AND BACKWARD SDE. Nizar Touzi Ecole Polytechnique Paris Département de Mathématiques Appliquées

Size: px
Start display at page:

Download "AND BACKWARD SDE. Nizar Touzi nizar.touzi@polytechnique.edu. Ecole Polytechnique Paris Département de Mathématiques Appliquées"

Transcription

1 OPIMAL SOCHASIC CONROL, SOCHASIC ARGE PROBLEMS, AND BACKWARD SDE Nizar ouzi Ecole Polyechnique Paris Déparemen de Mahémaiques Appliquées Chaper 12 by Agnès OURIN May 21

2 2

3 Conens 1 Condiional Expecaion and Linear Parabolic PDEs Sochasic differenial equaions Markov soluions of SDEs Connecion wih linear parial differenial equaions Generaor Cauchy problem and he Feynman-Kac represenaion Represenaion of he Dirichle problem he Black-Scholes model he coninuous-ime financial marke Porfolio and wealh process Admissible porfolios and no-arbirage Super-hedging and no-arbirage bounds he no-arbirage valuaion formula PDE characerizaion of he Black-Scholes price Sochasic Conrol and Dynamic Programming Sochasic conrol problems in sandard form he dynamic programming principle A weak dynamic programming principle Dynamic programming wihou measurable selecion he dynamic programming equaion On he regulariy of he value funcion Coninuiy of he value funcion for bounded conrols A deerminisic conrol problem wih non-smooh value funcion A sochasic conrol problem wih non-smooh value funcion Opimal Sopping and Dynamic Programming Opimal sopping problems he dynamic programming principle he dynamic programming equaion Regulariy of he value funcion Finie horizon opimal sopping

4 Infinie horizon opimal sopping An opimal sopping problem wih nonsmooh value Solving Conrol Problems by Verificaion he verificaion argumen for sochasic conrol problems Examples of conrol problems wih explici soluions Opimal porfolio allocaion Law of ieraed logarihm for double sochasic inegrals he verificaion argumen for opimal sopping problems Examples of opimal sopping problems wih explici soluions Perual American opions Finie horizon American opions Inroducion o Viscosiy Soluions Inuiion behind viscosiy soluions Definiion of viscosiy soluions Firs properies Comparison resul and uniqueness Comparison of classical soluions in a bounded domain Semijes definiion of viscosiy soluions he Crandall-Ishii s lemma Comparison of viscosiy soluions in a bounded domain Comparison in unbounded domains Useful applicaions Proof of he Crandall-Ishii s lemma Dynamic Programming Equaion in he Viscosiy Sense DPE for sochasic conrol problems DPE for opimal sopping problems A comparison resul for obsacle problems Sochasic arge Problems Sochasic arge problems Formulaion Geomeric dynamic programming principle he dynamic programming equaion Applicaion: hedging under porfolio consrains Sochasic arge problem wih conrolled probabiliy of success Reducion o a sochasic arge problem he dynamic programming equaion Applicaion: quanile hedging in he Black-Scholes model Second Order Sochasic arge Problems Superhedging under Gamma consrains Problem formulaion Hedging under upper Gamma consrain

5 Including he lower bound on he Gamma Second order arge problem Problem formulaion he geomeric dynamic programming he dynamic programming equaion Superhedging under illiquidiy cos Backward SDEs and Sochasic Conrol Moivaion and examples he sochasic Ponryagin maximum principle BSDEs and sochasic arge problems BSDEs and finance Wellposedness of BSDEs Maringale represenaion for zero generaor BSDEs wih affine generaor he main exisence and uniqueness resul Comparison and sabiliy BSDEs and sochasic conrol BSDEs and semilinear PDEs Appendix: essenial supremum Quadraic backward SDEs A priori esimaes and uniqueness A priori esimaes for bounded Y Some propeies of BMO maringales Uniqueness Exisence Exisence for small final condiion Exisence for bounded final condiion Porfolio opimizaion under consrains Problem formulaion BSDE characerizaion Ineracing invesors wih performance concern he Nash equilibrium problem he individual opimizaion problem he case of linear consrains Nash equilibrium under deerminisic coefficiens Probabilisic numerical mehods for nonlinear PDEs Discreizaion Convergence of he discree-ime approximaion Consisency, monooniciy and sabiliy he Barles-Souganidis monoone scheme

6 6 12 Inroducion o Finie differences mehods Overview of he Barles-Souganidis framework Firs examples he hea equaion: he classic explici and implici schemes he Black-Scholes-Meron PDE A nonlinear example: he Passpor Opion Problem formulaion Finie Difference approximaion Howard algorihm he Bonnans-Zidani [7] approximaion Working in a finie domain Variaional Inequaliies and spliing mehods he American opion

7 Inroducion hese noes have been prepared for he graduae course ough a he Fields Insiue, orono, during he hemaic program on quaniaive finance which was held from January o June, 21. I would like o hank all paricipans o hese lecures. I was a pleasure for me o share my experience on his subjec wih he excellen audience ha was offered by his special research semeser. In paricular, heir remarks and commens helped o improve pars of his documen, and o correc some misakes. My special hanks go o Bruno Bouchard, Mee Soner and Agnès ourin who acceped o ac as gues lecurers wihin his course. hese noes have also benefied from he discussions wih hem, and some pars are based on my previous work wih Bruno and Mee. hese noes benefied from careful reading by Maheus Grasselli and om Salisbury. I grealy appreciae heir help and hope here are no many misakes lef. I would like o express all my hanks o Maheus Grasselli, om Hurd, om Salisbury, and Sebasian Jaimungal for he warm hospialiy a he Fields Insiue, and heir regular aendance o my lecures. hese lecures presen he modern approach o sochasic conrol problems wih a special emphasis on he applicaion in financial mahemaics. For pedagogical reason, we resric he scope of he course o he conrol of diffusion processes, hus ignoring he presence of jumps. We firs review he main ools from sochasic analysis: Brownian moion and he corresponding sochasic inegraion heory. his already inroduces o he firs connecion wih parial differenial equaions (PDE). Indeed, by Iô s formula, a linear PDE pops up as he infiniesimal counerpar of he ower propery. Conversely, given a nicely behaved smooh soluion, he socalled Feynman-Kac formula provides a sochasic represenaion in erms of a condiional expecaion. We hen inroduce he class of sandard sochasic conrol problems where one wishes o maximize he expeced value of some gain funcional. he firs main ask is o derive an original weak dynamic programming principle which avoids he heavy measurable selecion argumens in ypical proofs of he dynamic programming principle when no a priori regulariy of he value funcion 7

8 8 CHAPER. INRODUCION is known. he infiniesimal counerpar of he dynamic programming principle is now a nonlinear PDE which is called dynamic programming equaion, or Hamilon-Jacobi-Bellman equaion. he hope is ha he dynamic programming equaion provides a complee characerizaion of he problem, once complemened wih appropriae boundary condiions. However, his requires srong smoohness condiions, which can be seen o be violaed in simple examples. A parallel picure can be drawn for opimal sopping problems and, in fac, for he more general conrol and sopping problems. In hese noes we do no rea such mixed conrol problem, and we raher analyze separaely hese wo classes of conrol problems. Here again, we derive he dynamic programming principle, and he corresponding dynamic programming equaion under srong smoohness condiions. In he presen case, he dynamic programming equaion akes he form of he obsacle problem in PDEs. When he dynamic programming equaion happens o have an explici smooh soluion, he verificaion argumen allows o verify wheher his candidae indeed coincides wih he value funcion of he conrol problem. he verificaion argumen provides as a by-produc an access o he opimal conrol, i.e. he soluion of he problem. Bu of course, such lucky cases are rare, and one should no coun on solving any sochasic conrol problem by verificaion. In he absence of any general a priori regulariy of he value funcion, he nex developmen of he heory is based on viscosiy soluions. his beauiful noion was inroduced by Crandal and Lions, and provides a weak noion of soluions o second order degenerae ellipic PDEs. We review he main ools from viscosiy soluions which are needed in sochasic conrol. In paricular, we provide a difficuly-incremenal presenaion of he comparison resul (i.e. maximum principle) which implies uniqueness. We nex show ha he weak dynamic programming equaion implies ha he value funcion is a viscosiy soluion of he corresponding dynamic programming equaion in a wide generaliy. In paricular, we do no assume ha he conrols are bounded. We emphasize ha in he presen seing, here is no apriori regulariy of he value funcion needed o derive he dynamic programming equaion: we only need i o be locally bounded! Given he general uniqueness resuls, viscosiy soluions provide a powerful ool for he characerizaion of sochasic conrol and opimal sopping problems. he remaining par of he lecures focus on he more recen lieraure on sochasic conrol, namely sochasic arge problems. hese problems are moivaed by he superhedging problem in financial mahemaics. Various exensions have been sudied in he lieraure. We focus on a paricular seing where he proofs are simplified while highlighing he main ideas. he use of viscosiy soluions is crucial for he reamen of sochasic arge problems. Indeed, deriving any a priori regulariy seems o be a very difficul ask. Moreover, by wriing formally he corresponding dynamic programming equaion and guessing an explici soluion (in some lucky case), here is no known direc verificaion argumen as in sandard sochasic conrol problems. Our approach is hen based on a dynamic programming principle suied o his class of problems, and called geomeric dynamic programming principle, due o

9 9 a furher exension of sochasic arge problems o fron propagaion problems in differenial geomery. he geomeric programming principle allows o obain a dynamic programming equaion in he sense of viscosiy soluions. We provide some examples where he analysis of he dynamic programming equaion leads o a complee soluion of he problem. We also presen an ineresing exension o sochasic arge problems wih conrolled probabiliy of success. A remarkable rick allows o reduce hese problems o sandard sochasic arge problems. By using his mehodology, we show how one can solve explicily he problem of quanile hedging which was previously solved by Föllmer and Leuker [21] by dualiy mehods in he sandard linear case in financial mahemaics. A furher exension of sochasic arge problems consiss in involving he quadraic variaion of he conrol process in he conrolled sae dynamics. hese problems are moivaed by examples from financial mahemaics relaed o marke illiquidiy, and are called second order sochasic arge problems. We follow he same line of argumens by formulaing a suiable geomeric dynamic programming principle, and deriving he corresponding dynamic programming equaion in he sense of viscosiy soluions. he main new difficuly here is o deal wih he shor ime asympoics of double sochasic inegrals. he final par of he lecures explores a special ype of sochasic arge problems in he non-markov framework. his leads o he heory of backward sochasic differenial equaions (BSDE) which was inroduced by Pardoux and Peng [33]. Here, in conras o sochasic arge problems, we insis on he exisence of a soluion o he sochasic arge problem. We provide he main exisence, uniqueness, sabiliy and comparison resuls. We also esablish he connecion wih sochasic conrol problems. We finally show he connecion wih semilinear PDEs in he Markov case. he exension of he heory of BSDEs o he case where he generaor is quadraic in he conrol variable is very imporan in view of he applicaions o porfolio opimizaion problems. However, he exisence and uniqueness can no be addressed as simply as in he Lipschiz case. he firs exisence and uniqueness resuls were esablished by Kobylanski [27] by adaping o he non- Markov framework echniques developed in he PDE lieraure. Insead of his hilghly echnical argumen, we repor he beauiful argumen recenly developed by evzadze [39], and provide applicaions in financial mahemaics. he final chaper is dedicaed o numerical mehods for nonlinear PDEs. We provide a complee proof of convergence based on he Barles-Souganidis moone scheme mehod. he laer is a beauiful and simple argumen which explois he sabiliy of viscosiy soluions. Sronger resuls are provided in he semilinear case by using echniques from BSDEs. Finally, I should like o express all my love o my family: Chrisine, our sons Ali and Héni, and our dougher Lilia, who accompanied me during his visi o orono,

10 1 CHAPER. INRODUCION all my hanks o hem for heir paience while I was preparing hese noes, and all my apologies for my absence even when I was physically presen...

11 Chaper 1 Condiional Expecaion and Linear Parabolic PDEs hroughou his chaper, (Ω, F, F, P ) is a filered probabiliy space wih filraion F = {F, } saisfying he usual condiions. Le W = {W, } be a Brownian moion valued in R d, defined on (Ω, F, F, P ). hroughou his chaper, a mauriy > will be fixed. By H 2, we denoe he collecion of all progressively [ measurble processes φ wih appropriae (finie) ] dimension such ha E φ 2 d <. 1.1 Sochasic differenial equaions In his secion, we recall he basic ools from sochasic differenial equaions dx = b (X )d + σ (X )dw, [, ], (1.1) where > is a given mauriy dae. Here, b and σ are F B(R n )-progressively measurable funcions from [, ] Ω R n o R n and M R (n, d), respecively. In paricular, for every fixed x R n, he processes {b (x), σ (x), [, ]} are F progressively measurable. Definiion 1.1. A srong soluion of (1.1) is an F progressively measurable process X such ha ( b (X ) + σ (X ) 2 )d <, a.s. and X = X + b s (X s )ds + σ s (X s )dw s, [, ]. Le us menion ha here is a noion of weak soluions which relaxes some condiions from he above definiion in order o allow for more general sochasic differenial equaions. Weak soluions, as opposed o srong soluions, are 11

12 12 CHAPER 1. CONDIIONAL EXPECAION AND LINEAR PDEs defined on some probabilisic srucure (which becomes par of he soluion), and no necessarily on (Ω, F, F, P, W ). hus, for a weak soluion we search for a probabiliy srucure ( Ω, F, F, P, W ) and a process X such ha he requiremen of he above definiion holds rue. Obviously, any srong soluion is a weak soluion, bu he opposie claim is false. he main exisence and uniqueness resul is he following. heorem 1.2. Le X L 2 be a r.v. independen of W. Assume ha he processes b. () and σ. () are in H 2, and ha for some K > : b (x) b (y) + σ (x) σ (y) K x y for all [, ], x, y R n. hen, for all >, here exiss a unique srong soluion of (1.1) in H 2. Moreover, [ ] E sup X 2 C ( 1 + E X 2) e C, (1.2) for some consan C = C(, K) depending on and K. Proof. We firs esablish he exisence and uniqueness resul, hen we prove he esimae (1.2). Sep 1 For a consan c >, o be fixed laer, we inroduce he norm [ 1/2 φ H 2 c := E e c φ d] 2 for every φ H 2. Clearly, he norms. H 2 and. H 2 c on he Hilber space H 2 are equivalen. Consider he map U on H 2 by: U(X) := X + b s (X s )ds + σ s (X s )dw s,. By he Lipschiz propery of b and σ in he x variable and he fac ha b. (), σ. () H 2, i follows ha his map is well defined on H 2. In order o prove exisence and uniqueness of a soluion for (1.1), we shall prove ha U(X) H 2 for all X H 2 and ha U is a conracing mapping wih respec o he norm. H 2 c for a convenien choice of he consan c >. 1- We firs prove ha U(X) H 2 for all X H 2. o see his, we decompose: [ U(X) 2 H 3 X 2 2 L + 3 E ] 2 2 b s (X s )ds d [ ] 2 +3E σ s (X s )dw s d By he Lipschiz-coninuiy of b and σ in x, uniformly in, we have b (x) 2 K(1 + b () 2 + x 2 ) for some consan K. We hen esimae he second erm

13 1.1. Sochasic differenial equaions 13 by: [ E ] 2 b s (X s )ds d [ ] K E (1 + b () 2 + X s 2 )ds <, since X H 2, and b(., ) L 2 ([, ]). As, for he hird erm, we use he Doob maximal inequaliy ogeher wih he fac ha σ (x) 2 K(1 + σ () 2 + x 2 ), a consequence of he Lipschiz propery on σ: [ ] [ ] E σ s (X s )dw s 2 d E 2 max σ s (X s )dw s d [ ] 4 E σ s (X s ) 2 ds [ ] 4 KE (1 + σ s () 2 + X s 2 )ds <. 2- o see ha U is a conracing mapping for he norm. H 2 c, for some convenien choice of c >, we consider wo process X, Y H 2 wih X = Y, and we esimae ha: E U(X) U(Y ) 2 2E (b s (X s ) b s (Y s )) ds = 2E (b s (X s ) b s (Y s )) ds 2E 2( + 1)K E + 2E b s (X s ) b s (Y s ) 2 ds + 2E E X s Y s 2 ds. (σ s (X s ) σ s (Y s )) dw s 2 σ s (X s ) σ s (Y s ) 2 ds σ s (X s ) σ s (Y s ) 2 ds 2K( + 1) Hence, U(X) U(Y ) c X Y c c, and herefore U is a conracing mapping for sufficienly large c. Sep 2 We nex prove he esimae (1.2). We shall alleviae he noaion wriing b s := b s (X s ) and σ s := σ s (X s ). We direcly esimae: [ ] [ u u ] 2 E sup X u 2 = E sup X + b s ds + σ s dw s u u ( [ ] [ u ]) 2 3 E X 2 + E b s 2 ds + E sup σ s dw s u ( [ ] [ ]) 3 E X 2 + E b s 2 ds + 4E σ s 2 ds

14 14 CHAPER 1. CONDIIONAL EXPECAION AND LINEAR PDEs where we used he Doob s maximal inequaliy. Since b and σ are Lipschizconinuous in x, uniformly in and ω, his provides: [ ] E sup X u 2 u ( C(K, ) 1 + E X 2 + [ ] ) E sup X u 2 ds u s and we conclude by using he Gronwall lemma. he following exercise shows ha he Lipschiz-coninuiy condiion on he coefficiens b and σ can be relaxed. We observe ha furher relaxaion of his assumpion is possible in he one-dimensional case, see e.g. Karazas and Shreve [24]. Exercise 1.3. In he conex of his secion, assume ha he coefficiens µ and σ are locally Lipschiz and linearly growing in x, uniformly in (, ω). By a localizaion argumen, prove ha srong exisence and uniqueness holds for he sochasic differenial equaion (1.1). In addiion o he esimae (1.2) of heorem 1.2, we have he following flow coninuiy resuls of he soluion of he SDE. In order o emphasize he dependence on he iniial dae, we denoe by {Xs,x, s } he soluion of he SDE (1.1) wih iniial condiion X,x = x. heorem 1.4. Le he condiions of heorem 1.2 hold rue, and consider some (, x) [, ) R n wih. (i) here is a consan C such ha: [ E sup X,x s s Xs,x 2 ] Ce C x x 2. (1.3) (ii) Assume furher ha B := sup < ( ) 1 E ( br () 2 + σ r () 2) dr <. hen for all [, ]: [ E sup s X,x s X,x s 2 ] Ce C (B + x 2 ). (1.4) Proof. (i) o simplify he noaions, we se X s := Xs,x and X s := Xs,x for all s [, ]. We also denoe δx := x x, δx := X X, δb := b(x) b(x ) and δσ := σ(x) σ(x ). We firs decompose: ( s δx s 2 3 δx 2 + δb u du 2 s ) 2 + δσ u dw u ( s s ) 3 δx 2 + (s ) δbu 2 2 du + δσ u dw u.

15 1.1. Sochasic differenial equaions 15 hen, i follows from he Doob maximal inequaliy and he Lipschiz propery of he coefficiens b and σ ha: [ ] h( ) := E sup δx s 2 s ( 3 δx 2 + (s ) s ( 3 δx 2 + K 2 ( + 4) ( 3 δx 2 + K 2 ( + 4) E s δbu 2 du + 4 E ) δσu 2 du s s ) E δx u 2 du ) h(u)du. hen he required esimae follows from he Gronwall inequaliy. (ii) We nex prove (1.4). We again simplify he noaion by seing X s := Xs,x, s [, ], and X s := X,x s, s [, ]. We also denoe δ :=, δx := X X, δb := b(x) b(x ) and δσ := σ(x) σ(x ). hen following he same argumens as in he previous sep, we obain for all u [, ]: [ ] h(u) := E sup δx s 2 s u Observe ha ( E X x 2 2 E 2 ( ( 3 E X x 2 + K 2 ( + 4) ( 3 E X x 2 + K 2 ( + 4) b r (X r )dr 2 + E u u ) σ r (X r )dr 2 E b r (X r ) 2 dr + E σ r (X r ) 2 dr ) E δx r 2 dr ) h(r)dr (1.5) ( 6( + 1) K 2 E X r x 2 + x 2 + E b r () 2) dr ) 6( + 1) (( )( x 2 + B) + K 2 E X r x 2 dr. By he Gronwall inequaliy, his shows ha E X x 2 C( x 2 + B) e C( ). Plugging his esimae in (1.5), we see ha: ( h(u) 3 C( x 2 + B) e C ( ) + K 2 ( + 4) u ) ) h(r)dr, (1.6) and he required esimae follows from he Gronwall inequaliy.

16 16 CHAPER 1. CONDIIONAL EXPECAION AND LINEAR PDEs 1.2 Markov soluions of SDEs In his secion, we resric he coefficiens b and σ o be deerminisic funcions of (, x). In his conex, we wrie b (x) = b(, x), σ (x) = σ(, x) for [, ], x R n, where b and σ are coninuous funcions, Lipschiz in x uniformly in. Le X.,x denoe he soluion of he sochasic differenial equaion X,x s = x + s b ( u, X,x u ) du + s he wo following properies are obvious: Clearly, Xs,x F. σ ( u, Xu,x ) dwu s = F (, x, s, (W. W ) u s ) for some deerminisic funcion For u s: Xs,x = X u,x,x u s. his follows from he pahwise uniqueness, and holds also when u is a sopping ime. Wih hese observaions, we have he following Markov propery for he soluions of sochasic differenial equaions. Proposiion 1.5. (Markov propery) For all s: E [Φ (X u, u s) F ] = E [Φ (X u, u s) X ] for all bounded funcion Φ : C([, s]) R. 1.3 Connecion wih linear parial differenial equaions Generaor Le {X,x s, s } be he unique srong soluion of X,x s = x + s b(u, X,x u )du + s σ(u, X,x u )dw u, s, where µ and σ saisfy he required condiion for exisence and uniqueness of a srong soluion. For a funcion f : R n R, we define he funcion Af by Af(, x) = lim E[f(X,x +h h h )] f(x) if he limi exiss. Clearly, Af is well-defined for all bounded C 2 funcion wih bounded derivaives and Af(, x) = b(, x) Df(x) r [ σσ (, x)d 2 f(x) ], (1.7)

17 1.3. Connecion wih PDE 17 where Df and D 2 f denoe he gradien and Hessian of f, respecively. (Exercise!). he linear differenial operaor A is called he generaor of X. I urns ou ha he process X can be compleely characerized by is generaor or, more precisely, by he generaor and he corresponding domain of definiion. As he following resul shows, he generaor provides an inimae connecion beween condiional expecaions and linear parial differenial equaions. Proposiion 1.6. Assume ha he funcion (, x) v(, x) := E [ g(x,x ] is C 1,2 ([, ) R n ). hen v solves he parial differenial equaion: v + Av = and v(,.) = g. Proof. Given (, x), le τ 1 := inf{s > : Xs,x x 1}. By he law of ieraed expecaion ogeher wih he Markov propery of he process X, i follows ha v(, x) = E [ v ( s τ 1, X,x s τ 1 )]. Since v C 1,2 ([, ), R n ), we may apply Iô s formula, and we obain by aking expecaions: [ s τ1 = E = E +E [ s τ1 [ s τ1 ( ) v + Av v x ( v + Av ] (u, Xu,x )du (u, X,x s ) σ(u, X,x ) ] (u, Xu,x )du u )dw u ] where he las equaliy follows from he boundedness of (u, Xu,x ) on [, s τ 1 ]. We now send s, and he required resul follows from he dominaed convergence heorem Cauchy problem and he Feynman-Kac represenaion In his secion, we consider he following linear parial differenial equaion, v + Av k(, x)v + f(, x) =, v(,.) = g (, x) [, ) Rd (1.8) where A is he generaor (1.7), g is a given funcion from R d o R, k and f are funcions from [, ] R d o R, b and σ are funcions from [, ] R d o R d and and M R (d, d), respecively. his is he so-called Cauchy problem. For example, when k = f, b, and σ is he ideniy marix, he above parial differenial equaion reduces o he hea equaion.

18 18 CHAPER 1. CONDIIONAL EXPECAION AND LINEAR PDEs Our objecive is o provide a represenaion of his purely deerminisic problem by means of sochasic differenial equaions. We hen assume ha b and σ saisfy he condiions of heorem 1.2, namely ha b, σ Lipschiz in x uniformly in, ( b(, ) 2 + σ(, ) 2) d <.(1.9) heorem 1.7. Le he coefficiens b, σ be coninuous and saisfy (1.9). Assume furher ha he funcion k is uniformly bounded from below, and f has quadraic growh in x uniformly in. Le v be a C 1,2 ( [, ), R d) soluion of (1.8) wih quadraic growh in x uniformly in. hen v(, x) = E where {X,x s and β,x s := e s [ β,x s f(s, X,x )ds + β,x g ( X,x ) ],, x R d, s, s } is he soluion of he SDE 1.1 wih iniial daa X,x = x, k(u,x,x u )du for s. Proof. We firs inroduce he sequence of sopping imes τ n := inf { s > : X,x s x n }, and we oberve ha τ n P a.s. Since v is smooh, i follows from Iô s formula ha for s < : d ( β,x s v ( s, X,x )) s ( = βs,x kv + v ) (s, + Av ) X,x s ds +βs,x ( ) ( ) s, X,x s σ s, X,x s dws = β,x s v ( x f(s, Xs,x )ds + v x by he PDE saisfied by v in (1.8). hen: = E E [ βτ,x n v ( τ n, Xτ,x )] n v(, x) [ τn ( βs,x f(s, X s )ds + v x ( s, X,x s ( s, X,x s ) σ ( s, X,x s ) σ ( s, X,x s )] ) dws. ) ) dws, Now observe ha he inegrands in he sochasic inegral is bounded by definiion of he sopping ime τ n, he smoohness of v, and he coninuiy of σ. hen he sochasic inegral has zero mean, and we deduce ha [ τn v(, x) = E β,x s f ( s, Xs,x ) ds + β,x τ n v ( τ n, Xτ,x ) ] n. (1.1)

19 1.3. Connecion wih PDE 19 Since τ n and he Brownian moion has coninuous sample pahs P a.s. i follows from he coninuiy of v ha, P a.s. τn β,x s f ( s, Xs,x ) ds + β,x τ n v ( τ n, X,x n βs,x = τ n ) f ( s, X,x ) ds + β,x β,x s s f ( s, X,x ) ds + β,x s v (, X,x ) g ( X,x ) (1.11) by he erminal condiion saisfied by v in (1.8). Moreover, since k is bounded from below and he funcions f and v have quadraic growh in x uniformly in, we have τn β,x s f ( s, Xs,x ) ds + β,x τ n v ( τ n, X,x ) ( τ n C 1 + max X 2 By he esimae saed in he exisence and uniqueness heorem 1.2, he laer bound is inegrable, and we deduce from he dominaed convergence heorem ha he convergence in (1.11) holds in L 1 (P), proving he required resul by aking limis in (1.1). he above Feynman-Kac represenaion formula has an imporan numerical implicaion. Indeed i opens he door o he use of Mone Carlo mehods in order o obain a numerical approximaion of he soluion of he parial differenial equaion (1.8). For sake of simpliciy, we provide he main idea in he case f = k =. Le ( X (1),..., X (k)) be an iid sample drawn in he disribuion of, and compue he mean: X,x ˆv k (, x) := 1 k k i=1 ( g X (i)). ). By he Law of Large Numbers, i follows ha ˆv k (, x) v(, x) P a.s. Moreover he error esimae is provided by he Cenral Limi heorem: k (ˆvk (, x) v(, x)) k N (, Var [ g ( X,x )]) in disribuion, and is remarkably independen of he dimension d of he variable X! Represenaion of he Dirichle problem Le D be an open subse of R d. he Dirichle problem is o find a funcion u solving: Au ku + f = on D and u = g on D, (1.12)

20 2 CHAPER 1. CONDIIONAL EXPECAION AND LINEAR PDEs where D denoes he boundary of D, and A is he generaor of he process X,x defined as he unique srong soluion of he sochasic differenial equaion X,x = x + µ(s, X,x s )ds + σ(s, X,x s )dw s,. Similarly o he he represenaion resul of he Cauchy problem obained in heorem 1.7, we have he following represenaion resul for he Dirichle problem. heorem 1.8. Le u be a C 2 soluion of he Dirichle problem (1.12). Assume ha k is nonnegaive, and { } E[τD] x <, x R d, where τd x := inf : X,x D. hen, we have he represenaion: [ ( u(x) = E g X,x τ x D ) e τx D k(x s)ds + τ x D ( f X,x ) e k(xs)ds d Exercise 1.9. Provide a proof of heorem 1.8 by imiaing he argumens in he proof of heorem 1.7. ]. 1.4 he Black-Scholes model he coninuous-ime financial marke Le be a finie horizon, and (Ω, F, P) be a complee probabiliy space supporing a Brownian moion W = {(W 1,..., W d ), } wih values in R d. We denoe by F = F W = {F, } he canonical augmened filraion of W, i.e. he canonical filraion augmened by zero measure ses of F. We consider a financial marke consising of d + 1 asses : (i) he firs asse S is locally riskless, and is defined by S ( ) = exp r u du,, where {r, [, ]} is a non-negaive adaped processes wih r d < a.s., and represens he insananeous ineres rae. (ii) he d remaining asses S i, i = 1,..., d, are risky asses wih price processes defined by he dynamics ds i S i = µ i d + d j=1 σ i,j dw j, [, ],

21 1.3. Connecion wih PDE 21 for 1 i d, where µ, σ are F adaped processes wih µi d+ σi,j 2 d < for all i, j = 1,..., d. I is convenien o use he marix noaions o represen he dynamics of he price vecor S = (S 1,..., S d ): ds = S (µ d + σ dw ), [, ], where, for wo vecors x, y R d, we denoe x y he vecor of R d wih componens (x y) i = x i y i, i = 1,..., d, and µ, σ are he R d vecor wih componens µ i s, and he M R (d, d) marix wih enries σ i,j. We assume ha he M R (d, d) marix σ is inverible for every [, ] a.s., and we inroduce he process λ := σ 1 (µ r 1),, called he risk premium process. Here 1 is he vecor of ones in R d. We shall frequenly make use of he discouned processes S := S S ( = S exp ) r u du, Using he above marix noaions, he dynamics of he process S are given by d S = S ( (µ r 1)d + σ dw ) = S σ (λ d + dw ) Porfolio and wealh process A porfolio sraegy is an F adaped process π = {π, } wih values in R d. For 1 i n and, π i is he amoun (in Euros) invesed in he risky asse S i. We nex recall he self-financing condiion in he presen framework. Le X π denoe he porfolio value, or wealh, process a ime induced by he porfolio sraegy π. hen, he amoun invesed in he non-risky asse is X π = X π π 1. Under he self-financing condiion, he dynamics of he wealh process is given by n i=1 πi dx π = n π i S i i=1 ds i + Xπ π 1 S ds. Le X π be he discouned wealh process X π ( := X π exp ) r u du,. hen, by an immediae applicaion of Iô s formula, we see ha d X = π σ (λ d + dw ),, (1.13)

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619 econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT

ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 94-9(5)634-4 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITY-LINKED LIFE INSURANCE

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

Stochastic Calculus and Option Pricing

Stochastic Calculus and Option Pricing Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes

More information

On the Role of the Growth Optimal Portfolio in Finance

On the Role of the Growth Optimal Portfolio in Finance QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 1441-8010 www.qfrc.us.edu.au

More information

Time Consistency in Portfolio Management

Time Consistency in Portfolio Management 1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process,

= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting Ornstein-Uhlenbeck or Vasicek process, Chaper 19 The Black-Scholes-Vasicek Model The Black-Scholes-Vasicek model is given by a sandard ime-dependen Black-Scholes model for he sock price process S, wih ime-dependen bu deerminisic volailiy σ

More information

Differential Equations in Finance and Life Insurance

Differential Equations in Finance and Life Insurance Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Optimal Time to Sell in Real Estate Portfolio Management

Optimal Time to Sell in Real Estate Portfolio Management Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

More information

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

More information

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen

More information

SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, E-mail: toronj333@yahoo.

SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, E-mail: toronj333@yahoo. SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.

More information

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model

OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model Appl Mah Opim (9) 59: 99 46 DOI.7/s45-8-95- OpimalCompensaionwihHiddenAcion and Lump-Sum Paymen in a Coninuous-Time Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract

Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium

More information

T ϕ t ds t + ψ t db t,

T ϕ t ds t + ψ t db t, 16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS

A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion

More information

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing

Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364-765X eissn 526-547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion

More information

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION

ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed

More information

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear

More information

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price

An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor

More information

LECTURE 7 Interest Rate Models I: Short Rate Models

LECTURE 7 Interest Rate Models I: Short Rate Models LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

nonlocal conditions.

nonlocal conditions. ISSN 1749-3889 prin, 1749-3897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.3-9 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed

More information

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

More information

Technical Appendix to Risk, Return, and Dividends

Technical Appendix to Risk, Return, and Dividends Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

A martingale approach applied to the management of life insurances.

A martingale approach applied to the management of life insurances. A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 Louvain-La-Neuve, Belgium.

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Multiprocessor Systems-on-Chips

Multiprocessor Systems-on-Chips Par of: Muliprocessor Sysems-on-Chips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,

More information

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu

More information

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.

A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B-1348, Louvain-La-Neuve

More information

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach Working Paper 5-81 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 Opion-Pricing in

More information

Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006

Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006 i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26 ii Conens Noaion vii 1 Srucural Approach 3 1.1 Basic Assumpions.....................................

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

Life insurance cash flows with policyholder behaviour

Life insurance cash flows with policyholder behaviour Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

More information

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts The Uncerain Moraliy Inensiy Framework: Pricing and Hedging Uni-Linked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy

More information

Merton Problem with Taxes: Characterization, Computation, and Approximation

Merton Problem with Taxes: Characterization, Computation, and Approximation SIAM J. FINANCIAL MATH. Vol. 1, pp. 366 395 c 21 Sociey for Indusrial and Applied Mahemaics Meron Problem wih Taxes: Characerizaion, Compuaion, and Approximaion Imen Ben Tahar, H. Mee Soner, and Nizar

More information

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach

Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,

More information

Longevity 11 Lyon 7-9 September 2015

Longevity 11 Lyon 7-9 September 2015 Longeviy 11 Lyon 7-9 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univ-lyon1.fr

More information

When to Cross the Spread? - Trading in Two-Sided Limit Order Books -

When to Cross the Spread? - Trading in Two-Sided Limit Order Books - When o Cross he Spread? - rading in wo-sided Limi Order Books - Ulrich Hors and Felix Naujoka Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden 6, 0099 Berlin Germany e-mail: {hors,naujoka}@mah.hu-berlin.de

More information

Optimal Life Insurance Purchase, Consumption and Investment

Optimal Life Insurance Purchase, Consumption and Investment Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.

More information

12. Market LIBOR Models

12. Market LIBOR Models 12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,

More information

Introduction to Arbitrage Pricing

Introduction to Arbitrage Pricing Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, -661 Warszawa, Poland

More information

Monte Carlo Observer for a Stochastic Model of Bioreactors

Monte Carlo Observer for a Stochastic Model of Bioreactors Mone Carlo Observer for a Sochasic Model of Bioreacors Marc Joannides, Irène Larramendy Valverde, and Vivien Rossi 2 Insiu de Mahémaiques e Modélisaion de Monpellier (I3M UMR 549 CNRS Place Eugène Baaillon

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

Modelling of Forward Libor and Swap Rates

Modelling of Forward Libor and Swap Rates Modelling of Forward Libor and Swap Raes Marek Rukowski Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology, -661 Warszawa, Poland Conens 1 Inroducion 2 2 Modelling of Forward Libor

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

1. Introduction. We consider a d-dimensional stochastic differential equation (SDE) defined by

1. Introduction. We consider a d-dimensional stochastic differential equation (SDE) defined by SIAM J. CONROL OPIM. Vol. 43, No. 5, pp. 1676 1713 c 5 Sociey for Indusrial and Applied Mahemaics SENSIIVIY ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES, AND APPLICAION O SOCHASIC OPIMAL CONROL

More information

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

PRICING and STATIC REPLICATION of FX QUANTO OPTIONS PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of

More information

Conditional Default Probability and Density

Conditional Default Probability and Density Condiional Defaul Probabiliy and Densiy N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Absrac This paper proposes differen mehods o consruc condiional survival processes, i.e, families of maringales decreasing

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Working Paper When to cross the spread: Curve following with singular control

Working Paper When to cross the spread: Curve following with singular control econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Naujoka, Felix; Hors,

More information

Almost-sure hedging with permanent price impact

Almost-sure hedging with permanent price impact Almos-sure hedging wih permanen price impac B. Bouchard and G. Loeper and Y. Zou November 3, 215 Absrac We consider a financial model wih permanen price impac. Coninuous ime rading dynamics are derived

More information

Lectures # 5 and 6: The Prime Number Theorem.

Lectures # 5 and 6: The Prime Number Theorem. Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

More information

Cash-Lock Comparison of Portfolio Insurance Strategies

Cash-Lock Comparison of Portfolio Insurance Strategies Cash-Lock Comparison of Porfolio Insurance Sraegies Sven Balder Anje B. Mahayni This version: May 3, 28 Deparmen of Banking and Finance, Universiy of Bonn, Adenauerallee 24 42, 533 Bonn. E-mail: sven.balder@uni-bonn.de

More information

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.

UNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment. UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information

On Valuing Equity-Linked Insurance and Reinsurance Contracts

On Valuing Equity-Linked Insurance and Reinsurance Contracts On Valuing Equiy-Linked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics,

More information

Applied Intertemporal Optimization

Applied Intertemporal Optimization . Applied Ineremporal Opimizaion Klaus Wälde Universiy of Mainz CESifo, Universiy of Brisol, UCL Louvain la Neuve www.waelde.com These lecure noes can freely be downloaded from www.waelde.com/aio. A prin

More information

Optimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach

Optimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach 28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 11-13, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

Optimal market dealing under constraints

Optimal market dealing under constraints Opimal marke dealing under consrains Eienne Chevalier M hamed Gaïgi Vahana Ly Vah Mohamed Mnif June 25, 2015 Absrac We consider a marke dealer acing as a liquidiy provider by coninuously seing bid and

More information

Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficient Risk Sharing with Limited Commitment and Hidden Storage Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing

More information

Market Completion and Robust Utility Maximization

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

More information

adaptive control; stochastic systems; certainty equivalence principle; long-term

adaptive control; stochastic systems; certainty equivalence principle; long-term COMMUICATIOS I IFORMATIO AD SYSTEMS c 2006 Inernaional Press Vol. 6, o. 4, pp. 299-320, 2006 003 ADAPTIVE COTROL OF LIEAR TIME IVARIAT SYSTEMS: THE BET O THE BEST PRICIPLE S. BITTATI AD M. C. CAMPI Absrac.

More information

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

arxiv:submit/1578408 [q-fin.pr] 3 Jun 2016

arxiv:submit/1578408 [q-fin.pr] 3 Jun 2016 Derivaive pricing for a muli-curve exension of he Gaussian, exponenially quadraic shor rae model Zorana Grbac and Laura Meneghello and Wolfgang J. Runggaldier arxiv:submi/578408 [q-fin.pr] 3 Jun 206 Absrac

More information

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and Black-Meron-Scholes

More information

Arbitrage-free pricing of Credit Index Options. The no-armageddon pricing measure and the role of correlation after the subprime crisis

Arbitrage-free pricing of Credit Index Options. The no-armageddon pricing measure and the role of correlation after the subprime crisis Arbirage-free pricing of Credi Index Opions. The no-armageddon pricing measure and he role of correlaion afer he subprime crisis Massimo Morini Banca IMI, Inesa-SanPaolo, and Dep. of uan. Mehods, Bocconi

More information

OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS

OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension

More information

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach

Optimal Consumption and Insurance: A Continuous-Time Markov Chain Approach Opimal Consumpion and Insurance: A Coninuous-Time Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Optimal Reinsurance/Investment Problems for General Insurance Models

Optimal Reinsurance/Investment Problems for General Insurance Models Opimal Reinsurance/Invesmen Problems for General Insurance Models Yuping Liu and Jin Ma Absrac. In his paper he uiliy opimizaion problem for a general insurance model is sudied. he reserve process of he

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

A Production-Inventory System with Markovian Capacity and Outsourcing Option

A Production-Inventory System with Markovian Capacity and Outsourcing Option OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030-364X eissn 1526-5463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A Producion-Invenory Sysem wih Markovian Capaciy

More information

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments

BALANCE OF PAYMENTS. First quarter 2008. Balance of payments BALANCE OF PAYMENTS DATE: 2008-05-30 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se

More information