BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH REFLECTION AND DYNKIN GAMES 1. By Jakša Cvitanić and Ioannis Karatzas Columbia University

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1 The Annals of Probabiliy 1996, Vol. 24, No. 4, BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH REFLECTION AND DYNKIN GAMES 1 By Jakša Cvianić and Ioannis Karazas Columbia Universiy We esablish exisence and uniqueness resuls for adaped soluions of backward sochasic differenial equaions (BSDE s wih wo reflecing barriers, generalizing he work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Exisence is proved firs by solving a relaed pair of coupled opimal sopping problems, and hen, under differen condiions, via a penalizaion mehod. I is also shown ha he soluion coincides wih he value of a cerain Dynkin game, a sochasic game of opimal sopping. Moreover, he connecion wih he backward SDE enables us o provide a pahwise (deerminisic approach o he game. 1. Inroducion. The noion of backward sochasic differenial equaion (BSDE was inroduced by Pardoux and Peng (199, who proved exisence and uniqueness of adaped soluions, under suiable square-inegrabiliy assumpions on he coefficiens and on he erminal condiion. Independenly, Duffie and Epsein (1992 inroduced sochasic differenial uiliies in economics models, as soluions o cerain BSDE s. More recenly, El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1995 generalized hese resuls o BSDE s wih reflecion, ha is, o a seing wih an addiional coninuous, increasing process added in he equaion; he funcion of his addiional process is o keep he soluion above a cerain prescribed lower-boundary process and o do so in a minimal fashion. Moreover, hese auhors make he crucial observaion ha he soluion is he value funcion of an opimal sopping problem; heir paper provided much of he inspiraion and moivaion for our work. We generalize hese resuls o he case of wo reflecing barrier processes, ha is, o a seing where, in addiion o agreeing wih a arge random variable ξ a he erminal ime = T, he soluion process of our BSDE has o remain beween wo prescribed upper- and lower-boundary processes, U and L, respecively, almos surely. This is accomplished by he cumulaive acion of wo coninuous, increasing reflecion processes, which keep he soluion wihin he prescribed bounds when i aemps o cross eiher of hem. We also esablish he connecion beween his problem and cerain sochasic games of sopping (Dynkin games, as well as wih a pair of coupled opimal sopping problems. The BSDE problem wih wo reflecing barriers is described in Secion 2 of he paper. Preliminary resuls on a relaed pair of coupled opimal sopping problems are obained in Secion 3. In Secion 4 i is shown ha any solu- Received Ocober 1995; revised March Work suppored in par by Army Research Office Gran DAAH AMS 1991 subjec classificaions. Primary 93E5, 6H1; secondary 6G4. Key words and phrases. Backward SDE s, reflecing barriers, Dynkin games, opimal sopping. 224

2 BACKWARD SDE S 225 ion of he BSDE wih wo reflecing barriers is also he value of a sochasic game of opimal sopping, usually called Dynkin game, herefore esablishing he uniqueness of such a soluion. Games of sopping have been sudied in Dynkin and Yushkevich (1968, Neveu (1975, Bensoussan and Friedman (1974, Bismu (1977, Sener (1982, Morimoo (1984, Alario-Nazare, Lepelier and Marchal (1982, Lepelier and Maingueneau (1984 and ohers. The game involves wo players, each of whom can decide o sop i a a random ime of his choice; upon erminaion a cerain amoun, which is random and depends on he ime of erminaion, is paid by one of he players o he oher. In Secion 5 we show ha he pair of coupled opimal sopping problems from Secion 3 has a soluion. This, in urn, implies direcly ha he BSDE wih wo reflecing barriers has a soluion (and ha he Dynkin game has a value, in he special case in which he drif does no depend on he soluion. The general case is reaed using his special case and a fixed poin argumen. An alernaive mehod for proving he exisence of a soluion o he BSDE wih wo reflecing barriers is presened in Secion 6: a sandard penalizaion mehod is applied, under a condiion which roughly says ha he barriers can be approximaed by semimaringales wih absoluely coninuous finie variaion pars. In his case, he reflecion processes which keep he soluion beween he barriers are also absoluely coninuous. In Secion 7 we presen a pahwise (deerminisic approach o he Dynkin game, much in he spiri of he pahwise reamen of he opimal sopping problem in Davis and Karazas (1994. I urns ou ha here is a game wih payoff equal o ha of he Dynkin game plus an exra nonadaped process ; his game can be solved pah-by-pah and has a (pah-dependen value whose condiional expecaion coincides wih he value of he Dynkin game and wih he soluion process of our BSDE. The nonadaped process is also obained direcly from he soluion of he BSDE. Moreover, he opimal sopping imes for he players are he same in boh games and are also opimal in ye anoher pair of pahwise, decoupled, opimal sopping problems. Finally, we collec in he Appendix some useful resuls on supermaringales, in paricular, on poenials and on Snell envelopes. In fuure work, we plan o rea he Markovian aspecs of his heory, including he associaed parial differenial equaions and variaional inequaliies. 2. Backward SDE wih wo reflecing barriers. On a given, complee probabiliy space P, le B = B 1 B d be a sandard d-dimensional Brownian moion on he finie inerval T, and denoe by F = T he augmenaion of he naural filraion F B, namely B = σ B s, s, T, generaed by B. We shall need he following noaion. For any given n N, le us inroduce he following spaces: Ln 2 of T -measurable random variables ξ Rn wih E ξ 2 < ; Hn 2 of F-predicable processes ϕ T Rn wih E ϕ 2 d < ; S k n of F-progressively measurable processes ϕ T Rn wih E sup T ϕ k <, k N;

3 226 J. CVITANIĆ AND I. KARATZAS S 2 ci of coninuous, increasing, F-adaped processes A T wih A =, E A 2 T <. Finally, we shall denoe by he σ-algebra of predicable ses in T. Problem 2.1 (Backward sochasic differenial equaion (BSDE wih upper and lower reflecing barriers. Le ξ be a given random variable in L1 2, and f T R R d R a given R R d -measurable funcion ha saisfies 2 1 (2.2 E f 2 ω d < f ω x y f ω x y k x x + y y ω T x x in R y y in R d for some < k <. Consider also wo coninuous processes L U in S 2 1 ha saisfy 2 3 L U T and L T ξ U T a.s. We say ha a riple X Y K of F-progressively measurable processes X T R, Y T R d and K T R is a soluion of he backward sochasic differenial equaion (BSDE wih reflecing barriers U, L (upper and lower, respecively, erminal condiion ξ and coefficien f, if he following hold: (i K = K + K, wih K ± S 2 ci ; (ii Y H 2 d, and 2 4 X = ξ + f s X s Y s ds + K + T K + K T K Y s db s T 2 5 L X U T 2 6 almos surely. X L dk + = U X dk = Discussion. In he seup of Problem 2.1 he processes L U play he role of reflecing barriers; hese are allowed o be random and ime-varying, and he sae-process X is no allowed o cross hem cf. (2.5 on is way o he prescribed erminal arge condiion X T = ξ cf. (2.4. The sae-process X is forced o say wihin he region enveloped by he lower and upper barriers L U, hanks o he cumulaive acion of he wo increasing reflecion processes K + K, respecively cf. (2.4; hese

4 BACKWARD SDE S 227 ac only when necessary o preven X from crossing he respecive boundary cf. (2.6 and, in his sense, heir acion can be considered minimal. Finally, i is he freedom o choose he inensiy of noise process Y = Y 1 Y d, Y i = d/d X B i, ha allows one o have an F-adaped soluion o (2.4, jus as in he case of unconsrained BSDE s Pardoux and Peng (199. When viewed backwards in ime, ha is, wih X θ = X T θ, Ỹ θ = Y T θ, K ± θ = K ± T K ± T θ and B θ = B T B T θ in (2.4 wrien as 2 4 X θ = ξ + + θ θ f T u X u Ỹ u du + K + θ K θ Ỹ u d B u θ T he effec of he increasing process K + resp., K is o push he saeprocess X upward (resp., downward, in order o preven i from crossing he lower boundary L = L T resp., he upper boundary Ũ = U T, and o do his wih minimal effor; ha is, 2 6 X θ L θ d K + θ = Ũ θ X θ d K θ = When (2.4 is viewed forwards in ime, in oher words as X = E ξ + dx = f X Y d dk + + dk + Y db f s X s Y s ds + K + T K T X T = ξ a s he effec of he increasing process K + resp., K is o push he saeprocess X downward (resp., upward, in order o preven overshooing, ha is, X T > ξ resp., undershooing, i.e., X T < ξ, he prescribed erminal arge ξ. 3. Analysis: a pair of coupled opimal sopping problems. In his secion and he nex, we shall assume ha here exiss a soluion X Y K o he BSDE of Problem 2.1 and will ry o derive some consequences and represenaions. We shall se 3 1 g ω = f ω X ω Y ω ω T from he assumpions on f and (2.1, (2.2, his defines a process g H1 2. For such g, we shall also inroduce he processes N = E ξ + g s ds g s ds 3 2 = E ξ + g s ds T

5 228 J. CVITANIĆ AND I. KARATZAS 3 3 L ξ = L 1 <T + ξ1 =T L = L ξ N T 3 4 U ξ = U 1 <T + ξ1 =T Ũ = U ξ N T Noe ha 3 5 N is coninuous on T a.s., and N S 2 1 (use he sochasic inegral represenaion propery of Brownian maringales and Doob s maximal inequaliy, ha 3 6 L Ũ belong o S 2 1 and are coninuous on T a.s. and ha we have, almos surely, 3 7 L Ũ T and L T L T = = Ũ T Ũ T Wih all his noaion, (2.4 and (2.5 give 3 8 L ξ X = ξ + g s ds + K + T K + K T K Y s db s U ξ on T, almos surely, and aking condiional expecaions wih respec o in (3.8, we have 3 9 L ξ X = N + π K + π K U ξ T almos surely. We have used he noaion of (A.2, (3.2 and he fac ha he sochasic inegral in (3.8 saisfies E Y s db s = a.s., hanks o he assumpion Y Hd 2. In paricular, (3.9 gives 3 1 π K + L + π K = η + π K Ũ + π K + = η Now K ± S 2 ci, so π K± belong o he space 2 c of Definiion A.2, by Corollary A.3 (Appendix. From his observaion and (3.6, he processes η ± of (3.1 are seen o be in S 2 1, and o have pahs which are coninuous on T and quasi-lef-coninuous on T, almos surely since, if hey have a jump a = T, his jump is upwards, by (3.7. Therefore, using he noaion of (A.8, he Snell envelopes 3 11 S η ± = ess sup E η ± τ T τ T are poenials in he space 2 c of Definiion A.2. Hence 3 12 S η ± = π A ± for suiable, uniquely deermined A ± S 2 ci

6 BACKWARD SDE S 229 hanks o Lemma A.4 and Corollary A.3. We have denoed by T he class of F-sopping imes τ T. Furhermore, 3 13 π K ± π A ± from (3.1, and 3 14 π A ± η ± da ± = a s from (A.13. In paricular, from (3.1, (3.3, (3.4 and he coninuiy of A ±, (3.14 reads 3 15 π A + π K + N L da + = U π K + π A + N da = a s a s We have esablished he following resul. Proposiion 3.1. For a given, fixed g H1 2, he mapping K+ K A + A of (3.12 and (3.1, namely 3 16 π A + = S L + π K π A = S Ũ + π K + maps S 2 ci S2 ci ino iself and saisfies (3.13 and (3.15. In Secion 5, we shall show ha his mapping has a fixed poin K + K S 2 ci S2 ci ; for his fixed poin, (3.13 holds as equaliy, and (3.15 is hen equivalen o (2.6, in ligh of he equaliy X = N + π K + π K in (3.9. Open quesion. Can we deduce π K ± = π A ±, hus also A ± = K ±, from (3.15, (3.13 and (2.6? This would show ha every soluion o he BSDE induces a fixed poin of he mapping of ( Analysis: a sochasic game of E. B. Dynkin. Our purpose in his secion is o show ha he exisence of a soluion X Y K o he BSDE of Problem 2.1 implies ha X is he value of a cerain sochasic game of sopping. Firs inroduced by Dynkin and Yushkevich (1968 and laer sudied, in differen conexs, by several auhors, including Neveu (1975, Bensoussan and Friedman (1974, Bismu (1977, Sener (1982, Morimoo (1984, Alario- Nazare, Lepelier and Marchal (1983, Lepelier and Maingueneau (1984 and ohers, such sochasic games are known as Dynkin games. As a corollary o our resuls, we shall presen in Secion 7 a very simple, pahwise approach o his game, in he spiri of a similar reamen in Davis and Karazas (1994 for he opimal sopping problem.

7 23 J. CVITANIĆ AND I. KARATZAS Theorem 4.1. Le X Y K be a soluion o he BSDE of Problem 2.1 and reain he noaion of (3.1. For any T and any wo sopping imes σ τ in he class T, consider he payoff 4 1 R σ τ = σ τ g u du + ξ1 σ τ=t + L τ 1 τ<t τ σ + U σ 1 σ<τ as well as he upper and lower values, respecively, 4 2 V = ess inf σ T ess sup E R σ τ τ T V = ess sup ess inf E R σ τ τ σ T T of a corresponding sochasic game. This game has value V, given by he sae-process X of he soluion o he BSDE, ha is, 4 3 V = V = V = X a.s. T as well as a saddlepoin ˆσ ˆτ T T given by 4 4 namely 4 5 ˆσ = inf s T /X s = U s T ˆτ = inf s T /X s = L s T E R ˆσ τ E R ˆσ ˆτ for every σ τ T T. = X E R σ ˆτ a.s. In he game of (4.1 and (4.2 wih =, player 1 chooses he sopping ime σ, player 2 chooses he sopping ime τ, and R σ τ represens he amoun paid by player 1 o player 2. I is he expecaion ER σ τ of his random payoff ha player 1 ries o minimize and player 2 ries o maximize. The game sops when one player decides o sop, ha is, a he sopping ime σ τ, or a T if σ = τ = T; he payoff R σ τ hen equals U σ if player 1 sops he game firs, σ τ L τ if player 2 sops he game firs or if boh sop g u du + he game simulaneously, ξ if neiher player sops he game before T. According o Theorem 4.1, he pair ˆσ ˆτ of (4.4 and (4.5 provides a saddlepoin of opimal sopping rules for he players, in he following sense: if player 1 chooses he rule ˆσ, hen ˆτ is an opimal sopping ime for player 2, and if player 2 chooses he rule ˆτ, hen ˆσ is an opimal sopping ime for player 1. Proof. I suffices o show (4.5, since (4.3 follows direcly from his.

8 BACKWARD SDE S 231 (i Firs, le us ake σ = ˆσ and arbirary τ T. On he even ˆσ < τ, we have U ˆσ = X ˆσ, K ˆσ = K from (4.4 and (2.6. Thus 4 6 R ˆσ τ = ˆσ ˆσ g u du + X ˆσ K ˆσ K g u du + X ˆσ + K + ˆσ K + K ˆσ K = ˆσ = X + g u du + X ˆσ + K ˆσ K ˆσ Y u db u almos surely, wih equaliy if τ = ˆτ, and on he even τ ˆσ we have, 4 7 R ˆσ τ = τ g u du + ξ1 τ=t + L τ 1 τ<t = K τ K = τ g u du + X τ + K + τ K + K τ K τ = X + g u du + X τ + K τ K τ Y u db u almos surely, wih equaliy if τ = ˆτ. Puing (4.6 and (4.7 ogeher, we obain 4 8 R ˆσ τ X + τ ˆσ Y u db u a.s. wih equaliy if τ = ˆτ, and aking condiional expecaions wih respec o, 4 9 E R ˆσ τ X = E R ˆσ ˆτ a.s. τ T because Y H 2 d. (ii Second, we ake τ = ˆτ and arbirary σ T. Argumens similar o hose of case (2, now lead o σ ˆτ 4 1 R σ ˆτ X + Y u db u which holds wih equaliy if σ = ˆσ, and o a.s E R σ ˆτ X = E R ˆσ ˆτ a.s. σ T

9 232 J. CVITANIĆ AND I. KARATZAS by aking condiional expecaions as before. Now (4.5 follows from (4.9 and (4.11. Corollary 4.2 (Uniqueness for Problem 2.1 in a special case.. ha, for some given g Hd 2, we have Suppose 4 12 f ω x y = g ω ω x y T R R d in he BSDE of Problem 2.1. Then his problem can have a mos one soluion. Proof. Suppose ha he riple X Y K solves he BSDE of Problem 2.1; hen he sae-process X of his riple is uniquely deermined, from (4.3 of Theorem 4.1, as he value of he Dynkin game (4.1 and (4.2. Bu X is also a coninuous semimaringale of he Brownian filraion F = F B, wih ( 2 4 X = X K + g u du + Y u db u T as is decomposiion; in i, boh K and Y are uniquely deermined as well. 5. Synhesis: exisence and uniqueness for he BSDE. We show in his secion how o consruc, for a given, fixed g H1 2, a (unique soluion X Y K for Problem 2.1 wih f g as in (4.12, saring from a fixed poin K + K of he mapping (3.16 in Proposiion 3.1. We address hen he quesions of exisence of such a fixed poin and of consrucing a (unique soluion o he BSDE of Problem 2.1 for general coefficien funcions f. Problem 5.1. Le ξ L1 2, g H2 1 be given, as well as wo coninuous processes L U in S 2 1 as in Problem 2.1. We say ha a riple X Y K of F-progressively measurable processes is a soluion of he backward sochasic equaion (BSE wih reflecing barriers U L, erminal condiion ξ and coefficien g, if K = K + K wih K ± S 2 ci, Y H2 d and if (2.5, (2.6 as well as 5 1 X = ξ + are saisfied almos surely. g u du + K T K Y u db u T Theorem 5.2. For a given g H1 2, suppose ha he mapping (3.16 in Proposiion 3.1 has a fixed poin K + K, namely, 5 2 π K + = S L + π K π K = S Ũ + π K + for some K + K S 2 ci S2 ci. Then he riple X Y K, wih 5 3 K = K + K X = N + π K + π K

10 BACKWARD SDE S 233 and wih Y Hd 2 uniquely deermined via E ξ + g s ds + K T = N + E K T 5 4 is he unique soluion o he BSE of Problem Y u db u T Proof. We have, from (5.3, (5.4 and (3.2, X + g u du + K = E ξ + g u du + K T = X + Y u db u for T, where X = N + E K T ; in paricular, X T = ξ and hus ξ + g u du + K T = X + Y u db u Subracing memberwise we obain (5.1. On he oher hand, (5.2 gives π K + L + π K π K Ũ + π K + and hus, in conjuncion wih (5.3 (3.3 and (3.4, we obain L N + L X = N + π K + π K Ũ + N U in oher words, (2.5 is saisfied. Finally, he fac ha he pair K + K solves (5.2 implies ha he equaliies of (3.15 hold wih A ± = K ± ; in conjuncion wih X = N + π K + π K, hese equaliies read X L dk + = U X dk = almos surely; ha is, (2.6 holds as well. Uniqueness follows from Corollary 4.2. I develops from Theorem 5.2 ha, in order o esablish he exisence and uniqueness of soluion o he BSE of Problem 5.1, i suffices o show he exisence of a pair K + K S 2 ci S2 ci ha solves he sysem (5.2, or equivalenly, from Corollary A.3, he exisence of a pair Z + Z 2 c 2 c ha solves 5 5 Z + = S L + Z Z = S Ũ + Z + in he noaion of (A.8. This sysem was inroduced by Bismu (1977 and was sudied by him and by Alario-Nazare (1982, among ohers, as a crucial

11 234 J. CVITANIĆ AND I. KARATZAS sep owards solving Dynkin games of he ype (4.1, (4.2 by reducing hem o a pair of coupled opimal sopping problems. For compleeness, we include a proof of he basic exisence resul in his direcion. Theorem 5.3. For he exisence of a soluion Z + Z 2 c 2 c o he sysem (5.5 equivalenly, of a soluion K + K S 2 ci S2 ci o he sysem (5.2, i is necessary ha 5 6 L ξ h θ + E ξ U ξ for some h 2 c, θ 2 c ; condiion (5.6 is also sufficien, provided ha 5 7 L < U < T holds almos surely. Remark. For any given g H1 2, he condiion (5.6 is equivalen o 5 6 L H Ũ for some H 2 c and 2 c This can be seen easily from (3.2 (3.4. Indeed, if 5 6 holds, hen we can ake h = H + E g + u du g + u du θ = + E g u du g u du T in (5.6, wih g + = g, g = g ; on he oher hand, if (5.6 holds, hen 5 6 holds as well, wih H = h + E g u du g u du = θ + E g + u du g + u du T Proof of Theorem 5.3. Suppose ha Z ± 2 c saisfy (5.5; hen we have Z + L + Z, Z Ũ + Z +, whence and 5 6 is saisfied. L Z + Z Ũ For he remainder of his proof, le us assume ha 5 6 holds and ry o esablish he exisence of a soluion o (5.5 by considering he ieraive scheme 5 8 Z + n+1 = S L + Z n Z n+1 = S Ũ + Z+ n n N and Z ±

12 BACKWARD SDE S 235 We claim ha 5 9 Z ± n 2 c n N and 5 1 L Z + n Z+ n+1 H Ũ Z n Z n+1 hold a.s. n N Proof of (5.9. Clearly Z ± 2 c. Suppose Z± n 2 c for some n N; hen boh L + Z n, Ũ + Z+ n are in S2 1 and have pahs which are a.s. coninuous on T and quasi-lef-coninuous on T, because of he coninuiy of Z ± n on T and (3.7. From Lemma A.4, (A.1 and Corollary A.3, we deduce Z ± n+1 2 c, and (5.9 follows. Proof of (5.1. The inequaliies Z + 1 = S L L and Z 1 = S Ũ Ũ, Z ± 1 are obvious, by he definiion (5.8. Clearly also, Z+ 2 = S L+Z 1 S L = Z + 1, Z 2 = S Ũ + Z+ 1 S Ũ = Z 1 ; assuming Z± n Z± n 1, we obain similarly Z + n+1 = S L + Z n S L + Z n 1 = Z+ n, Z n+1 = S Ũ + Z + n S Ũ + Z+ n 1 = Z n. This esablishes, by inducion, he monooniciy of Z ± n n N. For he las inequaliies in (5.1, observe from 5 6 ha H + L L, H Ũ Ũ; herefore, H S L = Z + 1 and S Ũ = Z 1. Suppose ha H Z + n, Z n for some n N; hen +Ũ H Z+ n, H L Z n and hus H S L + Z n = Z+ n+1, S Ũ + Z+ n = Z n+1, esablishing (by inducion he las inequaliies in (5.1. I follows from (5.1 and from Exercise 3.3, page 21 in Karazas and Shreve (1991 ha he poinwise, increasing limis 5 11 Z ± = lim n Z ± n are poenials, ha is, nonnegaive F-supermaringales, wih RCLL (Righ Coninuous wih Lef Limis pahs and Z ± T = a.s., and saisfy E sup T Z ± 2 <. From (5.8 and Lemma A.5 in he Appendix, i develops hen ha Z ± solve he sysem (5.5. In he remainder of he proof we show ha we can assume, wihou loss of generaliy, ha Z ± have coninuous pahs on T. Le us consider he evens on which he processes Z ± of (5.11 undergo a (lef jump a = T, namely, by El Karoui (1981, we have B ± = Z ± T > = Z ± T Z ± T B + Z + T = Z T + L T B Z T = Z + T Ũ T

13 236 J. CVITANIĆ AND I. KARATZAS and hus from (3.7, B = B + B L T = Ũ T = Z + T = Z T mod P Case 1. Suppose ha, in addiion o (5.7, we have as well P L T < U T = 1 or equivalenly P L T = Ũ T = = Then Alario-Nazare, Lepelier and Marchal (1982, page 3 prove ha he (lef jumps of Z + and Z occur on disjoin evens; his implies, as hey also show, he regulariy propery (A.4 for he poenials Z ±, and hence he Doob Meyer decomposiion Z ± = π K ± K ± S 2 ci cf. he Appendix, condiion (A.4, Lemma A.1(ii and Definiion A.2. Corollary A.3 gives hen Z ± 2 c, and his complees he proof. Case 2. Suppose now ha P L T = Ũ T = > Then he above argumen guaranees he coninuiy of Z ±, K ± only on T. To overcome his, we inroduce he random variable ζ = Z + T 1 B = Z T 1 B a s and he nonnegaive supermaringales { Z ± E ζ < T 5 12 Z ± = = T The nonnegaiviy follows easily from Faou s lemma and he supermaringale propery of Z ± since, for < T, we have Z ± = Z ± E Z ± T 1 B Z ± E Z ± T ( = Z ± E lim n Z± T n 1 Z ± lim inf n ( Z E ± T n 1 On he oher hand, he processes of (5.12 are easily seen o be supermaringales; heir pahs are coninuous on T and we have Z ± T = Z ± T 1 B c so ha Z + T >, Z T > = B c B + B =. In oher words, he (possible, lef jumps of Z ± a = T occur on disjoin evens. If we manage o show ha Z ± of (5.12 also solve he sysem (5.5, hen we can jus repea he argumens of Case 1, his ime for he new pair Z + Z, o obain Z ± 2 c and hereby complee he proof of he heorem. Now, le us observe from (5.12 ha Z + Z + L; his is obvious for = T since Z + T = = Z T + L T, whereas on T we have a.s. Z + = Z + E ζ Z + L E ζ = Z + L

14 BACKWARD SDE S 237 Le R be anoher RCLL supermaringale wih R Z + L; hen { R + E ζ < T R = R T = T is a supermaringale and saisfies R T L T = Z T + L T, as well as R Z + L + E ζ = Z + L on T almos surely. Therefore, R dominaes Z + L, and hus R S Z + L = Z + ; in oher words, R Z +, almos surely. We conclude ha Z + = S Z + L. A compleely similar argumen hen leads o Z = S Z + Ũ, and his leads o he fac ha he pair Z + Z solves he sysem (5.5. Puing Theorems 5.2 and 5.3 ogeher, we have he following conclusion Under condiions (5.6 and (5.7, he BSE of Problem 5.1 has a unique soluion. Le us discuss now he solvabiliy of our original problem, he BSDE of Problem 2.1. We shall do ha by adaping, o our siuaion a hand, a fixed poin mehod due o Pardoux and Peng (199 and modified by El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1995. Theorem 5.4. For fixed, given ξ L1 2 and coninuous L, U in S2 1 saisfying (2.3, suppose ha Problem 5.1 has a unique soluion for every g H1 2. Then here is also a unique soluion X Y K o he BSDE of Problem 2.1, and he sae process X admis he sochasic game represenaion (4.3 of Theorem 4.1. Corollary 5.5. Under condiions (5.6 and (5.7, he BSDE of Problem 2.1 has a unique soluion X Y K, and he represenaion (4.3 of Theorem 4.1 holds. This follows direcly from (5.13 and Theorem 5.4. Proof of Theorem 5.4. Le us sar wih a pair χ in he se = X Y S H2 d X has coninuous pahs wih L X U T and X T = ξ a s and define g H1 2 by seing g ω = f ω χ ω ω, where f T R R d R is he coefficien funcion of Problem 2.1. For his g H1 2, he BSE of Problem 5.1 has, by assumpion, a unique soluion X Y K ; in paricular, he pair X Y belongs o he se of (5.14. This way, we have consruced a mapping 5 15 ϕ via X Y = ϕ χ

15 238 J. CVITANIĆ AND I. KARATZAS In order o esablish he unique solvabiliy of he BSDE of Problem 2.1, i is clearly sufficien o show ha he mapping ϕ of (5.15 is a conracion wih respec o an appropriae norm in S 2 1 H2 d ; as such, we shall ake ( 1/ X Y β = E e β X 2 + Y 2 d for an appropriae β o be deermined in (5.18 below. Le χ be anoher pair in he se of (5.14, denoe by X Y K, X Y = ϕ χ, K = K + K, he unique soluion of he BSE of Problem 5.1 wih g ω = f ω χ ω ω, and define χ = χ χ = X = X X Ȳ = Y Y K = K K Clearly d X = f χ f χ d d K + Ȳ db, and from Iô s rule, d e β X 2 = e β β X 2 + Ȳ 2 d 2 X d K + 2 X Ȳ db we obain he bounds (as argued below: 5 17 e β E X 2 + E = 2E +2E 2kE 4k 2 E +2 X f χ f χ d e βu β X u 2 + Ȳ u 2 du e βu X u d K u 2E e βu X u Ȳ u db u e βu X u f u χ u u f u χ u u du e βu X u χ u + u du e βu X u 2 du E e βu χ u 2 + u 2 du where k is he Lipschiz consan of (2.2. We have used in (5.17 he elemenary inequaliy he bounds a + b 2k y a + b = 2 2k y 2 ( 1/2 E e 2βu X u 2 Ȳ u 2 du ( ( e βt E 1 2 eβt E sup X T sup X 2 + T 4k 2 y 2 + a2 + b 2 2 1/2 Ȳ u 2 du Ȳ u 2 du <

16 BACKWARD SDE S 239 which, ogeher wih he Burkholder Davis Gundy inequaliies e.g., Karazas and Shreve (1991, page 166 imply ha he sochasic inegral in (5.17 is a maringale and hus has zero expecaion, and he inequaliy X d K = X X dk X X dk = X X dk + + X X dk + X X dk + + X X dk = X L dk + + L X dk + + X U dk + U X dk + X L dk + + L X dk+ + X U dk + U X dk which is a consequence of (2.5 and (2.6. Now choose 5 18 β = 1 + 4k 2 in (5.17 o obain 5 19 E e βu X u 2 + Ȳ u 2 du 1 2 E e βu χ u 2 + u 2 du he conracion propery ha we sough for he norm of (5.16. The proof of Theorem 5.4 is complee. 6. Exisence by penalizaion. We shall presen in his secion a second approach o he quesion of exisence of soluions o he BSE of Problem 5.1, which complemens he resul of (5.13 as i esablishes exisence under slighly differen condiions han (5.6 and (5.7; see (6.3 and (6.4. Under hese condiions one shows, in fac, ha he reflecion processes K ± are absoluely coninuous wih respec o Lebesgue measure, almos surely. This new approach considers a sequence of penalized versions (6.1 X n = ξ n + n g s ds + n X n s U n s ds L n s X n s ds Y n s db s T for n N, of he backward sochasic equaion (5.1, wih suiable random funcions ξ n, L n, U n and n N. From he sandard heory of unconsrained BSDE s Pardoux and Peng (199, equaion (6.1 has, for every n N, a unique F-adaped soluion X n Y n H1 2 H2 d. Then, wih 6 2 K ± n = k ± n u du k+ n = n L n X n k n = n X n U n and K n = K + n K n he idea is o show ha X n Y n K n n N converges o a riple X Y K of processes wih X S 2 1, Y H2 d and K = K+ K, K ± S 2 ci (in fac, wih K±

17 24 J. CVITANIĆ AND I. KARATZAS absoluely coninuous wih respec o Lebesgue measure, as we shall show, which solves he BSE of Problem 5.1. We shall assume in his secion ha 6 3 g S 2 1 and ha 6 4 here exis sequences U n n N, L n n N of Iô processes du n = u n d + v n db, dl n = l n d + m n db wih u n n N, l n n N bounded in S 2 1, v n n N Hd 2, m n n N Hd 2 and ξ n n N L1 2, such ha L n U n, T and L n T ξ n U n T hold almos surely for every n N, and, as n, ξ n ξ, sup T U n U, sup T L n L boh almos surely and in L1 2. Here, of course, ξ and U, L are he daa (erminal condiion and barriers, respecively for Problem 5.1. Condiion (6.4 imposes some regulariy on he boundary processes U and L in he form of uniform approximaion by Iô processes; i is saisfied rivially, if U and L are hemselves Iô processes in S 2 1. Theorem 6.1. Under condiions (6.3 and (6.4, he BSE of Problem 5.1 has a unique soluion X Y K ; in paricular, K is absoluely coninuous wih respec o Lebesgue measure, and X admis he sochasic game represenaion (4.3. The uniqueness is again a consequence of Theorem 4.1 (arguing jus as in Corollary 4.2, so we shall devoe he res of his secion o proving exisence. From Theorem 3.1 in Pardoux and Peng (199, he unconsrained BSDE of (6.1 has a unique, F-adaped soluion X n Y n H1 2 H2 d, for every n N. By analogy wih (4.2, here is an inerpreaion of his soluion in erms of a suiable sochasic game, which, again, provides as a corollary he uniqueness of he soluion o (6.1. Proposiion 6.2. For every n N, le n denoe he class of F-progressively measurable processes µ T n, and inroduce he payoff: 6 5 ( R n µ ν = ξ n exp + ( exp µ u + ν u du s µ u + ν u du g s + µ s L n s + ν s U n s ds

18 for every µ n, ν n. We have hen 6 6 E R n BACKWARD SDE S 241 µ ˆν n E R n ˆµ n ˆν n for every T, µ ν n n, where = X n E R n ˆµ n ν a.s. 6 7 ˆµ n = n1 Xn <L n ˆν n = n1 Xn >U n Corollary 6.3. The pair ˆµ n ˆν n n n of (6.7 is a saddlepoin for he sochasic game wih upper- and lower-values: 6 8 V n = ess inf ν n ess sup µ n V n = ess sup ess inf µ ν n n E R n µ ν E R n µ ν respecively. This game has value, namely V n, given as 6 9 V n = V n = V n = X n = E R n ˆµ n ˆν n a.s. Proof. From (6.1, and by applying Iô s rule o he produc of he proceses X n and exp µ u + ν u du, we obain ( X n = E ξ n exp µ u + ν u du ( ( exp s µ u + ν u du g s + µ s L n s + ν s U n s ds ( s exp µ u + ν u du µ s X n s L n s + n L n s X n s ds ( s exp µ u + ν u du ν s X n s U n s n X n s U n s ds almos surely, for every T, µ n, ν n, afer aking condiional expecaions wih respec o and noing ha he condiional expecaion of he sochasic inegral vanishes, since Y n Hd 2. In he las wo erms of (6.1, he inegrands saisfy µ X n L n + n L n X n ν X n U n n X n U n wih equaliy for µ = ˆµ n wih equaliy for ν = ˆν n

19 242 J. CVITANIĆ AND I. KARATZAS Now (6.6 follows from hese observaions and from (6.5, and leads direcly o (6.9. Le us define now 6 11 X n = X n U n ḡ n = g u n and observe ha (6.12 X n = ξ n U n T + +n n ḡ n s ds L n s U n s X n s ds X n s ds + v n s Y n s db s T from (6.1 and (6.4, as well as ( X n = ess sup ess inf E ξ n U n T exp µ u + ν u du µ ν n n ( s ( exp µ u + ν u du ḡ n s + µ s L n s U n s ds a.s. by analogy wih he sochasic game represenaion (6.9. From (6.13, and wih g = sup n g u n L1 2 by (6.3 and (6.4, we have ( s X n ess sup ess inf E exp µ u + ν u du ḡ n s ds µ ν n n ( s ess sup E exp µ u + n du ḡ n s ds µ n E exp n s ḡ n s ds 1 n E g and from Doob s maximal inequaliy, 6 14 E sup X n U n 2 c T n 2 A similar analysis yields 6 15 E sup L n X n 2 c n N T n2

20 BACKWARD SDE S 243 whence 6 16 E E sup X 2 n c E K + n T 2 + K n T 2 c T sup K n 2 c n N T follow as well, from (6.2. Here and in he sequel, c > denoes a real consan, whose value may vary from line o line. Lemma 6.4. We have E sup X n X m 2 + sup K n K m 2 T T Y n Y m 2 d as m n. Proof. From (6.1, (6.2 and Iô s rule d X n X m 2 = 2 X n X m dk n dk m dk+ n dk+ m we have E X n X m 2 + (6.18 = 2E +2E + Y n Y m db + Y n Y m 2 d Y n s Y m s 2 ds X n s L n s X m s L m s + L n s L m s dk + n s dk+ m s X m s U m s X n s U n s + U m s U n s dk n s dk m s Now, from (6.2, X n L n dk + n /d and X n U n dk n /d, so we obain from (6.18 E X n X m 2 + Y n s Y m s 2 ds 2c n m 6 19 as m n T

21 244 J. CVITANIĆ AND I. KARATZAS because c n m = E sup E T sup T X n s X m s dk n s dk m s L n s X n s dk + m s + L m s X m s dk + n s + L n s L m s dk + n s + dk+ m s + U n s U m s dk n s + dk m s + X m s U m s dk n s + X n s U n s dk m s ds goes o zero as n, m, by virue of (6.14 (6.16 and (6.4. In paricular, (6.19 gives (6.2 sup E X n X m 2 + E T On he oher hand, again by Iô s rule we have E sup X n X m T E sup = 2E T sup T ( 2c n m + 2E ( X n X m 2 + Y n Y m 2 d as m n Y n s Y m s 2 ds ( X n s X m s dk n s dk m s + X n s X m s Y n s Y m s db s sup X n s X m s T Y n s Y m s db s By he Burkholder Davis Gundy inequaliies, his las erm is bounded by 6 22 ( 2cE X n X m 2 Y n Y m 2 d 1/2 ( 1/2 E sup X n X m 4c 2 Y n Y m 2 d T 1 2 E sup X n X m 2 + 2c 2 E Y n Y m 2 d T

22 BACKWARD SDE S 245 for a suiable consan < c < and i follows from (6.19 (6.22 ha 6 23 E sup X n X m 2 as m n T Finally, from basic properies of he Iô inegral and Doob s maximal inequaliy, ( 2 E sup Y n s db s Y m s db s ce Y n s Y m s 2 ds T as m n, in conjuncion wih (6.2, which leads o 6 24 E sup K n K m 2 as m n T along wih (6.23 and K n = X n X n + g u du + T. Y n s db s, We conclude from (6.17 ha here exis coninuous adaped processes X K in S 2 1, as well as a process Y in H2 d, such ha E sup X n X 2 + sup K n K 2 T T Y n Y 2 d as n Thus, passing o he limi as n in (6.14 and (6.15, we obain, hanks o (6.4 and (6.25, E sup X U 2 = E sup L X 2 = T T so ha he riple X Y K saisfies (2.5, almos surely. On he oher hand, passing o he limi as n in 6 1 X n = ξ n + g u du + K n T K n Y n u db u T we obain from (6.25 ha he BSE (5.1 is saisfied as well. Now (6.16 shows ha he sequences k ± n n N of nonnegaive, F-progressively measurable processes of (6.2 are bounded in L 2 T ; consequenly, here exis F- progressively measurable processes k ± T such ha (along a relabelled subsequence, 6 26 k ± n k± as n weakly in L 2 T

23 246 J. CVITANIĆ AND I. KARATZAS For hese processes, we have E X n L n k + n d X L k + d = E X n X k + n d + X L k + n k+ d E + E sup T X n X K + n T + E + sup T X L k + n k+ d L L n k + n d L n L K + n T as n hanks o (6.16, E sup T X n X 2 in (6.25, E sup T L n L 2 in (6.4 and X L L 2 T in conjuncion wih (6.26. In paricular, (6.27 bu E X n L n k + n d E X L k + d X L k + d = X n L n k + n d as n X n L n L n X n d holds almos surely, for every n N. Therefore, he processes k ± of (6.26 saisfy 6 28 X L k + d = hanks o (6.27, and a similar argumen gives 6 29 U X k d = In order o finish he proof of exisence in Theorem 6.1, i hus remains o show ha 6 3 K = k + u du a.s. a.s. k u du = K holds almos surely, for every fixed T, because, since boh K K have coninuous pahs, his implies K = K, a.s. Proof of (6.3. The key idea here is o urn he weak convergence of (6.26 ino srong by considering convex combinaions; namely, he Banach Mazur lemma Dunford and Schwarz (1963, page 422, Corollary 14; Ekeland

24 BACKWARD SDE S 247 and Temam (1976, page 6 shows ha here exis, for every n N, an ineger N n n and weighs λ n j, j = n N n wih N n j=n λ n j = 1, such ha 6 31 N n k ± n = j=n λ n j k± j k± as n in L 2 T For fixed T, (6.31 and Jensen s inequaliy, give ( 2 k ± n s k± s ds k ± n s k± s 2 ds ( 2 E k ± n s k± s ds T E Therefore, (6.32 E K n K 2 k ± n s k± s 2 ds as n where K n = as n k + n s k n s ds On he oher hand, denoing by η 2 = E η 2 he norm of L1 2, we have from (6.25 ha K n K 2 < ε holds for arbirary ε > and all sufficienly large n N; herefore, we have as well K n K 2 = so ha N n j=n λ n j K j K 2 N n j=n 6 33 E K n K 2 as n and (6.3 follows from (6.32 and (6.33. λ n j K j K 2 < ε The proof of Theorem 6.1 is complee. Le us end his secion by exending he resuls of Theorem 6.1 for he BSE of Problem 5.1 o he BSDE of Problem 2.1. Theorem 6.5. Under he condiions (6.4, 6 34 f ω x y f ω x does no depend on y R d and 2 1 E sup f 2 ω T < he BSDE of Problem 2.1 has a unique soluion X Y K and he sochasic game represenaion (4.3 of Theorem 4.1 holds, wih 3 1 g ω f ω X ω

25 248 J. CVITANIĆ AND I. KARATZAS Proof. For given ξ L1 2 and coninuous L U in S2 1 saisfying (2.3 and (6.4, Problem 5.1 has a unique soluion X Y K in S 2 1 H2 d S2 1, for every given g S 2 1 see Theorem 6.1. Arguing now as in Theorem 5.4 and Corollary 5.5, wih he addiional observaion ha he process of 3 1 belongs o he space S 2 1 hanks o he assumpions (6.34, 2 1 and (2.2 on f, as well as he fac X S 2 1, we conclude ha he BSDE of Problem 2.1 also has a unique soluion. 7. A pahwise approach o Dynkin s game. We shall show in his secion ha he heory we have developed allows for a pahwise (deerminisic approach o he (sochasic Dynkin game of Theorem 4.1, an approach analogous o he pahwise reamen of he opimal sopping problem in Davis and Karazas (1994. Le X Y K solve he BSE of Problem 5.1 for given g H1 2 or, le X Y K solve he BSDE of Problem 2.1, and hen define g in H1 2 by (3.1; we shall assume for simpliciy, 7 1 L T = ξ = U T a.s. We obain from (3.8, almos surely, (7.2 X L ( = ξ L + + K + T K + U X ( = U ξ + K T K g s ds K T K g s ds K + T K + + for all T. In he more suggesive backward noaion (7.3 x + θ = X T θ L T θ x θ = U T θ X T θ a ± θ = K ± T K ± T θ y + θ = ξ L T θ + T θ g s ds a θ y θ = U T θ ξ g s ds a + θ + T θ T θ T θ Y s db s Y s db s Y s db s Y s db s θ T

26 he equaions (7.2 read BACKWARD SDE S x ± θ = y ± θ + a ± θ θ T and (2.6 reads 7 5 x ± θ da ± θ = almos surely. Now all processes in (7.3 have coninuous pahs, and hose of a ± are also increasing wih a ± = ; i follows from (7.4 and (7.5 ha a ± solve he Skorohod reflecion problem e.g., Karazas and Shreve (1991, page 21 associaed wih y ±, and hus 7 6 a ± θ = max max u θ y± u since y ± =, or equivalenly ( K + T = K + + max L τ ξ τ T 7 7 = K T + K ξ + max τ T K T = K + max σ T = max u θ y± u τ g u du +K T K τ + g u du ( L τ K τ K + ( ξ U σ + σ τ g u du +K + T K + σ = K + T K + ξ + g u du ( min U σ + K + σ K + + σ T σ τ θ T Y u db u g u du + λ τ σ Y u db u g u du + λ σ where = λ T is he coninuous, nonadaped process 7 8 λ = M T M wih M = Theorem 7.1. wih payoff 7 9 Q σ τ ω = Y u db u T For given ω, consider he pahwise (deerminisic game σ τ g u ω du + ξ ω 1 σ τ=t + L τ ω 1 τ<t τ σ +U σ ω 1 σ<τ + λ σ τ ω = R σ τ ω + λ σ τ ω σ τ T

27 25 J. CVITANIĆ AND I. KARATZAS and upper and lower values 7 1 W ω = min σ T max τ T Q σ τ ω W ω = max τ T min σ T Q σ τ ω respecively. Then, for a.e. ω, he following condiions hold: (i This game has a value. Tha is, 7 11 W ω = W ω = W ω = X ω + λ ω (ii The pair ˆσ ω ˆτ ω T 2, as in (4.4, is a saddlepoin for his game. Tha is, 7 12 Q ˆσ ω τ ω Q ˆσ ω ˆτ ω ω = X ω + λ ω Q σ ˆτ ω ω σ τ T 2 (iii ˆσ ω aains 7 13 σ min U σ ω + K + σ ω K + ω + λ σ ω + g u ω du σ T (iv ˆτ ω aains 7 14 = X ω + λ ω = W ω max L τ ω K τ ω + K ω + λ τ ω + τ T = X ω + λ ω = W ω τ g u ω du Finally, he value of he sochasic (Dynkin game of Theorem 4.1 is given as he opional projecion 7 15 V = X = E W a.s. of he value of he pahwise game, for every T. In oher words, he effec of he nonadaped compensaor process of (7.8 is o enforce he nonanicipaiviy consrain in he passage from he pahwise o he Dynkin game o ensure ha he random imes ω ˆσ ω, ω ˆτ ω in he saddle-poin ˆσ ω ˆτ ω of he pahwise game are sopping imes. Then he pair ˆσ ˆτ T 2 provides a saddlepoin for he Dynkin game cf. (4.5, and he value of his game is simply he condiional expecaion of he value for he pahwise game, given cf. (7.15.

28 1 2 BACKWARD SDE S 251 On he oher hand, he pair ˆσ ω ˆτ ω T 2 is also a saddle-poin for ye anoher pahwise game, his one wih payoff σ U σ ω + K + σ ω K + ω + λ σ ω + g u ω du L τ ω K τ ω K ω + λ τ ω + τ g u ω du σ τ T separable in he wo variables σ and τ; from (7.13 and (7.14, he value of his game coincides wih he value W ω = X ω + λ ω of he original, pahwise game of (7.9 (7.11. Proof of Theorem 7.1. The relaions of (7.12 follow direcly from (4.8, (4.1 and he definiion (7.9 whereas (7.11 is a simple consequence of (7.12. On he oher hand, (7.15 follows from (7.11, (4.3 and he maringale propery of he process M in (7.8 since E λ = E M T M = I remains o esablish (7.13 and (7.14. Le us drop he dependence on ω in he noaion of wha follows. Observe ha equaion (5.1 reads 7 16 X + λ = X ρ + ρ in our noaion of (7.8 and recall 7 17 a.s. g u du + K ρ K + λ ρ X ˆσ = U ˆσ K ˆσ = K and X ˆτ = L ˆτ K + ˆτ = K + for ρ T from (4.4, (2.6 and (7.1. Now he firs claim in (7.14 is equivalen o 7 14 L τ K τ + K + λ τ + τ g u du ˆτ L ˆτ K ˆτ + K + λ ˆτ + g u du and, from (7.16 and (7.17, he righ-hand side of 7 14 equals X ˆτ + K ˆτ K + λ ˆτ + ˆτ g u du = X + λ = X τ + K τ K + λ τ + τ τ T g u du his clearly dominaes L τ K τ + K + λ τ + τ g u du, esablishing Similarly, he firs claim in (7.13 amouns o U σ + K + σ K + + λ σ + g u du (7.13 ˆσ U ˆσ + K + ˆσ K + + λ ˆσ + g u du σ σ T

29 252 J. CVITANIĆ AND I. KARATZAS again from (7.16 and (7.17, he righ-hand side of 7 13 equals X ˆσ + K ˆσ K + λ ˆσ + ˆσ g u du = X + λ = X σ + K σ K + λ σ + σ g u du which is dominaed by U σ +K + σ K + +λ σ + σ g u du, esablishing Finally, he equaliies in (7.13 and (7.14 are consequences of (7.7 and (7.6, respecively, as well as of (5.1 and (7.11. APPENDIX Consider a complee probabiliy space P equipped wih a filraion F = T which saisfies = mod P, as well as he usual condiions of righ-coninuiy and augmenaion by P-null ses. On his space, le Z = Z T be a poenial, ha is, an F-supermaringale wih pahs which are nonnegaive, RCLL righ-coninuous on T, wih lef-limis on T and saisfy Z T =, almos surely. If his poenial is of class T, ha is, if A 1 he family Z τ τ is uniformly inegrable where is he class of F-sopping imes τ T, hen here exiss a unique naural increasing process A = A T, adaped o F, wih righ-coninuous pahs and A =, EA T <, such ha Z is indisinguishable from he poenial A 2 π A = E A T A T generaed by A: A 3 Z = π A T almos surely. Furhermore, if Z is regular in he sense A 4 lim EZ τ n = EZ τ for any τ τ n wih τ n τ a.s. n hen A has coninuous pahs. This is he classical Doob Meyer decomposiion of supermaringales and is well known e.g., Secion 1.4 in Karazas and Shreve (1991. The following resul is also known cf. Dellacherie and Meyer (1975, (198, El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1995, bu we include a proof for he reader s convenience. Lemma A.1. (i Le A be a coninuous, adaped and increasing process, wih A = and E A 2 T < ; hen he poenial Z = π A of (A.2 is regular and saisfies A 5 E sup Z 2 T <

30 BACKWARD SDE S 253 (ii Conversely, le Z be a regular poenial ha saisfies (A.5; hen he coninuous increasing process A of is Doob Meyer decomposiion (A.3 saisfies E A 2 T < Proof. (i From (A.2 and Doob s maximal inequaliy e.g., Karazas and Shreve (1991 we have E sup π 2 A E sup M 2 4E M 2 T = 4E A 2 T < T T where M = E A T, T. Regulariy follows easily. (ii The condiion (A.5 implies (A.1. Tha is, ha Z is of class T. Togeher wih regulariy, his gives a unique Doob Meyer decomposiion (A.3 wih A adaped, increasing and coninuous; in paricular, A 6 Now E A 2 ρ n = 2E A T = lim n A ρ n where ρ n = inf T / A n T ρn A ρ n A da ρn = 2E E A ρ n A da ρn = 2E E Z Z ρ n da 2E 2E sup Z A ρ n T 2 ρn Z da E sup Z 2 T E A 2 ρ n whence E A 2 ρ n 4E sup T Z 2 = c < ; leing n, we obain, from (A.6 and monoone convergence, E A 2 T c <. Definiion A.2. Denoe by 2 c he space of poenials Z which have coninuous pahs and saisfy (A.5. Denoe also by S 2 ci he space of coninuous, increasing, adaped processes A wih A = and E A 2 T <. Corollary A.3. Suppose ha F coincides wih he filraion F B = B T generaed by some sandard (d-dimensional Brownian moion process B. Then he spaces 2 c, S2 ci of Definiion A.2 can be pu ino a one-o-one correspondence via Z = π A A S 2 ci Z 2 c The proof follows direcly from Lemma A.1 and he fac ha F B - (Brownian maringales are represenable as sochasic inegrals wih respec o B (and have, hus, coninuous pahs.

31 254 J. CVITANIĆ AND I. KARATZAS For he remainder of his secion, le η T R be a given, F-adaped process wih RCLL pahs and η T =, a.s. and assume A 7 η = sup η L 1 T The Snell envelope of η is he process S η given as A 8 where S η = ess sup E η τ T τ T A 8 θ = τ / τ θ a s for θ T Then S η is a poenial, of class T (as i is dominaed by he maringale E η T, and is he smalles nonnegaive supermaringale dominaing η. Thus, i has a Doob Meyer decomposiion of he form A 9 S η = E A T A T for a unique naural increasing and adaped process A = A η wih righconinuous pahs and A =, EA T <. Furhermore, S η is regular, hus A has coninuous pahs, if η is quasilef-coninuous: lim sup A 1 n η τ n η τ a.s., for every τ n wih τ n τ, a.s. In erms of he Snell envelope of (A.8, we can obain he soluion of he opimal sopping problem A 11 associaed wih η, as follows: A 12 u = sup τ T Eη τ u = ES η T T The sopping ime ρ = inf u T / η u = S u η T aains he supremum in (A.11, and is hus opimal for his problem: u = Eη ρ. If, furhermore, η is quasi-lef-coninuous, so ha A coninuous by (A.1, hen we also have A 13 S η η da = (in oher words, A increases only on S = η and A 14 u = Eη ν where ν = inf u T / A u > A T ρ All hese resuls are sandard in he general heory of opimal sopping e.g., Neveu (1975, El Karoui (1981 and Karazas (1993. We shall also need he following properies. a.s.

32 BACKWARD SDE S 255 Lemma A.4. If η has quasi-lef-coninuous pahs and E sup T η 2 <, hen we have E A 2 T < for he coninuous, increasing process of (A.9. Proof. A 15 In he noaion of (A.7, we have hen ( E sup S 2 η E sup E η 2 4E η 2 < T T from Doob s maximal inequaliy; he resul follows hen from (A.1 and Lemma A.1(ii. Lemma A.5. If η, η n n N are adaped, RCLL processes wih η T = η n T = and η n η, T almos surely, as well as E sup T η + η n < for all n N, hen S η n S η T a.s. Proof. Clearly S n = S η n η n and S n+1 = S η n+1 S η n = S n for all n N, so S n n N is an increasing (poinwise sequence of nonnegaive, RCLL supermaringales, of class T ; i converges o a nonnegaive, RCLL supermaringale S = lim n S n e.g., Karazas and Shreve (1991, page 21. Clearly S η, and hus S S η as well; on he oher hand, η n η implies S n = S η n S η, whence S S η. REFERENCES Alario-Nazare, M. (1982. Jeux de Dynkin. Ph.D. disseraion, Univ. Franche-Comé, Besançon. Alario-Nazare, M., Lepelier, J. P. and Marchal, B. (1982. Dynkin games. Lecure Noes in Conrol and Inform. Sci Springer, Berlin. Bensoussan, A. and Friedman, A. (1974. Non-linear variaional inequaliies and differenial games wih sopping imes. J. Func. Anal Bismu, J. M. (1977. Sur un problème de Dynkin. Z. Wahrsch. Verw. Gebiee Davis, M. H. A. and Karazas, I. (1994. A deerminisic approach o opimal sopping. In Probabiliy, Saisics and Opimizaion (F. P. Kelly, ed Wiley, New York. Dellacherie, C. and Meyer, P. A. (1975, (198. Probabiliés e Poeniel. Chaps. I IV, V VIII. Hermann, Paris. Duffie, D. and Epsein, L. (1992. Sochasic differenial uiliy. Economerica Dunford, N. and Schwarz, J. T. (1963. Linear Operaors. I: General Theory. Wiley, New York. Dynkin, E. B. and Yushkevich, A. A. (1968. Theorems and Problems in Markov Processes. Plenum Press, New York. Ekeland, I. and Temam, R. (1976. Convex Analysis and Variaional Problems. Norh-Holland, Amserdam. El Karoui, N. (1981. Les aspecs probabilises du conrôle sochasique. Ecole d Eé de Sain- Flour IX Lecure Noes in Mah Springer, Berlin. El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1995. Refleced soluions of backward SDE s and relaed obsacle problems for PDEs. Preprin. Karazas, I. (1993. Lecure Noes on Opimal Sopping Problems. Unpublished manuscrip. Karazas, I. and Shreve, S. E. (1991. Brownian Moion and Sochasic Calculus, 2nd ed. Springer, New York. Lepelier, J. P. and Maingueneau, M. A. (1984. Le jeu de Dynkin en héorie générale sans l hypohèse de Mokobodski. Sochasics

33 256 J. CVITANIĆ AND I. KARATZAS Morimoo, H. (1984. Dynkin games and maringale mehods. Sochasics Neveu, J. (1975. Discree-Parameer Maringales. Norh-Holland, Amserdam. Pardoux, E. and Peng, S. (199. Adaped soluion of a backward sochasic differenial equaion. Sysems Conrol Le Sener, L. (1982. Zero-sum Markov games wih sopping and impulsive sraegies. Appl. Mah. Opim Deparmen of Saisics Columbia Universiy New York, New York cj@sa.columbia.edu Deparmens of Mahemaics and Saisics Columbia Universiy New York, New York ik@mah.columbia.edu