Probabilistic Representation and Approximation for Coupled Systems of Variational Inequalities

Transcription

1 Probabiliic Repreenaion and Approximaion for Coupled Syem of Variaional nequaliie Romuald Elie, dri Kharroubi To cie hi verion: Romuald Elie, dri Kharroubi. Probabiliic Repreenaion and Approximaion for Coupled Syem of Variaional nequaliie. Saiic and Probabiliy Leer, Elevier, 2010, 80 (17-18), pp <hal v2> HAL d: hal hp://hal.archive-ouvere.fr/hal v2 Submied on 4 Mar 2011 HAL i a muli-diciplinary open acce archive for he depoi and dieminaion of cienific reearch documen, wheher hey are publihed or no. The documen may come from eaching and reearch iniuion in France or abroad, or from public or privae reearch cener. L archive ouvere pluridiciplinaire HAL, e deinée au dépô e à la diffuion de documen cienifique de niveau recherche, publié ou non, émanan de éabliemen d eneignemen e de recherche françai ou éranger, de laboraoire public ou privé.

2 Probabiliic Repreenaion and Approximaion for Coupled Syem of Variaional nequaliie Romuald ELE dri KHARROUB CEREMADE, CNRS, UMR 7534, LPMA, CNRS, UMR 7599, Univerié Pari-Dauphine, Univerié Pari 7, and CREST and CREST May 2010 Fir verion: Sepember 2009 Abrac Our udy i dedicaed o he probabiliic repreenaion and numerical approximaion of oluion of coupled yem of variaional inequaliie. We inerpre he unique vicoiy oluion of a coupled yem of variaional inequaliie a he oluion of a one-dimenional conrained BSDE wih jump. Thi new repreenaion allow for he inroducion of a naural probabiliic numerical cheme for he reoluion of hee yem. Key word: BSDE wih jump, variaional inequaliie, vicoiy oluion, Mone Carlo imulaion, wiching problem. MSC Claificaion (2000): 93E20, 60H10, 60H30, 35K85, 49L25. 1 nroducion Pardoux and Peng (1992) developed he heory of backward ochaic differenial equaion, providing a probabiliic repreenaion of oluion of quai-linear parabolic PDE. Coupling he diffuion proce wih a pure jump proce, Pardoux e al. (1997) exend hi repreenaion o yem of coupled emilinear PDE wih differen linear differenial operaor on each line. nroducing rericion on he domain of he backward proce, El Karoui e al. (1997) cover he cla of variaional inequaliie. Conraining inead he jump par of he oluion, Kharroubi e al. (2010) conider quailinear variaional inequaliie. The focu of hi noe i o exend hi ype of Feynman-Kac repreenaion o he more general cla of coupled yem of quailinear variaional inequaliie ariing, for example, in opimal impule or wiching problem. We will ypically conider yem of PDE of he form v i Li v i f(i,.,(v k ) 1 k m,σ(i,.) D x v i ) min 1 j m h(i,j,.,v i,v j,σ(i,.) D x v i ) = 0, (1.1) on 0,T) R d, wih erminal condiion v i (T,.) = g(i,.) on R d, (1.2) Acknowledgemen. We would like o hank boh anonymou referee for ueful commen. 1

3 where, for any i := {1,...,m}, L i i a linear econd order local operaor L i v i (,x) := b(i,x) D x v i (,x)+ 1 2 r(σσ (i,x)d 2 xv i (,x)), (1.3) and b, σ, f, h and g are Lipchiz coninuou funcion. A oberved by Bouchard (2009), hi PDE appear in he reoluion of opimal wiching problem a well a ochaic arge problem wih jump. The major difficuly arie from he coupling beween all he componen (v i ) i m of he oluion and he ue of differen linear operaor a each line. When m i large, he numerical reoluion of (1.1)-(1.2) by claical PDE approximaion mehod i very ricky and highly compuaional. We inend o provide here a probabiliic repreenaion o (1.1)-(1.2) leading o an efficien probabiliic numerical cheme. When b and σ are independen of he regime i and he conrain funcion are of he form h : (i,j,.,y i,y j,.) y i y j c i,j, Hu and Tang (2007) inerpre he vecor oluion o (1.1)-(1.2) a a muli-dimenional BSDE wih erminal condiion and oblique reflecion. The challenging derivaion of a convergen numerical approximaion for hi ype of BSDE i of grea inere and i currenly under udy. The approach of hi paper relie inead on a recen reinerpreaion of obliquely muli-dimenional refleced BSDE in erm of one-dimenional conrained BSDE wih jump, a inroduced in Elie and Kharroubi (2009). The idea i o conider, a in Pardoux e al. (1997), a random regime driven by a pure jump ranmuaion proce, allowing o rerieve imulaneouly ome informaion concerning all he componen of he oluion. Given ad-dimenional Brownian moion W andanindependenpoion meaureµon R +, we conider, for any iniial condiion e := (,i,x) 0,T R d =: E, he unique R d -valued oluion (,X e ) e of he SDE: { = i+ (j r )µ(dr,dj) X = x+ b( r,x r )dr + σ(, T. (1.4) r,x r ) dw r Formally, given a mooh oluion (v i ) i o (1.1)-(1.2), he proce Y := v e(.,x ẹ ) aifie Y = g( e T,X e T)+ T f( e,x e,y +U,Z )d+k T K T Z dw T U (j)µ(d,dj) on 0,T, where we denoe Z := σ ( e,x e )D x v e (,X e ), U (.) := v. (,X e ) v e (,X e ), and K := 0 v u e L e u v e u f(u e,.,(v k) 1 k m,σ (u e,.)d xv e u )(u,xu e )du. Since v aifie (1.1), we expec he following conrain o be aified: (1.5) h( e,j,xe,y,y +U (j),z ) 0, j, T. (1.6) The BSDE (1.5) combined wih conrain (1.6) fall ino he cla of conrained BSDE wih jump and admi a unique minimal oluion under mild condiion on he coefficien. We reinerpre he Y-componen of he oluion a he unique vicoiy oluion o he coupled yem of variaional inequaliie (1.1)-(1.2). Thi new Feynman-Kac repreenaion i meaningful o he BSDE lieraure ince: 2

4 exend he reul of Kharroubi e al. (2010) o more general conrain and driver funcion depending on U. Thi allow for a rong coupling beween he dynamic of he value funcion componen and give a minimaliy condiion in ome paricular cae. generelize he concluion of Peng and Xu (2007) derived in he no-jump cae. offer a PDE repreenaion o refleced BSDE wih inerconneced obacle inroduced in Hamadène and Zhang (2008) ince hey relae direcly o conrained BDSE wih jump, ee Elie and Kharroubi (2009). generalize he ue of diffuion-ranmuaion proce in Pardoux e al. (1997) o yem of variaional inequaliie. Thi repreenaion lead o a naural probabiliic algorihm for he reoluion of (1.1)-(1.2). The conrained BSDE wih jump i replaced by a penalized BSDE wih jump, which i approximaed by he dicree-ime cheme udied in Bouchard and Elie (2008) and Gobe e al. (2006).Thi lead o a convergen numerical cheme baed on ime dicreizaion, Mone Carlo imulaion and projecion. The re of he paper i organized a follow. n Secion 2, we dicu exience, uniquene and penalizaion, and give a minimaliy condiion for conrained BSDE wih jump (1.5)-(1.6). Secion 3 preen he vicoiy properie and he numerical approximaion i deailed in he la ecion. Noaion. Throughou hi paper, we are given a finie horizon T and a probabiliy pace (Ω,G,P) endowed wih a d-dimenional andard Brownian moion W = (W ) 0, and an independen Poion random meaure µ on R +, wih ineniy meaure λ(di)d for ome poiive finie meaure λ on := {1,...,m}. We denoe E := 0,T R d. For a mooh funcion ϕ : 0,T R d R, ϕ, D xϕ and Dx 2 ϕ denoe rep. he derivaive of ϕ w.r.., he gradien and he Heian marix of ϕ w.r.. x. The dependence in ω Ω i omied when i i obviou. 2 Conrained Forward Backward SDE wih jump We preen in hi ecion he conrained forward backward SDE wih jump and recall he exience and uniquene reul of Elie and Kharroubi (2009). We dicu he correpondence beween he value funcion aociaed o Y and he U componen of he oluion. Under addiional regulariy of he value funcion, we provide a Skorohod ype minimaliy condiion for he conidered BSDE. 2.1 Exience and uniquene of a minimal oluion via penalizaion A dicued above, he forward proce i a ranmuaion-diffuion proce compoed of a pure jump proce and a diffuion wihou jump X whoe dynamic depend on. For any iniial condiion e := (,i,x) E, ( e,x e ) i he unique oluion o (1.4) aring from (i,x) a ime. Foranyiniialcondiione E, aoluionoheconrainedbsdewihjumpiaquadruple (Y e,z e,u e,k e ) S 2 L 2 W L2 µ A2 aifying (1.5)-(1.6), where S 2 i he e of real valued G-adaped càdlàg procee Y on 0,T.. Y S 2 := E up 0 r T Y r 21 2 <, 3

5 ( L p )1 W i he e of progreive Rd -valued procee Z.. Z L p := E T W 0 Z r p p dr <, p 1, L p µ i he e of P σ() meaurable map U : Ω 0,T R.. U L 2 µ := E T 0 U (j) 2 λ(dj)d 1 p <, p 1, A 2 i he cloed ube of S 2 compoed by nondecreaing procee K wih K 0 = 0. Furhermore, (Y,Z,U,K) i referred o a he minimal oluion o (1.5)-(1.6) whenever we have Y Y a.., for any oher oluion (Y,Z,U,K ). n order o enure exience and uniquene of a minimal oluion o (1.5)-(1.6) for any iniial condiion, we make he following aumpion. (H0) The following hold: (i) There exi a conan L.. f(i,x,(u j ),z) f(i,x,(u j ),z ) L (z,(u j ) ) (z,(u j ) ), h(i,j,x,y,u j,z,j) h(i,j,x,y,u j,z ) L (y,z,u j ) (y,z,u j ), for all (x,i,j,y,z,u,y,z,u ) R d 2 R R d R 2, and f(i,x,(u j ),z) + h(i,j,x,y,u j,z) L ( 1+ (y,z,(u j ) ) ), for all (x,i,j,y,z,(u i ) i ) R d 2 R R d R. (ii) The funcion h(i,j,x,y,.,z) i non-increaing for all (i,x,y,z,j) R d R R d. (iii) There exi wo conan C 1 C 2 > 1 and a meaurable map γ : R d R R d R 2 C 2,C 1 uch ha, for any (i,x,y,z,u,u ) R d R R d R 2, f(i,x,y +u,z) f(i,x,y +u,z ) (u j u j )γ(i,x,y,z,u,u,j)λ(dj). (H1) For any e = (,i,x) E, here exi a quadruple (Ỹ e, Z e,ũe, K e ) S 2 L 2 W L2 µ A 2 oluion o (1.5)-(1.6), wih Ỹ e = ṽ e (,X), e for ome deerminiic funcion ṽ aifying ṽ i (,x) C(1+ x ) on E. We provide in Remark 3.2 a more racable ufficien condiion under which (H1) hold. The conrucion of he minimal oluion i done by penalizaion. For any iniial condiion e E and n N, we inroduce (Y e,n,z e,n,u e,n ) oluion o he following penalized BSDE T T T Y = g(t e,xe T )+ f( e,xe,y +U,Z )d U (j)µ(d,dj) Z dw T + n h(,j,x e,y e,y +U (j),z ) λ(dj)d, 0 T. (2.1) Under (H0), we ge from Barle e al. (1997) exience and uniquene of a oluion of (2.1). We inroduce K e,n :=. 0 h(e,j,xe e,n e,n,y,y +Ue,n (j),z e,n ) λ(dj)d, for any (e,n) E N. 4

6 Theorem 2.1. Suppoe (H0)-(H1) hold. For any e := (, i, x) E, here exi a unique quadruple (Y e,z e,u e,k e ) S 2 L 2 W L2 µ A2 minimal oluion o (1.5)-(1.6) wih K e predicable, and v i : (,x) Y,i,x define a deerminiic map from E ino R. Moreover (Y e,z e,u e ) i he limi of he (Y e,n,z e,n,u e,n ) n N in he following ene Y e,n Y e L 2 W + Z e,n Z e L p W + U e,n U e L p µ 0, n, 1 p < 2. Proof. Thi reul i a direc applicaion of Theorem 2.1 in Elie and Kharroubi (2009). Under addiional regulariy on Y e, we can improve he previou convergence up o p = 2. Propoiion 2.1. f (H0)-(H1) hold, (Y e,n ) n N converge increaingly o Y e, for any e E. Addiionally, if he proce Y e i quai-lef coninuou in ime, we have Y e Y e,n S 2 + Ze Z e,n L 2 + U e U e,n L + K e K e,n S 2 0, e E. (2.2) W 2 µ n Proof. Fix e E and oberve from Propoiion 2.1 in Elie and Kharroubi (2009) ha Y e,n converge increaingly o Y e. Since µ i a Poion meaure, he proce Y e,n i quai-lef coninuou. f Y e ha he ame regulariy, he predicable projecion of Y e and Y e,n are imply given by (Y e ) and (Y e,n ). Thi lead o Y e = lim n Y e,n. We deduce from he weak verion of Dini heorem, ee Dellacherie and Meyer (1980) p. 202, ha Y e,n converge uniformly o Y e on 0,T, and he dominaed convergence heorem give u Y e Y e,n S 2 0. Combined wih andard eimae of he form n Z e,n+p Z e,n 2 L 2 W + U e,n+p U e,n 2 L 2 µ + K e,n+p K e,n 2 S 2 C Y e,n+p Y e,n 2 S 2, hi implie ha he equence (Z n ), (U n ) and (K n ) are Cauchy and hence convergen. Remark 2.1. Under he addiional Aumpion (H2) below, (v i ) i i inerpreed a he unique vicoiy oluion o (1.1)-(1.2), ee Theorem 3.2. n hi cae, (v i ) i i coninuou, Y = v (,X ) i quai-lef coninuou and Propoiion 2.1 hold. We denoe by (v n ) n N he equence of deerminiic funcion defined by v n : e E Y e,n and we hall ue indifferenly he noaion v n (,i,x) or vi n (,x), for (,i,x) E. Under (H0)- (H1), we know from Propoiion 2.1 ha v i he poinwie limi of (v n ) n N. 2.2 Repreenaion of U and he minimaliy condiion Propoiion 2.2. Le (H0)-(H1) hold. For any e E and opping ime θ valued in,t, we have Yθ e = v θ e(θ,xe θ ), and he proce U repreen a U e (j) = v j (,X e ) v e (,X e ), j, T. (2.3) Proof. According o Propoiion 2.1, we imply need o provide imilar repreenaion for he penalized BSDE (2.1). Fix e E. For any opping ime θ valued in,t, uniquene of oluion of (2.1) and hemarkov properyof ( e,x e ) direcly give oy e,n θ = v n θ(θ,x e e θ ). Denoing Ũ e,n (j) := vj n(,xe ) v n e (,Xe ), for j and 0 T, we deduce from (2.1) ha 5

7 Ũ e,n (j)µ(d,dj) = Y e,n Y e,n = Ũ e,n (j)µ(d,dj), 0 T. T Therefore E 0 (Ue,n (j) U e,n (j)) 2 λ(dj)d = 0 and he proof i complee. Under an exra regulariy aumpion on he funcion v aified under Aumpion (H2) below, he previou repreenaion lead o a Skorohod ype minimaliy condiion for (1.5)-(1.6). Corollary 2.1. Le (H0)-(H1) hold. Suppoe (v i ) i i coninuou and he funcion h doe no depend on z. Then, for any e E, he minimal oluion (Y e,z e,u e,k e ) aifie T h( e,j,xe,y e,y e +Ue dk (j)) e = 0. (2.4) min Proof. Fixe E. Since(v i ) i iconinuou, heprocey e inherihequai-lefconinuiy of ( e,x e ). Combining (2.3) and Propoiion 2.1 lead o max U e (j) U e,n (j) 0. S 2 n We deduce from (2.2) and Lemma 5.8 in Gegoux-Pei and Pardoux (1995), which alo hold for càglàd funcion, ha T min h( e,j,xe Since T min e,n e,n,y,y +Ue,n (j)) h( e,j,x e,y e,n dk e,n T min n h( e,j,xe,y e,y e +Ue (j)) dk e. e,n,y +Ue,n (j)) dk e,n 0 and (1.6) hold, we ge (2.4). 3 Link wih coupled yem of variaional inequaliie n hi ecion, we inerpre he minimal oluion of (1.5)-(1.6) a he unique vicoiy oluion of he PDE (1.1)-(1.2), hu generalizing he repreenaion derived in Kharroubi e al. (2010), Pardoux e al. (1997) and Peng and Xu (2007). 3.1 Vicoiy properie of he penalized BSDE The penalized parabolic inegral parial differenial equaion (PDE) aociaed o (2.1) i naurally defined for each n N by ϕ i Li ϕ i f(i,.,(ϕ j ),σ ( (i,.)d x ϕ i ) n h i,j,.,ϕi,ϕ j,σi D ) λ(dj) xϕ i = 0 (3.1) on 0,T) R d, and v i (T,.) = g(i,.) on R d, where L i he m-dimenional Dynkin operaor aociaed o X, defined in (1.3). Since he penalized BSDE fall ino he cla of BSDE wih jump udied by Pardoux e al. (1997), we deduce he following Feynman-Kac repreenaion reul. Propoiion 3.1. Under (H0)-(H1), he funcion (v n ) n are coninuou vicoiy oluion of (3.1). ndeed, for any n N, v n (T,.) = g and, for any (i,,x) 0,T) R d and ϕ C 1,2 (0,T R d ) uch ha (,x) i a global minimum (rep. max.imum) of (vi n ϕ), we have ϕ Li ϕ f(i,.,(vj n ),σ (i,.)d x ϕ) n h(i,j,.,vi n,vj,σ n (i,.)d x ϕ) λ(dj) (,x) (rep. ) 0. 6

8 Proof. Fix n N. The coninuiy of v n follow from imilar argumen a in he proof of Lemma 2.1 in Pardoux e al. (1997). According o he repreenaion deailed in he proof of Propoiion 2.2, he vicoiy propery of v n fi in he framework of Theorem 4.1 in Pardoux e al. (1997), up o he comparion heorem for BSDE, which i replaced by Theorem 2.5 in Royer (2006). 3.2 Vicoiy properie of he conrained BSDE wih jump Formally, paing o he limi in (3.1) when n goe o infiniy, we expec v o be a oluion of (1.1) on 0,T) R d. A for he boundary condiion, we canno expec o have v(t,.) = g, and we hall conider he relaxed boundary condiion given by ( min v i g(i,.), min h i,j,.,v i,v j,σ (i,.)d x v i )(T,x) = 0 on R d. (3.2) Remark 3.1. n he paricular cae where he driver funcion f i independen of (y,z,u) and he conrain funcion i given by h : (i,j,x,y,y+v,z) c i,j v wih c a given co funcion, we rerieve he yem of variaional inequaliie aociaed o wiching problem min min v i v j c i,j v i Li v i f(i,.), min = 0, on 0,T) R d, (3.3) v i g(i,.), min vi v j c i,j (T,.) = 0, on R d. (3.4) Thu, if (3.4) aifie a comparion heorem, v(t,.) i he malle funcion greaer han g aifying (3.4). n paricular, we rerieve he erminal condiion v(t,.) = g propoed by Hu and Tang (2007) when he erminal condiion g aifie he co conrain. n order o define vicoiy oluion of (1.1)-(3.2), we inroduce, for any locally bounded vecor funcion (u i ) i on 0,T R d i lower emiconinuou and upper emiconinuou (lc and uc for hor) envelope u and u defined for (,x) 0,T R d by u (,x) = liminf (,x ) (,x), <T u(,x ), and u (,x) = limup u(,x ). (,x ) (,x), <T Definiion 3.1. A vecor funcion (u i ) i, lc (rep. uc) on 0,T) R d, i called a vicoiy uperoluion (rep. uboluion) o (1.1)-(3.2) if, for each (i,,x) 0,T R d and ϕ C 1,2 (0,T R d ) uch ha (,x) i a global minimum (rep. maximum) of (u i ϕ), we have, if < T, min ϕ if = T, min h(i,j,.,u i,u j,σ (i,.)d x ϕ) Li ϕ f(i,.,(u j ),σ (i,.)d x ϕ),min u i g(i,.), min h(i,j,.,u i,u j,σ (i,.)d x ϕ) (T,x) ( rep. ) 0. A locally bounded vecor funcion (u i ) i on 0,T) R d i called a vicoiy oluion o (1.1)-(3.2) if u and u are repecively vicoiy uperoluion and uboluion o (1.1)-(3.2). Theorem 3.1. Under (H0)-(H1), he funcion v i a (diconinuou) vicoiy oluion o (1.1)-(3.2). (,x) (rep. )0, 7

9 Proof. Fir, following he proof of Lemma 3.3 and Remark 3.2 in Kharroubi e al. (2010), andard eimae on he penalized BSDE (2.1) lead o E up 0,T Y e,n 2 ( C 1+E g(t,x e T) e 2 + T X e 2 d+ up ṽ e (,X) e 2), e E. 0,T Combining Faou lemma wih andard eimae on X and linear growh condiion on g and ṽ, ee (H1), wege ha up 0,T v i (,x) 2 C(1+ x 2 ) wihc > 0. Thu, v i locally bounded. We oberve ha he vicoiy propery of v in he inerior of he domain i baed on he ame argumen a he one preened in he proof of Theorem 4.1 of Kharroubi e al. (2010). The only difference come from he more general form of he coefficien f and h. Thi i no a relevan iue here ince hey are coninuou. n order o alleviae he preenaion of he paper, we chooe o omi i here and only prove he vicoiy propery (3.2) on he mauriy boundary. (i) Le u fir conider he uperoluion propery of v o (3.2). Le (i,x 0 ) R d and ϕ C 1,2 (0,T R d ) uch ha (T,x 0 ) i a null global minimum of (v i ϕ). Paing o he limi of he vicoiy properie of he penalized BSDE, we ge min h(i,j,x,v i,v j,σ (i,.)d x ϕ)(t,x 0 ) 0. Furhermore v n (T,.) = g, n N, o ha he monoonic propery of he equence of coninuou funcion (v n ) n N give v (T,.) g. Therefore v i a vicoiy uperoluion of (3.2). (ii) We now urn o he uboluion propery of v. We argue by conradicion and uppoe he exience of (i,x 0 ) R d and ϕ C 1,2 (0,T R d ) uch ha 0 = (vi ϕ)(t,x 0) = max (v 0,T R d i ϕ), (3.5) and min ϕ g(i,.), min h(i,j,.,ϕ,vj,σ (i,.)d x ϕ) (T,x 0 ) =: 2ε > 0. Theregulariy ofv, ϕ and D x ϕ a well a he monoonic propery of h lead o he exience of an open neighborhood O of (T,x 0 ) 0,T R d, and Υ,r > 0 uch ha for all (,x,η,η ) O ( Υ,Υ) B(0,r), we ge min ϕ η g(i,.), min h(i,j,.,ϕ η,v j,σ (i,.)d x ϕ+η ) (,x) ε. (3.6) Weinroduceaequence( k,x k ) k valuedin0,t) R d aifying( k,x k ) (T,x 0 )andv i ( k,x k ) vi (T,x 0). Le u chooe δ > 0 uch ha k,t B(x k,δ) O for k large enough, and inroduce he modified e funcion ϕ k given by ϕ k (,x) := ϕ(,x)+ (ζ x x k 2 ( ) x xk δ 2 +C k φ + ) T, δ where 0 < ζ < Υ δr, φ i a regular funcion in C 2 (R d ) uch ha φ B(0,1) 0, φ B(0,1) c > 0, φ(x) lim x 1+ x =, and C k > 0 i a conan o be deermined laer. Since (v ϕ k )(,x) (v ϕ)(,x) ζ x x k 2 for (,x) δ 2 k,t R d, we deduce from (3.5) ha (v ϕ k )(,x) ζ, for (,x) k,t B(x k,δ). Chooing C k large enough, he paricular form of he funcion φ lead o (v i ϕk )(,x) ζ 2, for (,x) B(x k,δ) c k,t. (3.7) 8

10 Thank o he T erm in he modified e funcion ϕ k, we deduce ha ϕk ( Li ϕ k f i,.,(vj +ϕk η vi 1 j=i),σ (i,.)d x ϕ k) (,x) 0, (3.8) for any (,x,η) k,t) B(x k,δ) ( Υ+ζ,Υ) andk large enough. We now chooe η < Υ ζ 2 ε and inroduce he opping ime θ k := inf { k ; X e k / B(x k,δ) or e k e k } T, where e k := ( k,i,x k ). Le u finally conider he proce (Y k,z k,u k,k k ) given on k,θ k by Y k := ϕ k (,X e k 1 ) η k,θ k ) +v e (θ k k,x e k θ k )1 =θk, Z k := σ ( e k,xe k )D xϕ k (,X e k ( ) ), U k := vj (,Xe k ) ϕk (,X e k 1 ) η e, j k K k := ( ϕ k e k k +L r ϕ )+f( k e k r,.,(vj +ϕk η v e k1 e ) r j= k,z k r r) (r,x e k r )dr k (ϕk η v j )(r,xe k r )µ(dr,dj) + ϕ k η v e k θ k (θ k,x e k θ k )1 =θk. One eaily check from (3.6)-(3.7)-(3.8) ha (Y k,z k,u k,k k ) i oluion o θk θk θk Y = v e k (θ k,x e k θ θ k )+ f( e k r,x e k r,y r +U r,z r )dr Z r dw r U r (j)µ(dr,dj) +K θk K r k on k,θ k, ogeher wih he conrain h( e k r,j,xe k r,y r,y r +U r (j),z r ) 0 a.e., j. Since (Y e k,z e k,u e k,k e k) i a minimal oluion o hi conrained BSDE wih jump, we deduce ϕ k ( k,x k ) η = ϕ( k,x k )+ T k η v i ( k,x k ), for all k large enough. Leing k go o infiniy, hi conradic (3.5) and conclude he proof. Remark 3.2. The main drawback of hi repreenaion i he neceiy of Aumpion (H1). Following imilar argumen a in he proof of Propoiion 6.3 in Kharroubi e al. (2010), oberve ha i i aified whenever here exi a Lipchiz funcion (w i ) i C 2 (R d ) uperoluion o (3.2) aifying a linear growh condiion, and here exi a conan C > 0 uch ha L i w i +f(.,(w j ),σ i Dw i) C on R d, i. 3.3 A comparion argumen n hi ecion, we provide ufficien condiion characerizing he value funcion v a he unique vicoiy oluion of (1.1)-(3.2). Thi give in paricular he coninuiy of v, leading o he rong convergence by penalizaion and he minimaliy condiion, preened in Secion 2. The proof relie a uual on a comparion argumen, which hold under he following addiional aumpion. (H2) The following hold: (i) For any i, f(i,.) i convex in ((y j ),z) and increaing in u i. (ii) For any i,j, h(i,j,.) i concave in (y i,y j,z) and decreaing in y i. (iii) There exi a nonnegaive vecor funcion (Λ i ) i C 2 (R d ) and a poiive conan ρ Λ uch ha, for all i, Λ i g i, lim i (x) x 1+ x = and we have : L i Λ i +f(i,.,(λ j ),σ (i,.)d x Λ i ) ρλ i and min h(i,j,.,λ i,λ j,σ (i,.)d x Λ i ) > 0. 9

11 An example where (H2) hold i given for he cae of opimal wiching in Bouchard (2009). Remark 3.3. A in Bouchard (2009), (iii) allow u o conruc a nice ric uperoluion of (1.1) allowing o conrol oluion of (1.1)-(3.2) by convex perurbaion. Following he approach of Kharroubi e al. (2010), he general form of f and h force u o add he exra convexiy aumpion (i) and (ii). Theorem 3.2. Le (H0)-(H1)-(H2) hold. Then, for any U lc (rep. V uc) vicoiy uperoluion (rep. uboluion) of (1.1)-(3.2) aifying U + V (,x) C(1+ x ) on 0,T R d, we have U i V i on 0,T R d, i. n paricular, v i coninuou and i i he unique vicoiy oluion of (1.1)-(3.2) aifying a linear growh condiion. We omi he proof of hi comparion heorem which i a naural exenion of Theorem 4.1 in Kharroubi e al. (2010). Following he argumen of he proof of Propoiion 3.3 in Peng and Xu (2007), v can ill be inerpreed a he minimal vicoiy oluion of (1.1)-(3.2) in he cla of funcion wih linear growh, whenever a comparion heorem for he PDE (3.1) hold. 4 Numerical iue The numerical reoluion of yem of variaional inequaliie of he form(1.1)-(1.2) uually relie on he ue of ieraed free boundary. We fir olve he yem wihou boundary condiion and conider recurively he yem conrained by he boundary condiion coming from he previou ieraion. n a wiching problem, we conrain he oluion aociaed o n + 1 poible wiche by he obacle buil from he oluion where only n wiche are allowed. Such a numerical approach i compuaionally demanding. We preen here a naural convergen algorihm baed on he approximaion of he oluion o he correponding conrained BSDE wih jump (1.5)- (1.6). We combine a penalizaion procedure wih he dicree-ime cheme udied by Bouchard and Elie (2008) and he aiical eimaion projecion preened in Gobe e al. (2006). Thank o he previou Feynman-Kac repreenaion, hi give rie o a convergen probabiliic algorihm olving coupled yem of variaional inequaliie. We fix an iniial condiion e E and omi i in he expreion for eae of preenaion. Suppoe ha (H0)-(H1)-(H2) hold. The algorihm i divided in hree ep. Sep 1. Approximaion by penalizaion. We fir approach he conrained BSDE wih jump (1.5)-(1.6) by i penalized verion (2.1) characerized by a driver f n := f nh a in Secion 2.1. We deduce from Propoiion 2.1 ha he penalizaion error converge o 0 a n goe o infiniy, ee (2.2). Sep 2. Time dicreizaion. Oberve ha he pure jump proce can be imulaed perfecly and denoe by (τ l ) l i jump ime on 0,T. We inroduce he Euler ime cheme approximaion X h of he forward proce X defined on he concaenaion ( l ) l of he regular ime grid { k := kh, k = 1,...,T/h} wih he jump (τ l ) of : X h 0 = X 0 and X h l+1 := X h l +b( l,x h l )( l+1 l )+σ( l,x h l )W l+1 W l. We deduce an approximaion Y n,h T of YT n a mauriy given by g T (XT h ). The penalized BSDE (2.1) can now be dicreized by an exenion of he cheme expoed in Bouchard and Elie (2008) 10

12 An approximaion of Y n a ime 0 i compued recurively following he backward cheme for k = T/h 1,,0 : Z n,h k := 1 h E k Y n,h k+1 (W k+1 W k ) U n,h k (i) := 1 h E k Y n,h µ(( k, k+1 {i}) k+1 λ(i), i Y n,h k := E k Y n,h k+1 + (4.1) k+1 k f n (,X h k,y n,h k+1,z n,h k,u n,h k )d where E k denoe he condiional expecaion wih repec o G k. Following he argumen of Secion 2.5 in Bouchard and Elie (2008) and idenifying (Y n,h,z n,h,u n,h ) a a proce conan on each inerval ( k, k+1, we verify he convergence of hi dicree-ime approximaion : Y n Y n,h S 2 + Zn Z n,h L 2 W + U n U n,h L 2 µ 0, n N. (4.2) h Sep 3. Approximaion of he condiional expecaion. The la ep coni in eimaing he condiional expecaion operaor E k ariing in (4.1). We adop here he approach of Longaff-Schwarz generalized in Gobe e al. (2006) relying on lea quare regreion. FixN NandimulaeN independencopieofhebrownianincremen(w j k+1 W j k ) 0 k T/h and he poion meaure ( µ j (( k, k+1 ) 0 k T/h. For each imulaion j N, define j N and a he rajecorie of and X h. By inducion, one can eaily verify he Markov propery of he proce (Y n,h,z n,h,u n,h ) defined in (4.1): X h,n j Y n,h i = c n,h k ( i,x h i ), Z n,h i = a n,h k ( i,x h i ), U n,h i = b n,h k ( i,x h i ), for ome deerminiic funcion (a n,h k,bn,h k,cn,h k ) k n. The idea i o approximae hee funcion uing Ordinary Lea Square (OLS) eimaor. Given L N, we inroduce a collecion of bai funcion (a L l,bl l,cl l ) 1 l L of R R d R d. For each rajecory j N, define he aociaed := g N (X h,n k,n j, n ). Now we define recurively (Z n,h,l,n,u n,h,l,n j, k ), erminal value given by Y n,h,l,n j, n j, k backward in ime for k = T/h 1,,0, by compuing he OLS approximaion a follow: 1 (ˆα 1,, ˆα L ) := arg min α 1,,α L N (ˆβ,, ˆβ 1 L )(i) := arg min β 1,,β L N N 1 h Y n,h,l,n W j k+1 W j k j=1 N 1 h Y n,h,l,n j=1 for i, leading o he approximaion Z n,h,l,n j, k := L l=1 k+1 j, k+1 µ j (( k, k+1 {i}) λ(i) ˆα l a L l (N j, k,x h,n j, k ) and U n,h,l,n j, k (i) := remain o inroduce (ˆγ 1,,ˆγ L ) he minimizer of he mean quare error 1 N N k+1 Y n,h,l,n j, k+1 + j=1 k f n ( N j,,x h,n j, k,y n,h,l,n k+1,z n,h,l,n k L l=1 L α l a L l (N j, k,x h,n 2, l=1 l=1,u n,h,l,n k )d j, k ) L β l b L l (N j, k,x h,n 2, j, k ) ˆβ l (j)b L l (N j, k,x h,n j, k ), i. L γ l c L l (N j, k,x h,n l=1 j, k ) 2 in order o deduce he OLS approximaion Y n,h,l,n j, k := L l=1 ˆγ lc L l (N j, k,x h,n j, k ). 11

13 WereferoGobe e al. (2006) forheconrol of heaiical errordueoheapproximaion of he condiional expecaion operaor by OLS projecion, and, by exenion, Y n,h Y n,hl,n S 2 + Zn,h Z n,h,l,n L 2 W + U n,h U n,h,l,n L 2 µ 0, n N, h > 0. (4.3) N,L The convergence of he algorihm follow from (2.2), (4.2) and (4.3). The derivaion of a convergence rae require preciion on he influence of n on he dicreizaion and aiical error, a well a a conrol of he penalizaion error. Thi challenging poin i lef o furher reearch. Reference 1 Barle, G., Buckdahn, R., Pardoux, E., Backward ochaic differenial equaion and inegral-parial differenial equaion, Sochaic, 60, Bouchard, B., A ochaic arge formulaion for opimal wiching problem in finie horizon, Sochaic, 81 (2), Bouchard, B., Elie, R., Dicree-ime approximaion of decoupled forward-backward SDE wih jump, Sochaic Procee and heir Applicaion, 118, Dellacherie, C., Meyer, P.A., Probabilié e Poeniel, V-V, Hermann. 5 El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C., Refleced oluion of BSDE and relaed obacle problem for PDE, The Annal of Probabiliy, 25(2), Elie, R., Kharroubi,., Conrained Backward SDE wih Jump: Applicaion o Opimal Swiching, Preprin. hp://arxiv.org/ab/ Gobe, E., Lemor, J.P., Warin, X., Rae of convergence of empirical regreion mehod for olving generalized BSDE. Bernoulli, 12 (5), Gegou-Pei, A., Pardoux, E., Equaion différenielle ochaique rérograde réfléchie dan un convexe. Sochaic and Sochaic Repor, 57, Hamadène, S., Zhang, J., The Saring and Sopping Problem under Knighian Uncerainy and Relaed Syem of Refleced BSDE. To appear in Sochaic Procee and heir Applicaion. 10 Hu, Y., Tang, S., Muli-dimenional BSDE wih oblique Reflecion and opimal wiching, To appear in Probabiliy Theory and Relaed Field. 11 Kharroubi,., Ma, J., Pham, H., Zhang, J., Backward SDE wih conrained jump and Quai-Variaional nequaliie. The Annal of Probabiliy, 38, 2, Pardoux, E., Peng, S., Backward SDE and Quailinear Parabolic Parial Differenial Equaion, Lecure Noe in CS, 176, Springer. 13 Pardoux, E., Pradeille, F., Rao, Z., Probabiliic inerpreaion of a yem of emilinear parabolic PDE, Ann. n. H. Poincaré, ec. B, 33 (4), Peng, S., Xu, M., Conrained BSDE and vicoiy oluion of variaional inequaliie, Preprin. hp://arxiv.org/ab/ Royer, M., Backward ochaic differenial equaion wih jump and relaed nonlinear expecaion, Sochaic Procee and heir Applicaion, 116,