ], (9.6) f(, z) = α 2 dis2 A ( z + 1 α θ ) + zθ + 1 2α θ 2. There exiss an opimal rading sraegy p A wih p Π A(ω)(Z + 1 α θ ), [, T ], P a.s. (9.7) The disance dis o and he projecion Π on a closed subse of R m defined on page 12. are Proof. In order o ge he exisence of soluions of he BSDE (9.6) we apply Theorem 2.3 of (Kob). As in Lemma 56, we see ha (f(, z)) [,T ] defines a predicable process for every z R m. A sufficien condiion for he exisence of a soluion is condiion (H1) in (Kob): here are consans c, c 1 such ha f(, z) c + c 1 z 2 for all z R n P a.s. (9.8)

9.1. THE EXPONENTIAL UTILITY 129 By means of (7.6) we ge for every z R m, [, T ] dis 2 A (z + 1 ) α θ 2 z 2 + 2( 1 α θ + k 1 ) 2. So (9.8) follows from he boundedness of θ. Theorem 2.3 in (Kob) saes ha he BSDE (9.6) possesses a leas one soluion (Y, Z) H (R) H 2 (R m ). The proof of uniqueness is similar o he proof of Lemma 54. To find he value funcion of our opimizaion problem, we proceed wih he unique soluion (Y, Z) H (R) H 2 (R m ) of (9.6). Wih he same approach as in Lemma 56 we can find a (maybe non unique) predicable process p ha saisfies p Π A (Z + 1 α θ ) This rading sraegy p urns ou o be opimal, because Ã(p ) (ω) = 1 for λ P almos all (, ω). Furhermore, (p s Z s )dw s is a P BMO maringale, hus R (p ) is uniformly inegrable (Theorem 2.3 in (Kaz94)). Since, moreover, Y is a bounded process, we obain he uniform inegrabiliy of he family {exp( αx (p ) τ ) : τ sopping ime in [, T ]}. Therefore p A. Hence R (p ) is a maringale and [ ( ( ))] R (p ) = E exp α x + p s(dw s + θ s ds) F = exp( α(x Y )). I remains o show ha R (p) is a supermaringale for all p A. Since p A, he process M = M E( α (p s Z s )dw s ) is a local maringale. Hence here exiss a sequence of sopping imes (τ n ) n N saisfying lim n τ n = T P a.s. such ha (M τn ) is a posiive maringale for each n N. The process Ã(p) is decreasing. Thus R (p) τ n = M τn à (p) τ n is a supermaringale, i.e. for s For any se A F s we have E[R (p) τ n F s ] R (p) s τ n. E[R (p) τ n 1 A ] E[R (p) s τ n 1 A ]. Since {R (p) τ n } n and {R (p) s τ n } n are uniformly inegrable by he definiion of admissibiliy and he boundedness of Y, we may le n end o o obain E[R (p) 1 A ] E[R s (p) 1 A ]. This implies he claimed supermaringale propery of R (p).

13 CHAPTER 9. UTILITY MAXIMIZATION Remark 82 If he process p sdw s is a BMO maringale and E[exp( α(x (p) T F ))] <, a varian of an argumen of he above proof can be used o see ha p A. In fac, we see ha M (p) is a uniformly inegrable maringale, while A (p) is decreasing. Hence R (p) is a supermaringale. This jus saes ha for sopping imes τ Consequenly exp( α(x (p) τ Y τ )) E[ exp( α(x (p) T F )) F τ ]. exp( αx τ (p) ) exp( αy τ ) E[exp( α(x (p) T F )) F τ ]. This clearly implies uniform inegrabiliy of {exp( αx (p) τ ) : τ sopping ime in [, T ]}. The uiliy indifference price of F is he addiional iniial capial x F ha he invesor needs o ge he same maximal uiliy wih he liabiliy o pay ou F han wihou his liabiliy, see Remark 51. Le (Y F, Z F ) denoe he soluion of (9.6) wih erminal value F and (Y, Z ) he soluion of (9.6) wih erminal value. The uiliy indifference price is x F = Y F Y. (9.9) Observe ha he uiliy indifference price depends on he preferences. We can show ha he sraegy p is opimal in a wider sense. In fac, an invesor who has chosen a ime he sraegy p will sick o his decision if he sars solving he opimizaion problem a some laer ime beween and T. For his purpose, le us formulae he opimizaion problem more generally for a sopping ime τ T and an F τ measurable random variable which describes he capial a ime τ, i.e. X τ = Xτ p for some p A. So we consider he maximizaion problem [ V (τ, X τ ) = ess sup p A E exp ( α ( ))] X τ + p s (dw s + θ s ds) F. τ (9.1) Proposiion 83 (Dynamic Principle) The value funcion V (x) of he maximizaion problem (9.1) saisfies he dynamic programming principle, i.e. V (τ, X τ ) = exp( α(x τ Y τ )) for all sopping imes τ T where Y τ belongs o a soluion of he BSDE (9.6). An opimal sraegy ha aains he essenial supremum in (9.1) is given by p, he opimal sraegy consruced in Theorem 81.

9.2. POWER UTILITY 131 Proof. For [, T ], se R = exp( α(x Y ))E ( ) α(p s Z s )dw s exp( v(s, p s, Z s )ds) and apply he opional sopping heorem o he sochasic exponenial. The claim follows as in Theorem 81. Remark 84 If he consrain à on he sraegies is a convex cone, he value funcion V and he opimal sraegy p boh consruced in Theorem 81 are equivalen o hose deermined in (Sek2) and (EKR). Sekine considers he uiliy funcion x 1 exp( αx). He obains he value α funcion V (x) = 1 exp( αx + Ȳ) α saring wih he BSDE where Ȳ = αf z s dw s f(s, θ s, z s )ds, [, T ], f(, θ, z) = θ Π A ( z + θ ) 1 2 z Π A ( z + θ ) 2. We evidenly have o show ha Ȳ = αy for [, T ] or equivalenly αf(, θ, z α ) = f(, θ, z). Noe ha for a convex se C, he projecion Π C (a) is unique. If C is a convex cone and β >, hen βπ C (a) = Π C (βa). The equaliy for he funcions f and f herefore follows. El Karoui and Rouge (EKR) have obained he same BSDE and value funcion before Sekine. 9.2 Power uiliy In his secion we calculae he value funcion and characerize he opimal sraegy for he uiliy maximizaion problem wih respec o U γ (x) = 1 γ xγ, x, γ (, 1). This ime, our invesor maximizes he expeced uiliy of his wealh a ime T wihou an addiional liabiliy. The rading sraegies are consrained o ake values in a closed se à Rd. In his secion, we shall use a somewha

132 CHAPTER 9. UTILITY MAXIMIZATION differen noion of rading sraegy: ρ = ( ρ i ) i=1,...,d denoes he par of he wealh invesed in sock i. The number of shares of sock i is given by ρi X. A S i d dimensional F predicable process ρ = ( ρ ) T is called rading sraegy (par of wealh) if he following wealh process is well defined: X ( ρ) = x + d i=1 X s ( ρ) ρ i,s ds i,s = x + S i,s X ( ρ) s ρ s σ s (dw s + θ s ds), (9.11) and he iniial capial x is posiive. The wealh process X ( ρ) can be wrien as: ( ) X ( ρ) = xe ρ s σ s (dw s + θ s ds), [, T ]. I is more convenien o inroduce ρ = ρ σ, [, T ]. Accordingly, ρ is consrained o ake is values in A (ω) = Ãσ (ω) [, T ], ω Ω. The ses A saisfy (7.6). In order o formulae he opimizaion problem we firs define he se of admissible rading sraegies. Definiion 85 The se of admissible rading sraegies à consiss of all d dimensional predicable processes ρ = (ρ ) T ha saisfy ρ A (ω) P λ a.s and ρ s 2 ds < P a.s. Define he probabiliy measure Q P by ( ) dq dp = E θ s dw s. The se of admissible rading sraegies is free of arbirage because for every ρ Ã, he wealh process X(ρ) is a local Q maringale bounded from below, hence a Q supermaringale. Since Q is equivalen o P, he se of admissible rading sraegies à is free of arbirage. The invesor faces he maximizaion problem [ ( )] V (x) = sup E U X ( ρ) T. (9.12) ρ à In order o find he value funcion and an opimal sraegy we apply he same mehod as for he exponenial uiliy funcion. We herefore have o consruc a sochasic process R (ρ) wih erminal value R (ρ) T = U ( x + T X s ρ s ds s S s ),

9.2. POWER UTILITY 133 (ρ) and an iniial value R = R x ha does no depend on ρ, R(ρ) is a supermaringale for all ρ Ã and a maringale for a ρ Ã. Then ρ is he opimal sraegy and he value funcion given by V (x) = R. x Applying he uiliy funcion o he wealh process yields ( (X ρ,x ) γ = x γ exp γρ s dw s + γρ s θ s ds 1 2 This equaion suggess he following choice: R (ρ) ( = x γ exp γρ s dw s + where (Y, Z) is a soluion of he BSDE Y = Z s dw s γρ s θ s ds 1 2 f(s, Z s )ds, [, T ]. ) γ ρ s 2 ds, [, T ]. γ ρ s 2 ds + Y ), (9.13) In order o ge he supermaringale propery of R(ρ) we have o consruc f(, z) such ha for [, T ] γρ θ 1 2 γ ρ 2 + f(, Z ) 1 2 γρ + Z 2 for all ρ Ã. (9.14) R (ρ ) will even be a maringale if equaliy holds for ρ Ã. This is equivalen o f(, Z ) 1 2 γ(1 γ) ρ 1 2 1 γ (Z + θ ) 1 γ Z + θ 2 1 2 1 γ 2 Z 2. Hence he appropriae choice for f is f(, z) = ( ) γ(1 γ) 1 dis 2 A 2 1 γ (z + θ ) γ z + θ 2 2(1 γ) 1 2 z 2, and a candidae for he opimal sraegy mus saisfy ρ Π A(ω) ( ) 1 1 γ (Z + θ ), [, T ]. In he following Theorem boh value funcion and opimal sraegy are described.

134 CHAPTER 9. UTILITY MAXIMIZATION Theorem 86 The value funcion of he opimizaion problem is given by V (x) = x γ exp(y ) for x >, where Y is defined by he unique soluion (Y, Z) H (R) H 2 (R m ) of he BSDE wih f(, z) = Y = Z s dw s f(s, Z s )ds, [, T ], (9.15) ( ) γ(1 γ) 1 dis 2 A 2 1 γ (z + θ ) γ z + θ 2 2(1 γ) 1 2 z 2. There exiss an opimal rading sraegy ρ Ã wih he propery ( ) 1 ρ Π A(ω) 1 γ (Z + θ ). (9.16) Proof. As in Lemma 56, (f(, z)) [,T ] is a predicable sochasic process which also depends on σ. Due o (7.6) and he boundedness of θ, Condiion (H1) for Theorem 2.3 in (Kob) is fulfilled. We obain he exisence of a soluion (Y, Z) H (R) H 2 (R m ) for he BSDE (9.15). Uniqueness follows from he comparison argumens in he uniqueness par of he proof of Theorem 81. Le ρ denoe he predicable process saisfying ρ 1 Π A ( 1 γ Z + θ ). (see Lemma 56) Lemma 89 below shows ha ρ Ã. By Theorem 2.3 in (Kaz94), he process R (ρ ) is a maringale wih erminal value R (ρ ) T ( = x γ exp γρ sdw s + γρ sθ s ds 1 ) γ ρ 2 s 2 ds. This is he power uiliy from erminal wealh of he rading sraegy ρ. Therefore he expeced uiliy of ρ (ρ,x) is equal o R = x γ exp(y ). To show ha his provides he value funcion le ρ Ã. (9.14) yields ( R (ρ) = x γ exp(y )E (γρ s + Z s )dw s ) for a process v wih v s λ P a.s. ( ) exp v s ds, [, T ],

9.2. POWER UTILITY 135 The sochasic exponenial is a local maringale. There exiss a sequence of sopping imes (τ n ) n N, lim n τ n = T such ha E[ R (ρ) τ n F s ] R (ρ) s τ n, s for every n N. Furhermore, R(ρ) is bounded from below by. Passing o he limi and applying Faou s lemma yields ha R (ρ) is a supermaringale. (ρ) The erminal value R T is he uiliy of he erminal wealh of he rading sraegy ρ. Consequenly E[U(X (ρ,x) T )] R (x) = x γ exp(y ) for all ρ Ã. Again we can show ha an invesor saring o ac a some sopping ime in he rading inerval [, T ] will perceive he sraegy ρ jus consruced as opimal. Le τ T denoe a sopping ime and X τ an F τ measurable random variable which describes he capial a ime τ, i.e. X τ = Xτ ρ for a ρ à and an iniial capial x >. Consider he maximizaion problem V (τ, X τ ) = ess sup ρ Aτ E [ ( )] U X τ + X s ρ s (dw s + θ s ds). (9.17) τ Proposiion 87 (Dynamic Principle) The value funcion x γ exp(y) saisfies he dynamic programming principle, i.e. V (τ, X τ ) = (X τ ) γ exp(y τ ) for all sopping imes τ T, where Y τ is given by he unique soluion (Y, Z) of he BSDE (9.15). An opimal sraegy which aains he essenial supremum in (9.17) is given by ρ consruced in Theorem 86. Proof. See Proposiion 83. Remark 88 Suppose ha he consrain se à is a convex cone. Then he opimal sraegy ρ consruced in Theorem 86 is he same as in (Sek2). Sekine uses he uiliy funcion x 1 γ xγ and obains he value funcion Ṽ (x) = 1 γ xγ exp((1 γ)ỹ),

136 CHAPTER 9. UTILITY MAXIMIZATION where Ỹ is defined by he unique soluion (Ỹ, Z) H (R) H 2 (R m ) of he BSDE Ỹ = Z s dw s g(s, Z s )ds, [, T ]. Here g(, z) = θ 2 2 1 ( 2 θ Π A z + θ ) 2 1 γ ( 1 γ 2 z Π A z + θ ) 2. 1 γ As for he exponenial uiliy funcion we have o show (1 γ)ỹ = Y or z equivalenly (1 γ)g(, ) = f(, z). In fac, we have 1 γ ( ) [ z θ 2 (1 γ)g, = (1 γ) 1 γ 2 1 ( ) ] z + 2 θ θ 2 Π A 1 γ ( ) (1 γ)2 z z + 2 1 γ Π θ 2 A 1 γ 1 = θ Π A (z + θ ) 2(1 γ) Π A (z + θ ) 2 To obain he las equaliy, we use (see (9.18) below). For he funcion f we obain 1 2 z 2 + zπ A (z + θ ) 1 2 Π A (z + θ ) 2 = (z + θ )Π A (z + θ ) 2 γ 2(1 γ) Π A (z + θ ) 2 1 2 z 2 γ = 2(1 γ) Π A (z + θ ) 2 1 2 z 2. (z + θ )Π A (z + θ ) = Π A (z + θ ) 2 f(, z) = γ(1 γ) 1 2 1 γ (z + θ ) Π A γ (z + θ ) 2 2 (1 γ) 1 2 z 2 = γ 1 γ (z + θ )Π A (z + θ ) + = γ 2(1 γ) Π A (z + θ ) 2 1 2 z 2. ( ) 1 2 1 γ (z + θ ) γ 2(1 γ) Π A (z + θ ) 2 1 2 z 2

9.3. LOGARITHMIC UTILITY 137 For [, T ], z R m we herefore have (1 γ)g(, z ) = f(, z). 1 γ I remains o prove ha for a convex cone C and a R m he following equaliy holds: Π C (a)(a Π C (a)) =. (9.18) If Π C (a) = hen he ideniy is saisfied. If no, consider he half line λπ C (a), λ. This half line is par of he cone C, so Π C (a) is also he projecion of a on he half line. Lemma 89 Le (Y, Z) H (R) H 2 (R m ) be a soluion of he BSDE (9.15), and le ρ be given by (9.16). Then he processes are P BMO maringales. Z s dw s, ρ sdw s Proof. We may ake a lower bound k for Y, and apply Iô s formula o Y k 2, o conclude in he same manner as before. 9.3 Logarihmic Uiliy To complee he specrum of imporan uiliy funcions, in his secion we shall consider logarihmic uiliy. As in he preceding secion, he agen has no liabiliy a ime T. Trading sraegies and wealh process have he same meaning as in secion 9.2 (see (9.11)). The rading sraegies ρ are consrained o ake values in a closed se à Rd. For ρ = ρ σ he consrains are described by A = Ãσ, [, T ]. In order o compare he logarihmic uiliy of he erminal wealh of wo rading sraegies we have o impose a mild inegrabiliy condiion on ρ. Recall ha ρ i > 1 means ha he invesor has o borrow money in order o buy sock i and if ρ i < hen he invesor has a negaive number of sock i. In order o find he maximal uiliy, we impose an inegrabiliy condiion on ρ ha is no resricive. Definiion 9 The se of admissible rading sraegies A l consiss of all R d valued predicable processes ρ saisfying E[ ρ s 2 ds] < and ρ A P λ a.s.

138 CHAPTER 9. UTILITY MAXIMIZATION For he logarihmic uiliy funcion U(x) = log(x), x >, we obain a paricularly simple BSDE ha leads o he value funcion and he opimal sraegy. The opimizaion problem is given by V (x) = sup E[log(X (ρ) T )] (9.19) ρ A l [ = log(x) + sup E ρ s dw s + (ρ s θ s 1 ] ρ A l 2 ρ s 2 )ds, (9.2) where he iniial capial x is posiive again. We aim o deermine a process R (ρ) wih R (ρ) T = log(x (ρ) T ), and an iniial value ha does no depend on ρ. Furhermore, R (ρ) is a supermaringale for all ρ A l, and here exiss a ρ A l such ha R (ρ ) is a maringale. The sraegy ρ is he opimal sraegy and R ρ is he value funcion of he opimizaion problem (9.19). We can choose for [, T ] R (ρ) = log x + Y + (ρ s + Z s )dw s + ( 1 2 ρ s θ s 2 + 1 ) 2 θ2 s + f(s) ds, where f() = 1 2 dis2 A (θ ) 1 2 θ 2, [, T ], and (Y, Z ) is he unique soluion of he following BSDE: Y = Z s dw s f(s)ds, [, T ]. Due o definiion 9, he boundedness of θ and (7.6), he sochasic inegral in R (ρ) is a maringale for all ρ A l. Hence R (ρ) is a supermaringale for all ρ A l. An opimal rading sraegy ρ which saisfies ρ Π A (θ ), P λ a.e. (ω, ) (9.21) can be consruced by means of Lemma 56. The iniial value Y saisfies [ Y = E We summarize our resuls in a heorem: ] f(s)ds.

9.3. LOGARITHMIC UTILITY 139 Theorem 91 There exiss a rading sraegy ρ A aaining he supremum in 9.19. This rading sraegy is saed in (9.21). The maximal uiliy for he iniial capial x > is [ ] V (x) = R ρ (x) = log(x) + E f(s)ds. In paricular ρ only depends on θ, σ and he se A describing he consrains on he rading sraegies, i.e. he value ha hose processes ake a ime.

14 CHAPTER 9. UTILITY MAXIMIZATION

Appendix A BMO maringales Here we recall and collec a few well known facs from he heory of maringales of bounded mean oscillaion, briefly called BMO maringales. We follow he exposiion in (Kaz94). The saemens will be made for infinie ime horizon. In he ex hey will be applied o he simpler framework of finie horizon, replacing wih T. Definiion 92 Le M = (M ) be a uniformly inegrable maringale wih respec o a probabiliy measure P and a complee, righ coninuous filraion F saisfying M =. For 1 p < se M BMOp := sup E[ M M τ p F τ ] 1/p. τ F sopping ime (A.1) The normed linear space {M : M BMOp < } wih norm M BMOp (aken wih respec o P ) is denoed by BMO p (Kazamaki (Kaz94), p. 25). By Corollary 2.1 in (Kaz94), p. 28, we have for all 1 p < M BMO 1 iff M BMO p. BM O(P ) denoes all uniformly inegrable P maringales such ha M BMO1 <. The norm in BMO 2 (P ) can be alernaively expressed as M BMO2 = sup E[ M M τ F τ ] 1/2. (A.2) τ F sopping ime The combined inequaliies of Doob and Burkholder Davis Gundy read for p > 1 ( p p 1 )p E[ M p ] E[ sup M p ] c p E[ M p/2 ] 141 (A.3)

142 APPENDIX A. BMO MARTINGALES wih a universal posiive consan c p. Therefore for any BMO maringale M we obain M L p (P ) for all p > 1, [, ]. BMO-maringales possess he convenien propery of generaing uniformly inegrable exponenials according o he following Theorem. Theorem 93 (Theorem 2.3 (Kaz94)) If M BMO, hen E(M) is a uniformly inegrable maringale. According o he following Theorem, he BMO propery is preserved by equivalen changes of measure. In fac, le M BMO(P ) and ˆP given by he measure change d ˆP = E(M) dp. Define φ : X ˆX = X, M X. Theorem 94 (Theorem 3.6 (Kaz94)) If M BMO(P ), hen φ : X ˆX is an isomorphism of BMO(P ) ono BMO( ˆP ).

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Selbsändigkeiserklärung Hiermi erkläre ich, dass ich die vorliegende Arbei selbs andig ohne unerlaube Hilfe verfaß und nur die angegebene Lieraur und Hilfsmiel verwende habe. Kapiel 2 und 3 beruhen auf der Arbei Parial equilibrium and marke compleion mi Ying Hu und Peer Imkeller. In Kapiel 9 sehen die Ergebnisse der Arbei Uiliy maximizaion in incomplee markes, die ebenfalls mi Ying Hu und Peer Imkeller geschrieben wurde. Beide Arbeien wurden von den Auoren jeweils zu gleichen Teilen geschrieben. Mahias Müller 24. Februar 25 149