Sochasic Processes and heir Applicaions 76 1998) 191 215 Bacward sochasic dierenial equaions wih subdierenial operaor and relaed variaional inequaliies Eienne Pardoux a;, Aurel Rascanu b;1 a Universie de Provence, LATP, URA-CNRS no. 225, 39 rue F. Jolio-Curie, 13453 Marseille, Cedex 13, France b Deparmen of Mahemaics, Al.I.Cuza Universiy, 66, Iasi, Romania Received 17 July 1997; received in revised form 3 April 1998; acceped 9 April 1998 Absrac The exisence and uniqueness of he soluion of a bacward SDE, on a random possibly innie) ime inerval, involving a subdierenial operaor is proved. We hen obain a probabilisic inerpreaion for he viscosiy soluion of some parabolic and ellipic variaional inequaliies. c 1998 Elsevier Science B.V. All righs reserved. Keywords: Bacward sochasic equaions; Subdierenial operaors; Variaional inequaliies; Viscosiy soluions; Probabilisic formulae for PDE. Inroducion Bacward sochasic dierenial equaions BSDE) provide probabilisic formulae for he viscosiy soluion of semilinear parial dierenial equaions PDE) see, in paricular, Pardoux, 1997; Pardoux and Peng, 1992, and heir references). In his paper one gives such formulae for parabolic variaional inequaliies on he whole space and also for he soluion of a Dirichle problem for an ellipic variaional inequaliy. We resric ourselves o variaional inequaliies for PDEs, and no sysems of PDEs. The only diculy in reaing general sysems concerns he diculy of giving a deniion of viscosiy soluion for such sysems. In he rs par of his paper we sudy BSDEs on a random possibly innie) ime inerval, whose coecien conains he subdierenial of a convex funcion. BSDEs wih subdierenial operaors include as a special case BSDEs whose soluion is reeced a he boundary of a convex subse of R. In he one-dimensional case, BDSEs wih one-sided reecion have been sudied in El Karoui e al. 1997), ogeher wih he associaed opimal sopping ime=opimal conrol problem, and an obsacle problem for a PDE also called variaional inequaliy ). BSDEs wih wo-sided Corresponding auhor. 1 The wor of his auhor was done during a visi o he Universiy of Provence & INRIA, whose generous suppor is graefully acnowledged. 34-4149/98/$19. c 1998 Elsevier Science B.V. All righs reserved PII: S34-414998)3-1
192 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 reecion, ogeher wih he associaed sochasic game of opimal sopping, are sudied in Cvianic and Karazas 1996). Muli-dimensional BSDEs reeced a he boundary of a convex se is sudied in Gegou-Pei and Pardoux 1996). Our BSDEs include his las class as a special case. Also, we prove ha he bounded variaion process o be added is absoluely coninuous, a resul which was no formulaed for all convex ses in Gegou-Pei and Pardoux 1996). However, our resuls do no include hose in El Karoui e al. 1997) and Cvianic and Karazas 1996), since hose resuls allow randomly moving boundaries, while our convex funcion is xed. Also, we do no sudy he sochasic conrol problem associaed wih our BSDE. The paper is organized as follows. The BSDEs and he resuls concerning hem are formulaed in Secion 1. Secion 2 is concerned wih a priori esimaes for sequences of penalized approximaions of our equaions. We prove in Secion 3 he resuls saed in Secion 1. In Secion 4, we prove ha he soluion of a BSDE provides he unique soluion of a cerain parabolic variaional inequaliy. Finally, in Secion 4 we sudy he connecion beween our BSDEs and he Dirichle problem for an ellipic variaional inequaliy. 1. Bacward sochasic variaional inequaliies: exisence and uniqueness resuls Le ; F;P;{F : }) be a complee righ coninuous sochasic basis. We will assume ha F = {B s : 6s6}) N; where N is he class of P-null ses of F and B is a d-dimensional sandard Brownian moion. Le R; ;d N and be a sopping ime. We inroduce he noaions: [; is he Banach space of coninuous F -progressively measurable sochasic processes f : [; ) R such ha [ ) 1=2 f S = E sup e f) 2 66 S 2; and M 2; [; is he Hilber space of F -progressively measurable sochasic processes f : [; ) R such ha [ 1=2 f M = E e s fs) ds) 2 : In he sequel, we shall omi he indices ; ; whenever, respecively, =1;= and = : For example, S 2 = S 2; 1 [; ) and M 2[;=M2; [;: The rs goal of his paper is o sudy he exisence and uniqueness of he soluion of he bacward sochasic dierenial equaion dy + F; Y ;Z )d @ Y )d+z db ; Y = ; 66; 1.1)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 193 where H 1 ) : [; ) is an a.s. nie F -sopping ime, he funcion F : [; ) R R d R saises for some R; ; ;and an F -progressively measurable process: H 2 ) i) F ; ;y;z)isf -progressively measurable; ii) y F!; ; y; z):r R is coninuous; iii) y y ;F; y; z) F; y ;z) 6 y y 2 ; F; y; z) F; y; z ) 6 z z ; F; y; ) 6)+ y for all ; y;y R ; z;z R d ; P-a.s.; @ is he subdierenial see below) of he funcion : R ; + which saises H 3 ) i) is a proper + ) convex lower-semiconinuous funcion; ii) y) )= 2 and nally is an R -valued F -measurable random variable, and here exiss 2+ 2, such ha H 4 ) Denoe i) ii) E[e 2 + ) ) ; ) E e s s) 2 ds : Dom = {u R : u) }; @ u)={u R :u ;v u)+ u)6 v); v R }; Dom @ )={u R : @ u) }; u; u ) @ u Dom @ ); u @ u): We remar ha he subdierenial operaor @ : R 2 R operaor, i.e. ha is a maximal monoone u v ;u v) ; u; u ); v; v ) @ : 1.2) In all wha follows, C denoes a consan, which may depend only on ;, and, which may vary from line o line. 2 This assumpion is no a resricion since we can replace u) by u+u ) u ) u ;u) where u ;u ) @ :
194 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 The main resul is given in he following heorem: Theorem 1.1. Le he assumpions H 1 ) H 4 ) be saised. Then here exiss a unique riple Y; Z; U) such ha Y S 2; [; M 2; [;; Z M 2; d [;; E e s Y s )ds + ; U M2; [;; 1.3a) 1.3b) Y ;U ) @ ; dp d a:e: on [;; 1.3c) Y + U s ds = + Fs; Y s ;Z s )ds Moreover; for any sopping ime ; 66; his soluion saises [ E e s Y s 2 + Z s 2 )ds 6C 1 ; ); where Z s db s ; ; a:s: 1.3d) 1.4a) [ E sup e Y 2 6C 1 ; ); 1.4b) 66 E[e Y )6C 2 ; ); [ E e s U s 2 ds 6C 2 ; ); [ 1; )=E e 2 + [ 2; )=E e 2 + )) + 1.4c) 1.4d) e s Fs; ; ) 2 ds ; 1.5a) e s s) 2 ds : 1.5b) The riple Y; Z; U) which saises Eqs. 1.3a), 1.3b), 1.3c) and 1.3d) will be called a soluion of BSDE 1:1) and we shall wrie Y; Z; U) BSDE ; ; ; F). Proposiion 1.1. Under he condiions of Theorem 1:1; if Y; Z; U) BSDE; ; ; F) and Ỹ; Z;Ũ) BSDE ; ; ; F); we have [ E e s Y s Ỹ s 2 + Z s Z s 2 )ds 6C); 1.6a) [ E sup e Y Ỹ 2 6C); 1.6b) 66 where [ )=E e 2 + e s Fs; Y s ;Z s ) Fs; Y s ;Z s ) 2 ds : 1.7)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 195 Remar 1.1. In he case where = T is a nie xed number, he same resuls hold, wih he same assumpions excep ha we need no assume ha 2 + 2, and we can choose =. Corollary 1.1. Le assumpions H 2 ); H 3 ) and 2 + 2 ; E e s s) 2 ds be saised. Then here exiss a unique riple Y; Z; U) S 2; M 2; ) M 2; such ha: T M 2; Y + T U s ds=y T + T lim Ee Y 2 )=; Y ;U ) @ ; dp d a:e: Moreover; [ ) E sup e Y 2 + sup Ee Y )+E T Fs; Y s ;Z s )ds Z s db s ; e s Y s 2 + Z s 2 )ds 6CE e s U s 2 6CE e s s) 2 ds: 1.8a) 1.8b) d 66T; P a:s: 1.9a) e s Fs; ; ) 2 ds; 1.9b) 1.9c) 1.1a) 1.1b) 2. A priori esimaes on a penalized equaion The exisence resul for Theorem 1.1 will be obained via an approximaion of he funcion by a convex C 1 -funcion ;, dened by u) = inf { 1 2 u v 2 + v): v R } = 1 2 u J u 2 + J u); 2.1) where J u =I+ @ ) 1 u): For he reader s convenience we menion some properies of his approximaion see Barbu, 1976 or Brezis, 1973 for more deails): 1 D u)= 1 @ u)= 1 u J u) @ J u); 2.2a) J u J v 6 u v and lim J u =Pr Dom u) 2.2b) for all u; v R ;. We rs noe ha he convexiy of implies ha for all u R, ) u)+d u); u):
196 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 Bu from H 3 ii) and he deniion of i follows easily ha u) = ). Hence, for all u R, 6 u)6d u);u): 2.2c) By Eq. 2.2a) and he monooniciy of he operaor @ we have 1 6 D u) 1 ) D v);j u J v 1 = D u) 1 ) D v);u v 1 D u) 2 1 1 D v) 2 + + 1 ) D u);d v)) and hen 1 D u) 1 ) 1 D v);u v + 1 ) D u) D v) 2.3) for all u; v R ; ;. Consider he approximaing equaion Y + 1 D Y s )ds=+ Fs; Y z ;Z s)ds Z s db s ; ; P a:s: 2.4) I follows from he resuls in Darling and Pardoux 1997) ha Eq. 2.4) has a unique soluion Y ;Z ) S 2; [; M 2; [;) M 2; d [;: Proposiion 2.1. Le assumpions H 1 ) H 4 ) be saised and le be a sopping ime such ha 66: Then [ E sup e Y 2 + e s Y s 2 + Z s 2 )ds 6C 1 ; ) 2.5) 66 wih 1 dened by Eq. 1.5a). Proof. Iô s formula for e Y 2 yields e ) Y 2 + e s Y s 2 + Z s 2 )ds+ 2 =e 2 +2 e s Fs; Y s ;Z s);y s )ds 2 Bu from Eq. 2.2c) ) 1 D y);y and from Schwarz s inequaliy e s D Y s );Y s )ds e s Y s ;Z s db s ): 2Fs; y; z);y)62 +1+r) 2 +r) y 2 + 1 1+r z 2 + 1 r Fs; ; ) 2 :
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 197 Hence, e ) Y 2 + 6e 2 + 1 r We choose 2 + 2 ; r 2 + 2 ) 1+ 2 1: e s [ 2 2 r1 + 2 )) Y s 2 + e s Fs; ; ) 2 ds 2 e s Y s ;Z s db s ); r 1+r Z s 2 ds ; a:s: 2.6) The resul wihou he sup in he expecaion follows by aing he expecaion in he above inequaliy. Finally, he resul follows by a combinaion wih Burholder Davis Gundy s inequaliy. Indeed, he rs sep yields, in paricular, ha E e s Z s 2 ds6c; and one hen obains sup e Y 2 6e 2 + 1 e s Fs; ; ) 2 ds + 2 sup 66 r Then, from Burholder Davis Gundy s inequaliy, E sup e Y )6C 2 1 +2E 66 and he resul follows. 6C 1 + 1 2 E sup 66 sup 66 66 ) e s Y s ;Z s db s ) e Y 2 ) + C 2 E e s Y s ;Z s db s ) : e s Z s 2 ds Proposiion 2.2. Under he condiions of Proposiion 2.1, here exiss a posiive consan C such ha for any sopping ime ; 2 1 E e s D Y s ) ) ds6c 2 ; ); 2.7a) Ee J Y )+E e s J Y )ds6c 2; ); Ee Y J Y ) 2 )6 2 C 2 ; ); 2.7b) 2.7c) where 2; ) is given by Eq. 1.5b). Proof. Wriing he subdierenial inequaliy e s Y s ) e s e r ) Y s )+e r Y r )+e r D Y r );Y s Y r )
198 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 for s = i+1 ; r = i ; where = 1 2 and i+1 i =1=n, summing up over i, and passing o he limi as n, we deduce: e Y )+ e s Y s )ds+ 1 e s D Y s ) 2 ds 6e )+ e s D Y s );Fs; Y s ;Z s)) ds e s D Y s );Z s db s ); ; a:s: 2.8) The resul follows by combining his wih he following inequaliies and Eq. 2.5) he righ side of he second inequaliy follows from Eq. 2.2c)) 1 2 D y) 2 + J y)= y); J y)6 y); y)6 y)6 D y);y); )6 ); D y); y + Fs; y; z)) 6 1 2 D y) 2 + y + Fs; y; z) )2 2 6 1 2 D y) 2 + 2 y 2 + Fs; y; z) 2 ) 6 1 2 D y) 2 + [ 2 y 2 +4 2 z 2 + 2 y 2 + 2 s)): Proposiion 2.3. Le assumpions H 1 ) H 4 ) be saised and ;. Then [ E e s Y s Ys 2 + Z s Zs 2 )ds 6 +)C ); where 2.9a) E sup e Y Y )6 2 + )C ); 2.9b) 66 [ )=E e 2 + )) + e s Fs; ; ) 2 ds : 2.1) Proof. By Iô s formula e ) Y Y 2 + e s [ Y s Y s 2 + Z s Z s 2 ds +2 e s Y s Ys ; 1 D Y s ) 1 ) D Ys ) ds
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 199 =2 Y s Ys ;Fs; Y s ;Z s) Fs; Ys ;Zs)) ds 2 We have, moreover, Y s Ys ; Z s Zs )db s ): 2Y Y ;Fs; Y ;Z ) Fs; Y ;Z )) 62 +1+r) 2 ) Y Y 2 + 1 1+r Z Z 2 and by Eq. 2:3b) i follows ha e ) Y Y 2 + e s [ 2 2 r 2 ) Y s Y s 2 + r 1 1+r Z s Zs 2 ds62 + 1 ) e s D Y s D Ys ) ds Now, from Eq. 2.7a) 1 2 + 1 ) E 2 e s Y s Ys ; Z s Zs )db s ): 2.11) e s D Y s D Y s ) ds6c + ) ): Eq. 2.9a) hen follows by aing he expecaion in Eq. 2.11), and Eq. 2.9b) follows from Eqs. 2.11), 2.9a) and Burholder Davis Gundy s inequaliy. 3. Proofs of he exisence and uniqueness of he soluion We begin wih he Proof of Proposiion 1.1. From Iô s formula we have e ) Y Ỹ 2 + +2 e s U s Ũ s ;Y s Ỹ s )ds e s Y s Ỹ s 2 + Z s Z s 2 )ds =e 2 +2 e s Y s Ỹ s ;Fs; Y s ;Z s ) Fs; Ỹ s ; Z s )) ds 2 e s Y s Ỹ s ;Z s Z s )db s ): 3.1)
2 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 Bu 2U s Ũ s ;Y s Ỹ s ) ; dp ds a:e:; 2Y Ỹ;Fs; Y; Z) Fs; Ỹ; Z))6 2 +1+r) 2 +r) Y Ỹ 2 + 1 1+r Z Z 2 + 1 r Fs; Y; Z) Fs; Y; Z) 2 wih r given by Eq. 2.6), where, are replaced by,. Wih hese inequaliies and Eq. 3.1), aing he expecaion, we clearly have Eq. 1.6a). Then in a sandard manner from Eqs. 3.1) and 1:6a), via Burholder Davis Gundy s inequaliy, we obain easily Eq. 1.6b). Proof of Theorem 1.1. Uniqueness is a consequence of Proposiion 1.1. The exisence of he soluion Y; Z; U) is obained as limi of he riple Y s ;Z s; 1 D Y s )): From Proposiion 2.3 we have Y S 2; [; M 2; [;; Z M 2; d s:: lim Y = Y in S 2; [; M 2; [;; lim Z = Z in M 2; d ; 3.2) and Eqs. 1.4a) and 1:4b) follows by passing o he limi in Eq. 2.5). Also, from Eqs. 2.7a) and 2:7c) we have lim J Y )=Y in M 2; [;; lim Ee J Y ) Y 2 )= for any sopping ime, 66: Eqs. 1.3b) and 1:4c) follow from Eqs. 2.7b), 2:9b) and he fac ha is l.s.c. Hence, he limi pair Y; Z) saises Eqs. 1.3a), 1:3b) and 1.4a) 1.4c). For each, dene U =1= )D Y ) and U = U s ds. I follows from our convergence resuls and Eq. 2.4) ha here exiss a progressively measurable R -valued process { U ; 66} such ha for all T, ) E sup U U 2 ; : 66T Moreover, from Eq. 2.7a), sup E e U 2 d : From his, i follows ha for each T, U is bounded in he space L 2 ; H 1 ;T )), and a leas along a susequence i converges wealy o a limi in ha space. The limi is necessarily U, hence he whole sequence converges wealy, and U L 2 ; H 1 ;T )), in paricular, i is a.s. absoluely coninuous, U aes he form U = U s ds, where {U ; 66} is progressively measurable. Now, Eq. 1.4d)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 21 follows from he above inequaliy and Faou s lemma. Moreover, i follows e.g. from Lemma 5.8 in Gegou-Pei and Pardoux 1996) ha for all 6a b6t, V M 2 a; b), b a U ;V Y )d a a U ;V Y )d in probabiliy, and from Eq. 2.7a) b a U ;J Y ) Y )d. Now, since U @ J Y )), b b b U ;V J Y )) d + J Y )) d6 V )d; a a a and aing he lim inf in probabiliy in he above, we obain ha b b b U ;V Y )d+ Y )d6 V )d: a a a Since a, b and he process V are arbirary, his esablishes Eq. 1.3c). Eqs. 1.3d) has also been proved. Proof of Corollary 1.1. For each n 1, le Y n ;Z n ;U n ) BSDE;n; ; F). From he esimae 1:4) in Theorem 1.1 we have n E e s Ys n 2 + Zs n 2 )ds6c 1 E e s Fs; ; ) 2 ds; [ E sup e s Ys n 2 ) 6s6n 6C 1 E e s Fs; ; ) 2 ds; E[e Y n )6C 2 E e s s) 2 ds; [ n E e s Us n 2 ds 6C 2 E e s s) 2 ds; and Ys n = Yn n =;Zs n =;Us n =; for s n: Le m n. We have Y m + n U m s ds = Y m n + n n Fs; Ys m ;Zs m )ds Zs m for all [;n;!-a.s., and from Proposiion 1.1 [ n E e s Ys n Ys m 2 + Zs n Zs m 2 )ds 6Ce n E Yn m 2 ; E sup e s Ys n Ys m )6Ce 2 n E Yn m 2 : 6s6n From Eq. 1.4b), e T E YT m 2 )6E sup e s Ys m )6C 2 1 E T66m T db s ) e s Fs; ; ) 2 ds
22 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 as T = n : Hence, Y S 2; M 2; ; Z M 2; d for all T, and U M 2; such ha as n Y n Y in S 2; ;T) M 2; ;T); e TE Y T 2 6CE T e F; ; ) 2 d; Z n Z in M 2; d ;T); U n U in S 2; ;T); where U n = U s n ds, U is absoluely coninuous and Y; Z; U) saises he asserions of Corollary 1.1, where U =d U=d. The soluion is unique since if Y; Z; U) and Ỹ; Z;Ũ) are wo soluions of Eqs. 1.9a), 1.9b) and 1.9c) hen from Proposiion 1.1 ) n E sup e s Y s Ỹ s 2 + E e s Y s Ỹ s 2 ds 6s6n +E n e s Z s Z s 2 ds6c 1 Ee n Y n Ỹ n 2 ) and for n we ge Y = Ỹ; Z= Z; U is uniquely dened by Eq. 1.9a). 4. Connecion wih parabolic variaional inequaliies In his secion we will show ha he BSDE sudied in he previous secions allows us o give a probabilisic represenaion of soluions of a parabolic variaional inequaliy. Le ; F;P;F ;B ) be a R d -valued Wiener process, F = {B s :6s6}) N; and b :[;T R d R d ; :[;T R d R d R d be coninuous mappings such ha b; x) b; x) + ; x) ; x) 6L 1 x x ; [;T; x; x R d 4.1) for some consan L 1 ). For each ; x) [;T R d, le {Xs x ; 6s6T} be he unique soluion of he SDE s s Xs x =x+ br; Xr x )dr+ r; Xr x )db r : 4.2) We have see Friedman, 1976) for [;T; x; x R d : X x s =x; s [;; Xs x S p d [;T; p 2; 4.3a) 4.3b)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 23 E sup Xs x )6C1 p + x p ); 4.3c) s [;T E sup Xs x X x s )6C1 p + x p + x p ) p=2 + x x p ); 4.3d) s [;T where C = Cp; T; L 1 ;K 1 ); K 1 = sup [;T { b; ) + ; ) }: We now consider he BSDE Eq. 1.1) in he case = 1, wih he daa ; ; ; F) of he form = T;!)=gXT x!)); F!; s; y; z)=fs; Xs x!);y;z); 4.4) where g; f saises g CR d ; R) and M ; q N such ha gx) 6M1 + x q ); for all x R d ; f C[;T R d R R d ) and ; ; ; p N such ha f; x; y; ) 61 + x p + y ) y ỹ)f; x; y; z) f; x; ỹ; z))6 y ỹ 2 ; f; x; ; z) f; x; y; z) 6 z z 4.5) 4.6) for all [;T; x R d ; y; ỹ R; z; z R d ; and : R [; + is a proper; convex l:s:c: funcion; s:: y) ) = 4.7) and M ; m N such ha gx)) 6M1 + x m ); x R d : 4.8) For each we denoe by {Fs ; s [; T } he naural lraion of he Brownian moion {B s B ; s [; T } argumened wih he P-null ses of F: Under he assumpions 4.4) 4.8) i follows from Theorem 1.1 see Remar 1.1) ha for each ; x) [;T R d here exiss a unique Fs -progressively measurable riple Y x ;Z x ;U x ) S 2 [; T Md 2[; T M 2 [; T such ha T Ys x + s T Ur x dr=gxt x )+ s s [; T ; P a:s: fr; X x r T ;Yr x ;Zr x )ds s Z x r db r ; 4.9)
24 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 and Y x s ;U x s ) @ ; dp ds a:e: on [; T : 4.1) We shall exend Ys x ;Zs x ;Us x, for s [;T by choosing Ys x s [;. Hence T Ys x + s Ur x dr = gx x T 1 [;T r)fr; X x r ;Y x r ;Z x r )dr T )+ s T Zr x db r ; s [;T; a:s: s and Eq. 4.1) is saised a.e. on [; T. Proposiion 4.1. Under assumpions 4:1); 4:4) 4:8) we have = Y x ;Zs x =, U x s =; ) E sup Ys x 2 6C1 + x p ) 4.11) s [;T and ) E sup Ys x Y x s 2 6 C[E gxt x ) gx x T ) 2 s [;T T + E 1 [; T r)fr; Xr x ;Yr x ;Zr x ) 1 [ ;Tr)fr; X x r ;Yr x ;Zr x ) 2 dr 4.12) for all ; [;T, x; x R d C and p N are consans independen of ; [;T and x; x R d ). Proof. From inequaliy 1:4b), wih = ; = T ) in Theorem 1:1, T ) E sup Ys x 2 6C E gxt x ) 2 + E fr; Xr x ; ; ) 2 dr ; s [;T where C is independen of [;T and x R d ; which yields Eq. 4.11) using he assumpions on f and g and Eq. 4.3c). Eq. 4.12) follows from Eq. 1.6b) in Proposiion 1:1. We dene u; x)=y x ; ; x) [;T R d ; 4.14) which is a deerminisic quaniy since Y x -algebra. is F -measurable, and F is a rivial
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 25 Corollary 4.1. Under assumpions 4:1) and 4:4) 4:8) he funcion u saises: u; x) Dom ; ; x) [;T R d ; 4.15a) u; x) 6CT )1 + x p=2 ); ; x) [;T R d ; 4.15b) u C[;T R d ); 4.15c) where CT) ;p N are consans independen of and x. Proof. We have u; x)) = E Y x ) + ; Eq. 4.15a) follows, Eq. 4.15b) follows from Eq. 4.11). Le n ;x n ) ; x). Then u n ;x n ) u; x) 2 = E Y nxn n Y x 2 6 2E sup Ys nxn Ys x 2 +2E Y x n Y x 2 : s [;T Using Eqs. 4:12), 4:3c) and 4:3d), we obain ha u n x n ) u; x) as n ;x n ) ; x). In he sequel, we shall prove ha he funcion u dened by Eq. 4.14) is a viscosiy soluion of he parabolic variaional inequaliy PVI): where @u; x) + L u; x)+f; x; u; x); u); x)) @ u; x)); @ [;T; x R d ; ut; x)=gx); x R d ; 4.16) L = 1 2 d @ 2 ) ij ; x) + @x i @x j i; j=1 Remar ha a every poin y Dom @ y)=[ y); +y); d i=1 b i ; x) @ @x i : where y) and +y) are he lef derivaive and he righ derivaive, respecively, a he poin y. We shall dene he noion of viscosiy soluion in he language of sub- and superjes, following Crandall Ishii Lions 1992). Sd) will denoe below he se of d d symmeric non-negaive marices. Deniion 4.1. Le u C[;T R d ) and ; x) [;T R d. We denoe by P 2+ u; x) he parabolic superje of u a ; x)) he se of riples p; q; X ) R R d Sd) which are such ha us; y) 6 u; x)+ps )+q; y x) + 1 2 X y x);y x)+o s + y x 2 ):
26 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 P 2 u; x) he parabolic subje of u a ; x)) is dened similarly as he se of riples p; q; X ) R R d Sd) which are such ha us; y) u; x)+ps )+q; y x) + 1 2 X y x);y x)+o s + y x 2 ): We can give now he deniion of a viscosiy soluion of he parabolic variaional inequaliy 4:16): Deniion 4.2. Le u C[;T R d ) which saises ut; x)=gx). a) u is a viscosiy subsoluion of 4:16) if: u; x) Dom ; ; x) [;T R d and a any poin ; x) ;T) R d, for any p; q; X ) P 2+ u; x) p 1 2 Tr ); x)x ) b; x);q) f; x; u; x);q; x)) 6 u; x)): 4.17) b) u is a viscosiy supersoluion of Eq. 4.16) if: u; x) Dom ; ; x) [;T R d ; and a any poin ; x) ;T) R d, for any p; q; X ) P 2 u; x) p 1 2 Tr ; x)x ) b; x);q) f; x; u; x);q; x)) +u; x)): 4.18) c) u is a viscosiy soluion of Eq. 4.16) if i is boh a viscosiy sub- and supersoluion. Theorem 4.1. Le assumpions 4:1) and 4:4) 4:8) be saised. Then he funcion u; x) dened by Eq. 4.14) is a viscosiy soluion of Eq. 4.16). Proof. For each ; x) [;T R d ; ; 1; le Y ; x s;z ; x s), s [; T ; he soluion of BSDE Y x ; s + T s 1 T T D Y ; x r)dr=gxt x )+ fr; Xr x ;Y ; x r;z ; x r)dr Z ; x r db r : s s I is nown see Pardoux, 1997) ha u ; x)=y x ; ; [;T; x R d is he viscosiy soluion of he parabolic dierenial equaion: @u ; x) + L u ; x)+f; x; u ; x); u ); x)) = 1 @ D u ; x)); u T; x)=gx); [;T; x R d : 4.19)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 27 From Proposiion 2:3 we have u ; x) u; x) 2 6E sup Y ; x s Ys x 2 6C1 + x p ) s [; T for all ; x) [;T R d C and p N are consans independen of and ; x) [;T R d ). Firs, we shall show ha u is a subsoluion. From Lemma 6:1 in Crandall Ishii Lions 1992), if ; x) [;T R d and p; q; X ) P 2+ u; x), hen here exis sequences n n ;x n ) [;T R d ; p n ;q n ;X n ) P 2+ u n n ;x n ); such ha n ;x n ;u n n ;x n );p n ;q n ;X n ) ; x; u; x);p;q;x) as n : Bu for any n: p n 1 2 Tr ) n ;x n )X n ) b n ;x n );q n ) f n ;x n ;u n n ;x n );q n n ;x n ))6 1 n D n u n n ;x n )): 4.2) We can assume ha u; x) inf Dom ) since for u; x) = inf Dom ) we have u; x)) = and inequaliy 4:17) in Deniion 4:2 is clearly saised. Le y Dom, y u; x). The uniformly convergence u u on compacs implies ha n = n ; x; y) such ha y u n n ;x n ); n n. We muliply Eq. 4.2) by u n n ;x n ) y, one follows: [ p n 1 2 Tr ) n ;x n )X n ) b n ;x n );q n ) f n ;x n ;u n n ;x n );q n n ;x n )u n n ;x n )) y) + J n u n n ;x n )))6 y): 4.21) Passing o lim inf n in Eq. 4.21) we obain [ p 1 2 Tr ); x)x ) b; x);q)f; x; u; x);q; x))u; x) y) + u; x))6 y); hence, p 1 2 Tr ); x)x ) b; x);q) f; x; u; x);q; x)) u; x)) y) 6 ; u; x) y for all y u; x), which implies Eq. 4.17). Le us show ha u is a supersoluion. Similarly, given ; x) [;T R d p; q; X ) P 2 u; x) here exis he sequences n n ;x n ) [;T R d ; p n ;q n ;X n ) P 2 u n n ;x n ); and
28 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 such ha n ;x n ;u n n ;x n );p n ;q n ;X n ) ; x; u; x);p;q;x) as n : For any n: p 1 2 Tr ) n ;x n )X n ) b n ;x n );q n ) f n ;x n ;u n n ;x n );q n n ;x n )) 1 D n u n n ;x n )): 4.22) n We can assume ha u; x) supdom ) since for u; x) = supdom ) we have +u; x))=+ and Eq. 4.18) is saised. Le y Dom, u; x) y. Then here exiss n = n ; x; y) such ha u n n ;x n ) y; n n. We muliply Eq. 4.22) by y u n n ;x n ), and we have [ p n 1 2 Tr ) n ;x n )X n ) b n ;x n );q n ) f n ;x n ;u n n ;x n );q n n ;x n ))y u n n ;x n ))) J n u n n ;x n ))) y); y u; x); from where passing o lim inf n inequaliy 4.18) follows. We can now improve Eq. 4.15a). Corollary 4.2. a) u; x) Dom @ ); ; x) [;T R d. b) Y x s Dom @ ); s [;T; P-a:s:!. s; X x s Proof. b) follows from a) since Ys ; x = Ys =us; Xs x ). To prove a) we have wo cases. c 1 ) Dom @ ) = Dom and in his case, by Eq. 4.15a), u; x) Dom @ ); ; x) [;T R d. c 2 ) Dom @ ) Dom. Le b Dom \Dom @ ). Then b = supdom ) orb= inf Dom. Ifb= supdom ) and u; x)=b, hen ; ; ) P 2+ u; x) since us; y)6u; x)+o s + y x 2 ) and from Eq. 4.17) i follows b)= u; x)) and consequenly b Dom @ ); a conradicion which shows ha u; x) b. We argue similarly in he case b = inf Dom ). In order o esablish a uniqueness resul, we need o impose he following addiional assumpion. For each R here exiss a coninuous funcion m R : R + R +, m R )= such ha f; x; r; p) f; y; r; p) 6m R x y 1 + p )); [;T; x ; y 6R; p R d : 4.23)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 29 Theorem 4.2. Under assumpions 4:1); 4:4) 4:8) and 4:23) he PVI 4:16) has a unique viscosiy soluion in he class of coninuous funcions which grow a mos polynomially a inniy. Proof. The exisence is proved by Theorem 4:1. The proof of uniqueness is based on he ideas in El Karoui e al. 1997). I suces o show ha if u is a subsoluion and v a supersoluion such ha ut; x)=vt; x)=gx); x R d, hen u6v. We perform he ransformaion u; x):=u; x)e 1 + x 2 ) =2 ; v; x):= v; x)+ ) e 1 + x 2 ) =2 as in he proof of Theorem 8:6 in El Karoui e al. 1997). For he simpliciy of noaions, we will wrie below u; v insead of u; v. Hence, he ransformed) u and v saisfy in he viscosiy sense) @u @ + F; x; u; x);du; x);d2 u; x))6 e x)u; x)); @v @ + F; x; v; x);dv; x);d2 v; x)) 2 + e x) v; x) )) wih F dened as in El Karoui e al. 1997) and x)=1+ x 2 ) =2. Exacly as in El Karoui e al. 1997), we need only o show ha for any R, if B R := { x R}, sup u v) + 6 sup u v) + ; ;T) B R ;T @B R since he righ-hand side ends o zero as R. To prove his fac we assume here exiss R; such ha for some ;x ) ;T) B R =u ;x ) v ;x ) = sup u v) + sup u v) + ; ;T) B R ;T @B R and we shall arrive a a conradicion. We dene ˆ; ˆx; ŷ) as being a poin in [;T B R B R where he funcion ; x; y)=u; x) v; x) x y 2 2 achieves is maximum. Then by Lemma 8:7 from El Karoui e al. 1997): for large enough; ˆ; ˆx; ŷ) ;T) B R B R ; 4.24a) ˆx ŷ 2 and ˆx ŷ 2 as ; 4.24b) uˆ; ˆx) vˆ;ŷ)+: Then for large enough e ˆ ˆx)uˆ; ˆx) e ˆ ŷ) vˆ;ŷ) ˆ ) and, consequenly, e ˆ ˆx)uˆ; ˆx))6 + e ˆ ŷ) vˆ;ŷ) ˆ )) and he proof coninues exacly as in El Karoui e al. 1997). 4.24c)
21 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 5. Connecion wih ellipic variaional inequaliies We consider he following ellipic variaional inequaliy EVI): Lux)+@ ux)) fx; ux); u)x)); u @D = g; or equivalenly: x D; 5.1) Lux)+fx; ux)) u)x) [ ux)); +ux)); u @D x)=gx); x @D: Here D is a bounded domain of R d of he form D = {x R d : x) }; x D; 5.1 ) 5.2) where C 2 R d ); x) ; x @D {x R d :x)=}: We assume ha g CR d ); 5.3) f CR d R R d ) and R; ; such ha fx; y; ) 61 + y ); 5.4a) y ỹ)fx; y; z) fx; ỹ; z))6 y ỹ 2 ; 5.4b) fx; y; z) fx; y; z) 6 z z ; 5.4c) for all x R d ; y;ỹ R; z; z R d, and : R ;+ is a proper convex l:s:c: funcion s:: : y) )=; 5.5a) M : gx)) 6M; x D; 5.5b) and L is he inniesimal generaor of he Marov diusion process X : X x = x + bxs x )ds+ Xs x )db s ; ; i.e. L = 1 2 d @ 2 ) ij x) + @x i @x j i; j=1 d i=1 b i x) @ @x i : Here ; F;P;F ) ;B )isad-dimensional Brownian moion as in Secion 4 and b : R d R d ; :R d R d d are Lipschiz coninuous on D: 5.6)
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 211 Dene he sopping-ime: x = inf { : X ) = D}: We assume ha P x )=1; x D; 5.7a) = {x @D: P x )=} is a closed subse of @D; 5.7b) sup Ee x ) for some 2 + 2 : 3 5.7c) x D Consider now for each x D he one-dimensional BSDE: Y x + x x U x s ds = gx x x )+ Y x x ;U x x ) @ ; x x fx x s;y x s ;Z x s)ds x Zs x db s ; x ;! a:s:; 5.8) a:e: on [;: I follows from Theorem 1.1 ha he BSDE 5.8) has a unique soluion Y x ;Z x ;U x ) S 2; [; x M 2; [; x ) M 2; d [; x M 2; [; x : As in Darling and Pardoux 1997) we can show ha x x is a:s: coninuous; 5.9a) ux)=y x ; x D; is a deerminis coninuous funcion; 5.9b) Y x = ux x ); 66 x ; a:s: 5.9c) Proposiion 5.1. If he Dirichle problem 5:1) has a classical soluion u C 2 D) C D); hen ux)=y x ; x D; where Y x ;Z x ;U x ) is he soluion of BSDE 5.8). Proof. Le u x)=lux)+fx; ux); u)x)) @ ux));x D: Applying Iô s formula o e ux x ) we have x e x) ux x )+ e s [ ux s )+LuX s ) ds x x =e x ux x )+ e s ux s );X s )db s ); x and, consequenly, e x) ux x )+ x + fx s ;ux s ); u)x s )) ds + x e s u X s )ds=e x gx x )+ x x x e s [ux s ) x e s ux s );X s )db s ): 3 If for some 16i6d; inf x D ) ii x), hen such ha Eq. 5.7c) holds see Srooc and Varadhan, 1972).
212 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 Hence, by uniqueness: Y x = ux x x ); Z x = u)x x x ); U x =u X x x ): Under he assumpions given above, we canno hope for a classical soluion o exis in general. Tha is why we dene he noion of viscosiy soluion. P 2+ ux) he ellipic superje) and P 2 ux) he ellipic subje) are dened similarly as in Deniion 4.1. Le u C D) and x D; hen q; X ) P 2; + ux) if D uy)6ux)+q; y x)+ 1 2 Xy x);y x)) + o y x 2 ); y D and q; X ) P 2 ux) if D uy) ux)+q; y x)+ 1 2 Xy x);y x)) + o y x 2 ); y D: Deniion 5.1. a) A funcion u C D) is a viscosiy subsoluion of Eq. 5.1) if x D; q; X ) PD 2+ ux), ux) Dom ; 5.1a) V x; q; X ) def = 1 2 Tr )x)x) bx);q) fx; ux);qx)+ ux))6 if x D; min{v x; q; X );ux) gx)}6if x @D: 5.1b) 5.1c) b) u C D) is a viscosiy supersoluion of Eq. 5.1), if x D; q; X ) P 2 D ux), ux) Dom ; 5.11a) V + x; q; X ) def = 1 2 Tr )x)x) bx);q) fx; ux);qx)) + +ux)) if x D; 5.11b) max{v + x; q; X );ux) gx)} if x @D: 5.11c) c) u C D) is a viscosiy soluion of Eq. 5.1) if i is boh a viscosiy subsoluion and a viscosiy supersoluion. Theorem 5.1. Under assumpions 5.2) 5.7) he funcion u C D) given by ux)= Y x is a viscosiy soluion of Eq. 5.1). Moreover, ux) Dom @ ); x D;. Proof. Assuming ha u is a viscosiy soluion of Eq. 5.1), we deduce as in Corollary 4.2 ha ux) Dom @ ). In order o prove ha ux)=y x is a viscosiy soluion we could use as in he previous secion an argumen based on penalizaion. Le us, however, give a direc proof of he fac ha u is a viscosiy subsoluion.
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 213 Le x D and q; X ) PD 2+ ux). From he 1 law, here are wo possible cases: a) x!)= a.s. Then x @D; ux)=y x =gx) and consequenly 5.1a) is saised. b) x a.s. We wan o show ha in his case V x; q; X )6, which will conclude he proof. Suppose his is no he case. Then V x; q; X ). I follows by coninuiy of f; u; b and, lef coninuiy and monooniciy of ha here exiss ; such ha for all y x 6, 1 2 Tr[ y)x + I) by);q+x + I)y x)) fy; uy); [q +X + I)y x)y)) + uy)) : 5.13) Now, since q; X ) PD 2+ ux) here exiss 6 such ha uy) y), for all y D such ha y x 6, where Le y):=ux)+q; y x)+ 1 2 X + I)y x);y x)): := inf { ; X x x } x 1: We noe ha Y ; Z ):=Y x ;1 [; )Z x ); 661; solves he BSDE Y = ux x )+ 1 1 1 [; s)[fxs x ;uxs x ); Z s ) Us x ds Y ;U x ) @ ; dp d a:e: on [;: Moreover, i follows from Iô s formula ha Yˆ ; Ẑ ):= X);1 x [; ) )X x )); 661; saises ˆ Y = X x ) 1 1 [; s)l X x s )ds Le Ỹ ; Z ):=ˆ Y Y ;Ẑ Z ). We have 1 Ẑ s db s ; 661: Z s db s ; Le Ỹ = X x ) ux x )+ 1 1 Z s db s ;661: 1 [; s)[ L X x s ) fx x s ;ux x s ); Z s )+U x s ds s := [L X x s )+fx x s;ux x s ); Z s )1 [; s); ˆ s := [L X x s )+fx x s;ux x s );Ẑ s )1 [; s):
214 E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 Since ˆ s s 6C Ẑ s Z s, here exiss a bounded d-dimensional progressively measurable process { s ;6s61} such ha Now, ˆ s s = s ; Z s ): Ỹ = X x ) ux x )+ 1 [ ˆ s +U x s 1 + s ; Z s ) ds Z s db s : I is easily seen see e.g. he proof of Theorem 1.6 in Pardoux, 1997) ha Ỹ aes he form [ Ỹ = E X x ) ux x )) + sus x ˆ s )ds ; where = exp s; db s 1 2 We rs noe ha Y x ;U x ux x ))6U x s 2 ds): ) @ implies ha and his holds dp d a.e. Moreover, he choice of and implies ha ux x ) X x ); a.e. and for 66, i follows from Eq. 5.13) ha ˆ ux x )): All hese inequaliies and he above formula for Ỹ imply ha Ỹ, i.e. x) ux), which conradics he deniion of. Hence, V x; q; X )6. Remar 5.1. Under appropriae addiional assumpions, namely Eq. 4.23) and he fac ha is large enough, one can show ha he above ellipic variaional inequaliy has a unique viscosiy soluion, adaping he proof in Crandall Ishii Lions 1992). References Barbu, V., 1976. Nonlinear Semigroups and Dierenial Equaions in Banach Spaces. Ed. Academiei Române & Noordho Inernaional Publishing, Leiden. Brezis, H., 1973. Operaeurs Maximaux Monoones e Semigroupes de Conracions Dans les Espaces de Hilber. Norh-Holland, Amserdam. Crandall, M., Ishii, H., Lions, P.L., 1972. User s guide o he viscosiy soluions of second order parial dierenial equaions. Bull. A.M.S. 27, 1 67. Cvianic, J., Karazas, I., 1996. Bacward sochasic dierenial equaions wih reecion and Dynin games. Ann. Probab. 24, 224 256. Darling, R.W.R., Pardoux, E., 1997. Bacward SDE wih random erminal ime and applicaions o semilinear ellipic PDE. Ann. Probab. 25, 1135 1159. El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C., 1997. Reeced soluions of bacward SDE s and relaed obsacle problems for PDE s. Ann. Probab. 25, 72 737. Friedman, A., 1976. Sochasic Dierenial Equaions and Applicaions. Academic Press, New Yor. Gegou-Pei, A., Pardoux, E., 1996. Equaions dierenielles sochasiques rerogrades reechies dans un convexe. Sochasics Sochasic Rep. 57, 111 128.
E. Pardoux, A. Rascanu / Sochasic Processes and heir Applicaions 76 1998) 191 215 215 Pardoux, E., 1997. Bacward sochasic dierenial equaions and viscosiy soluions of sysems of semilinear parabolic and elleipic PDEs of second order. In: Decreusefond, L., Gjerde, J., sendal, B., Usunel, A.S., Eds.), Sochasic Analysis and Relaed Topics VI: The Geilo Worshop, 1996, Birhauser, Basel, pp. 79 128. Pardoux, E., Peng, S., 1992. Bacward SDE s and quasilinear PDE s. In: Rozovsi, B.L., Sowers, R.B., Eds.), Sochasic PDE and Their Applicaions, Lecure Noes in Compuer Science vol. 176, Springer. Srooc, D.W., Varadhan, S.R.S., 1972. On degenerae ellipic parabolic operaors of second order and heir associaed diusions. Comm. Pure Appl. Mah. 25, 651 713.