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

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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

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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)