REFLECTED SOLUTIONS OF BACKWARD SDE S, AND RELATED OBSTACLE PROBLEMS FOR PDE S

Transcription

1 The Annals of Probabiliy 1997, Vol. 25, No. 2, REFLECTED SOLUTIONS OF BACKWARD SDE S, AND RELATED OBSTACLE PROBLEMS FOR PDE S By N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez Universié Pierre e Marie Curie, Ecole Normale Supérieure, Lyon, Universié de Provence, Shandong Universiy and Universié de Marne la Vallée We sudy refleced soluions of one-dimensional backward sochasic differenial equaions. The reflecion keeps he soluion above a given sochasic process. We prove uniqueness and exisence boh by a fixed poin argumen and by approximaion via penalizaion. We show ha when he coefficien has a special form, hen he soluion of our problem is he value funcion of a mixed opimal sopping opimal sochasic conrol problem. We finally show ha, when pu in a Markovian framework, he soluion of our refleced BSDE provides a probabilisic formula for he unique viscosiy soluion of an obsacle problem for a parabolic parial differenial equaion. 1. Inroducion. Backward sochasic differenial equaions, BSDE s in shor, were firs inroduced by Pardoux and Peng [17. I has been since widely recognized ha hey provide a useful framework for formulaing many problems in mahemaical finance; see in paricular [9 and [13. They also appear o be useful for problems in sochasic conrol and differenial games see [13 and [14), for consrucing Ɣ-maringales on manifolds wih prescribed limis see [5) and providing probabilisic formulas for soluions of sysems of quasi-linear parial differenial equaions see [18). In his paper, we sudy he case where he soluion is forced o say above a given sochasic process, called he obsacle. An increasing process is inroduced which pushes he soluion upwards, so ha i may remain above he obsacle. The problem is formulaed in deail in Secion 2. We show ha he soluion can be associaed wih a classical deerminisic Skorohod problem. From his, i is easy o derive ha he increasing process of he refleced BSDE can be expressed as an infimum. Furhermore, we sae ha he soluion of he BSDE corresponds o he value of an opimal sopping ime problem. In Secion 3, we sae some esimaes of he soluions from which we derive some inegrabiliy properies of he soluion. We also give some a priori esimaes on he spread of he soluions of wo RBSDE s. In Secion 4, we prove a comparison heorem, similar o ha in [13 and [19, for nonrefleced BSDE s. Then, we give some properies of he increasing process associaed wih he RBSDE. Received July 1995; revised July AMS 1991 subjec classificaions. 6H1, 6H3, 35K85. Key words and phrases. Backward sochasic differenial equaion, probabilisic represenaion of soluion of second order parabolic PDE, obsacle problems for second order parabolic PDE. 72

2 SOLUTIONS OF BACKWARD SDE S 73 In Secions 5 and 6, exisence is esablished via wo differen approximaion schemes. The firs one sudied in Secion 5 is a Picard-ype ieraive procedure. The definiion of he sequence requires a each sep he soluion of an opimal sopping ime problem, which is solved wih he help of he noion of he Snell envelope. The second approximaion is consruced by penalizaion of he consrain in Secion 6. In Secion 7, we resric ourselves o concave coefficiens, in which case he soluion of he RBSDE is shown o be he value funcion of a mixed opimal sopping opimal sochasic conrol problem. Finally, in Secion 8, we show ha, provided he problem is formulaed wihin a Markovian framework, he soluion of he refleced BSDE provides a probabilisic represenaion for he unique viscosiy soluion of an obsacle problem for a nonlinear parabolic parial differenial equaion. We noe ha obsacle problems for linear parial derivaive equaions appear as Hamilon Jacobi Bellman equaions for opimal sopping problems; see, for example, [3. This inerpreaion is generalized here o nonlinear PDE s. I has been noiced in [18 ha soluions of BSDE s are naurally conneced wih viscosiy soluions of possibly degenerae parabolic PDE s. The noion of viscosiy soluion, invened by M. Crandall and P. L. Lions, is a powerful ool for sudying PDE s wihou smoohness requiremen on he soluion. We refer he reader o he survey paper of Crandall, Ishii and Lions [4, from which we have borrowed several noions and resuls. We have also used some echniques from Barles [1 and Barles and Burdeau [2 for proving he uniqueness resul in Secion 8. Le us menion ha he main resul of his paper has already been applied o a financial problem in [ Refleced BSDE, Skorohod problem and sopping ime problem. Le B T be a d-dimensional sandard Brownian moion defined on a probabiliy space P. Le T be he naural filraion of B, where conains all P-null ses of and le be he σ-algebra of predicable subses of T. Le us inroduce some noaion. L 2 = { ξ is an T -measurable random variable s.. E ξ 2 < + } { } H 2 = ϕ T is a predicable process s.. E ϕ 2 d < + 2 = { ϕ T is a predicable process s.. E sup T We are given hree objecs: he firs is a erminal value ξ s.. i) ξ L 2. The second is a coefficien f, which is a map f T R R d R ϕ 2) } < +

3 74 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ such ha ii) y z R R d f y z H 2 iii) for some K> and all y, y R, z z R d, a.s. f y z f y z K y y + z z and he hird is an obsacle S T, which is a coninuous progressively measurable real-valued process saisfying iv) E S + 2) <. sup T We shall always assume ha S T ξ a.s. In he las secion, in order o ge a probabilisic represenaion for an obsacle problem for PDE s, we shall assume ha ξ, f and S are given funcions of a diffusion process X T. Le us now inroduce our refleced BSDE. The soluion of our RBSDE is a riple Y Z K, T of progressively measurable processes aking values in R, R d and R +, respecively, and saisfying: v) Z H 2, in paricular E v ) Y 2 and K T L 2 ; vi) Y = ξ + vii) Y S T; Z 2 d < f s Y s Z s ds + K T K viii) K is coninuous and increasing, K = and Z s db s T Y S dk =. Acually, a general soluion of our RBSDE should saisfy assumpions vi) o viii). Bu we will, above all, consider soluions which saisfy inegrabiliy assumpions, ha is, v) and v ). We will see laer in Secion 3 ha v ) follows from v) and furhermore see Remark 3.2) ha, wihou loss of generaliy, condiion iv) can be replaced by E sup T S 2 < Noe ha from vi) and viii) i follows ha Y is coninuous. Inuiively, dk /d represens he amoun of push upwards ha we add o dy /d, so ha he consrain vii) is saisfied. Condiion viii) says ha he push is minimal, in he sense ha we push only when he consrain is sauraed, ha is, when Y = S. Noice ha in a deerminisic framework, his corresponds o he Skorohod problem. Consequenly, we will be able o apply some well known properies of he Skorohod problem. Recall he Skorohod lemma see, e.g., [11 and [2, page 229). Lemma 2.1. Le x be a real-valued coninuous funcion on such ha x. There exiss a unique pair y k of funcions on such ha a) y = x+k, b) y is posiive and c) k is coninuous and increasing, k =

4 SOLUTIONS OF BACKWARD SDE S 75 and y dk =. The pair y k is said o be he soluion of he Skorohod problem. The funcion k is moreover given by k = sup x s s Now, our problem involves a Skorohod problem and consequenly, he increasing process can be wrien as a supremum. More precisely, we give he following proposiion. Proposiion 2.2. Le Y Z K T be a soluion of he above RBSDE saisfying condiions vi) o viii). Then for each T, ) 1) K T K = sup ξ + f s Y s Z s ds Z s db s S u u T u u Proof. Noice ha Y T ω S T ω K T ω K T ω T is he soluion of a Skorohod problem. Applying he Skorohod lemma wih x = ξ + f s Y s Z s ds Z s db s S T ) ω T T k = K T K T ω and y = Y T S T ω, we derive he desired resul. I is no a all clear from 1) ha K will be -adaped. The adapedness of Y K will come from he adjusmen of he process Z. In oher words, Z is he process which has he effec of making Y K adaped. In he following proposiion, we show ha he square-inegrable soluion Y of he RBSDE corresponds o he value of an opimal sopping ime problem. Proposiion 2.3. Le Y Z K T be a soluion of he above RBSDE saisfying condiions v) o viii). Then for each T, [ v 2) Y = ess sup E f s Y s Z s ds + S v 1 v<t + ξ1 v=t v where is he se of all sopping imes dominaed by T, and = v v T Proof. Le v. From v) and v ), we may ake he condiional expecaion in vi) wrien beween imes and v, hence [ v Y = E f s Y s Z s ds + Y v + K v K [ v E f s Y s Z s ds + S v 1 v<t + ξ1 v=t

5 76 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ We now choose an opimal elemen of in order o ge he reversed inequaliy. Le D = inf u T Y u = S u wih he convenion ha D = T if Y u >S u, u T. Now he condiion Y S dk = and he coninuiy of K imply ha K D K = I follows ha [ D Y = E f s Y s Z s ds + S D 1 D <T + ξ1 D =T Hence, he resul follows. Remark 2.4. Noe ha in he paricular case where f =, S T = ξ, i follows from he previous proposiions ha Y = E ξ + K T [ = E ξ + sup S + M T M ξ + Hence, since S T = ξ [ Y = sup v E S v =E sup S + M T M where M = Z s db s. The las ideniy has already been esablished in [6 for a quie general filraion no necessarily Brownian) and process S no even quasi-lef-coninuous). 3. Some a priori esimaes. We will now give some esimaes of Y in order o derive some inegrabiliy properies of Y, when Z is supposed o be square-inegrable. In oher words, we wan o prove ha condiion v) implies condiion v ). Firs, we show ha Y is smaller han a square-inegrable process soluion of a forward SDE which depends on he process Z and has iniial condiion Y. Second, we show ha Y is greaer han a square-inegrable process soluion of a backward SDE which depends on he processes Y and Z. Proposiion 3.1. a) Le Y Z K T be he soluion of he above RBSDE saisfying assumpions vi) o viii). Le us consider Y T, he square-inegrable) soluion of he forward SDE Then Y = Y Y Y f s Y s Z s ds + T a.s. Z s db s

6 SOLUTIONS OF BACKWARD SDE S 77 If assumpion v) Z H 2 is saisfied and using he assumpion ha is rivial), hen Y 2 and consequenly, Y + 2. b) Le Y Z K T be he soluion of he above RBSDE saisfying assumpions vi) o viii) and assumpion v). Le β be he bounded process defined by 3) β = f Y Z f Z Y if Y and β = oherwise. Le Y Z be he square-inegrable) soluion of he classical backward SDE: 4) Then dy = β Y + f Z d Z db Y T = ξ Y Y T a.s. Proof. by Noice ha Y T is soluion of he forward SDE given Y = Y f s Y s Z s ds K + Z s db s The resul follows by applying he comparison heorem for ordinary differenial equaions. More precisely, we have Y Y = α s Y s Y s ds + K where α s = f s Y s Z s f s Y s Z s /Y s Y s if Y s Y s and oherwise. From ha and from he fac ha f is Lipschiz wih respec o y, and hence α is bounded, i follows ha Y Y. Noe ha when Z is square-inegrable, he square-inegrabiliy of Y follows from he fac ha Y is deerminisic and hence square-inegrable. I remains o show he second esimae. The mehod will consis in linearizing he equaion wih respec o Z, and exploiing some echniques used in [13 for esablishing he comparison heorem. Firs, noice ha Y Z saisfies dy = β Y + f Z d + dk Z db where β is he process defined by 3). Noice ha, since f is Lipschiz wih respec o y, he process β is bounded. Define R = exp β s ds, and inroduce he discouned processes: Ỹ = R Y ; Z = R Z ; K = R s dk s. Applying Iô s formula o R Y, we easily prove ha 5) Consequenly, 6) Ỹ = R T ξ + Ỹ R T ξ + R s f s Z s ds + K T K R s f s Z s ds Z s db s Z s db s

7 78 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Tha is, Ỹ is greaer han a square-inegrable process. Hence, using he esimae a), i follows ha Y is square-inegrable. Thus, by aking condiional expecaion in inequaliy 6), we prove ha Ỹ is greaer han he squareinegrable process R Y, where Y is soluion of BSDE 4). Esimae b) follows easily. Remark 3.2. Furhermore, we have ha if Y Z K T is he soluion of he above RBSDE saisfying assumpions vi) o viii) and he inegrabiliy assumpion v), hen, Y Y, T, where Y Z corresponds o he soluion of he BSDE wihou consrain, 7) Y = ξ + f s Y s Z s ds Z s db s So, we can replace S by S Y ; consequenly, we may assume wihou loss of generaliy ha E sup T S 2 <, ha is, ha S 2. Furhermore, we have shown ha if he process Z is square-inegrable, hen Y and K are also square-inegrable. More precisely, we sae he corollary. Corollary 3.3. Le Y Z K, T be a soluion of he above RBSDE saisfying assumpions vi) o viii) and he inegrabiliy assumpion v) on Z. Then condiion v ) is saisfied; ha is, [ α E Y 2 + K2 T < ha is, Y H 2 K T L 2 sup T β { } Y s Z s db s T is a uniformly inegrable maringale. Proof. Le us prove he second claim [ ) 1/2 [ E Y 2 Z 2 d E sup Y T 1 2 E sup Y 2 T ) 1/2 Z 2 d ) + 12 E Z 2 d and β follows from he Davis Burkholder Gundy inequaliy for he firs momen of he supremum of a maringale. Remark 3.4. Recall ha he square-inegrabiliy of Y in Proposiion 3.1 was esablished by using he fac ha he σ-algebra is rivial, which implies ha Y is deerminisic and hence square-inegrable. Anoher proof of Corollary 3.3 can be given which does no use he fac ha Y is deerminisic. We have jus showed ha Ỹ Z K is a soluion of equaion 5); more precisely, Ỹ Z K is a soluion of he refleced BSDE

8 SOLUTIONS OF BACKWARD SDE S 79 associaed wih he coefficien R f Z, he erminal condiion R T ξ and he obsacle S = R S. Then, applying Proposiion 2.2, we have K T = sup R T ξ + u T u R s f s Z s ds u Z s db s S u ) and hence K T R T ξ + R s f s Z s ds + sup u T u Z s db s + sup u T S + u Using he Burkolder Davis Gundy inequaliy, i is easy o prove ha E K 2 T < +. Furhermore, by equaion 5), we conclude ha E sup T Y 2 < + We now give a more precise a priori esimae on he norm of he soluion. Proposiion 3.5. Le Y Z K, T be a soluion of he above RBSDE. Then here exiss a consan C such ha E sup Y 2 + T Z 2 d + K 2 T ) CE ξ 2 + f 2 d + sup S + ) 2 T Proof. Applying Iô s formula o he process Y and he funcion y y 2 yields Y 2 + Z s 2 ds = ξ = ξ Y s f s Y s Z s ds Y s dk s 2 Y s f s Y s Z s ds S s dk s 2 Y s Z s db s Y s Z s db s where we have used he ideniy Y S dk =.

9 71 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Using Corollary 2.2 and he Lipschiz propery of f, we have ha, wih c = 1 + 2K + 2K 2, ) E Y 2 + Z s 2 ds [ = E ξ Y s f s Y s Z s ds + 2 S s dk s [ E ξ K [ E ξ 2 + Y s f s ds Y s 2 + Y s Z s ds + 2 f s 2 ds + 2 Gronwall s lemma applied o Y gives: 8) I follows ha 9) E E Y 2 CE [ξ 2 + [ Z 2 s ds C E ξ 2 + S s dk s + c f s 2 ds + 2 S s dk s f s 2 ds + 2 We now give an esimae of E K 2 T. From he equaion K T = Y ξ f Y Z d + Y 2 s ds S s dk s S s dk s Z db and esimaes 8) and 9), we show he following inequaliies: E K 2 T [ξ CE 2 + f s 2 ds + 2 S s dk s Consequenly, [ CE ξ 2 + f s 2 ds + 2C 2 E [ sup s T Z 2 s ds S + s E K2 T E K 2 T [ξ CE 2 + f 2 s ds + sup S + s 2 s T I follows easily ha for each T, ) E Y 2 + Z 2 d + K 2 T CE ξ 2 + f 2 d + sup S + ) 2 T The resul hen follows easily from Burkholder s inequaliy. We can now esimae he variaion in he soluion induced by a variaion in he daa.

10 SOLUTIONS OF BACKWARD SDE S 711 Proposiion 3.6. Le ξ f S and ξ f S be wo riples saisfying he above assumpions, in paricular i), ii), iii) and iv). Suppose Y Z K is a soluion of he RBSDE ξ f S and Y Z K is a soluion of he RBSDE ξ f S. Define ξ = ξ ξ f = f f S = S S Y = Y Y Z = Z Z K = K K Then here exiss a consan c such ha ) E sup Y 2 + Z 2 d + K T 2 T ) ce ξ 2 + f Y Z 2 d where [ T = E ξ 2 + [ + c E sup T S 2) 1/2 1/2 T f 2 d+ sup S + 2 +ξ 2 + T f 2 d+ sup S + 2 T Proof. The compuaions are similar o hose in he previous proof, so we shall only skech he argumen. Since Y s S s d K s, E Y 2 + E Z s 2 ds E ξ E + 2E f s Y s Z s Y s ds f s Y s Z s f s Y s Z s Y s ds S s d K s Argumens already used in he previous proof lead o [ E Y E Z s 2 ds ce ξ 2 + f s Y s Z s 2 + Y s 2 ds + sup T S ) K T + K T I remains o use Gronwall s lemma, Proposiion 2.3 and he Burkholder Davis Gundy inequaliy. We deduce immediaely he following uniqueness resul from he Proposiion 3.6 wih ξ = ξ, f = f and S = S. Corollary 3.7. Under he assumpions i), ii), iii) and iv), here exiss a mos one progressively measurable riple Y Z K, T, which saisfies v), vi), vii) and viii).

11 712 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Remark 3.8. Insead of saying ha a riple Y Z K T of R R d R + -valued progressively measurable processes is a soluion of our RBSDE, we could say ha a pair Y Z T of R R d -valued progressively measurable processes saisfying v) and vii) is a soluion of our RBSDE, meaning ha, if K T is defined by vi), hen he pair Y K also saisfies viii). In ha sense, i follows from Corollary 3.7 ha here exiss a mos one pair Y Z T of progressively measurable processes which solves he RBSDE. 4. Comparison heorem and properies of he increasing process. We nex prove a comparison heorem, similar o ha of [19 and [13 for nonrefleced BSDE s. Theorem 4.1. Le ξ f S and ξ f S be wo ses of daa, each one saisfying all he assumpions i), ii), iii) and iv) [wih he excepion ha he Lipschiz condiion iii) could be saisfied by eiher f or f only, and suppose in addiion he following: i) ξ ξ a.s., ii) f y z f y z dp d a.e., y z R R d, iii) S S, T, a.s. Le Y Z K be a soluion of he RBSDE wih daa ξ f S and Y Z K a soluion of he RBSDE wih daa ξ f S. Then Y Y T a.s. Proof. Applying Iô s formula o Y Y + 2, and aking he expecaion see Corollary 3.3), we have: E Y Y E 2E + 2E 1 Ys >Y s Z s Z s 2 ds Y s Y s + f s Y s Z s f s Y s Z s ds Y s Y s + dk s dk s Since on Y >Y, Y >S S,wehave Y s Y s + dk s dk s = Y s Y s + dk s

12 SOLUTIONS OF BACKWARD SDE S 713 Assume now ha he Lipschiz condiion in he saemen applies o f. Then Hence E Y Y E 2E 2KE E 1 Ys >Y s Z s Z s 2 ds Y s Y s + f s Y s Z s f s Y s Z s ds Y s Y s + Y s Y s + Z s Z s ds 1 Ys >Y s Z s Z s 2 ds + KE E Y Y + 2 KE Y s Y s + 2 ds and from Gronwall s lemma, Y Y + =, T. Y s Y s + 2 ds We noe ha our noion of RBSDE has much similariy wih he classical noion of refleced forward) SDE. However, we shall give a proposiion and proof exhibiing he main difference beween he wo noions: a leas in case of a regular obsacle, he increasing process is absoluely coninuous. Proposiion 4.2. Assume he condiions i) iv) on he daa, and moreover ha S is a semimaringale of he form S = S + U s ds + V s db s where U and V are, respecively, R and R d -valued progressively measurable processes saisfying U + V 2 d < a.s. Le Y Z K be a soluion of he RBSDE. Then 1) and Z = V dp d a.e. on he se Y = S 11) dk 1 Y =S f S V +U d Proof. I follows from vi) and he assumpion ha d Y S = f Y Z +U d dk + Z V db If we denoe by L T he local ime a of he coninuous semimaringale Y S, i follows from he Iô Tanaka formula ha d Y S + = 1 Y >S f Y Z +U d + 1 Y >S Z V db dl

13 714 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Bu Y S + Y S, from vii). Hence he wo above differenials coincide, and so do he maringale and bounded variaion pars. Consequenly, 1 Y =S Z V db = from which he firs saemen follows, and 12) dk dl = 1 Y =S f Y Z +U d = 1 Y =S f S V +U d Hence 13) dk dl = 1 Y =S f S V +U d The second resul follows from he fac ha K is increasing. Noe ha we have proved ha he local ime a of Y S is absoluely coninuous. Remark 4.3. This propery can be generalized easily o an obsacle S which is a more general semimaringale. S = S + U s ds + A + V s db s where A is a coninuous process of inegrable variaion such ha he measure da is singular wih respec o d and which admis as a decomposiion A = A + A, where A+ and A are increasing processes. Also, U and V are, respecively, R and R d -valued progressively measurable processes saisfying: U d + V 2 d +A + T + A T < The firs equaliy 1) is sill saisfied and he second esimae 11) or, more precisely, equaion 12) is replaced by a.s. dk dl = 1 Y =S f S V d + U d + da = 1 Y =S f S V +U d + da I follows ha here exiss a predicable process α T such ha α 1 and dk = α 1 Y =S f S V +U d + da Remark 4.4. The local ime L aofy S is no always idenically equal o zero. Tha is, he process α is no always equal o 1 as is shown by a counerexample given by Jacka [15. Le B be a Brownian moion on he filered space P wih = σ B s s. Le l b be he local ime a b of B. Define S = B a B +a for some fixed a>.

14 SOLUTIONS OF BACKWARD SDE S 715 Noice ha by Tanaka s formula, he semimaringale S admis he following Doob Meyer decomposiion: S = S + A + sgn B s a db s sgn B s + a db s where he finie variaion process A is given by A = l a l a. In his example, he coefficien f is aken o be equal o and he erminal condiion ξ is equal o S T. From Proposiion 2.3, he process Y T associaed wih he RBSDE corresponding o hose parameers is he Snell envelope of S ; ha is, wih he decomposiion Y = ess sup v E S v / dy = α 1 Y =S dl a Z dw Y T = S T Noice ha he funcion x x a x + a is bounded above by 2a and achieves is maximum a any x a. If B > a, le us inroduce D = inf s /B s a T. Recall ha D = inf s /Y = S T is he opimal ime sopping for Y T. Le us show ha D = D. I is sufficien o show ha B a = Y = S. Firs, i is clear ha B a Y = S. Le us show he inverse inclusion: suppose ha B > a, hen, Y S E l a D la l a D l a. Now, i is clear ha l a D l a =. Furhermore, E l a D l a > since here is a posiive probabiliy ha l a will increase on D. I follows ha Y S >. Consequenly, B a = Y = S and hence D = D.We have [ Y = E l a T l a T + α 1 Y =S dl a [ = E l a T 1 α 1 Y =S dl a Since Y = E l a D, wehave E [ l a T la D = E [ 1 α 1 Y =S dl a Now, E l a T la D > and hence, he process α is no idenically equal o 1. Jacka [15 has compued α explicily: α = 2φ 2a/ T 1/2 1 where φ is he sandard normal disribuion funcion.

15 716 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ 5. Exisence of a soluion of he RBSDE by Picard ieraion. One approach o he soluion of forward) refleced SDE s is o use he soluion of he Skorohod problem for consrucing a Picard-ype ieraive approximaion o he refleced equaion, see, for example, [11. We shall use he same approach here for our RBSDE. Noe ha in he forward case he soluion of he Skorohod problem is given explicily. Here, he Skorohod problem is replaced by a more complicaed problem which involves opimal sopping and which we shall call he backward reflecion problem, BRP in shor. I is as follows. Suppose ha f does no depend on y z ; ha is, i is a given progressively measurable process saisfying ii E f 2 d < A soluion o he BRP is a riple Y Z K which saisfies v), vii), viii) and vi ) Y = ξ + f s ds + K T K Assuming w.l.o.g. ha K =, we deduce ha Y + f s ds = Y K + Z s db s Z s db s T T Hence Y + f s ds T is a supermaringale, which from vii) dominaes he process S + f s ds T. We now esablish he following proposiion. Proposiion 5.1. Under he assumpions i), ii) and iv), he BRP v), vi ), vii) and viii), has a unique soluion Y Z K T. Proof. Uniqueness follows from Corollary 3.7. We now prove exisence. From Proposiion 2.3, le us inroduce he process Y T defined by [ v Y = ess sup E f s ds + S v 1 v<t + ξ1 v=t T v The process Y + f s ds is he value funcion of an opimal sopping ime problem wih payoff: H = f s ds + S 1 <T + ξ1 =T By he heory of Snell envelopes cf. [1 and [16), i is also he smalles coninuous supermaringale which dominaes H. The coninuiy of Y follows from ha of H on he inerval T, and he assumpion ha he jump of H a ime T is posiive. We have moreover ha [ Y E ξ + f d + sup S T

16 Hence, by Burkholder s inequaliy, ) E ce ξ 2 + sup T SOLUTIONS OF BACKWARD SDE S 717 Y 2 Denoe by D he sopping ime Then D is opimal, in he sense ha ) f 2 d + sup S 2 T D = inf u T Y u S u T 14) [ D Y = E f s ds + S D 1 D <T + ξ1 D =T T Le us now inroduce he Doob Meyer decomposiion of he coninuous supermaringale Y + f s ds. There exiss an adaped increasing coninuous process K and a coninuous uniformly inegrable maringale M such ha Y = M f s ds K K = and K = K D. Indeed, by condiion vi), we have ha [ D Y = E f s ds + S D 1 D <T + ξ1 D =T + K D K T I hen follows from 14) ha E K D K = and hence K D equivalenly Y S dk =. I remains o prove some inegrabiliy resuls. Since { } Y + f s ds T = K,or is a square-inegrable supermaringale which dominaes he square-inegrable maringale { } E f s ds + ξ ) T i follows from Theorem VII.8 in Delacherie and Meyer [8 ha K T is squareinegrable. Hence he maringale ) M = E M T =E ξ + f s ds K T is also square-inegrable. Finally, since is a Brownian filraion, M = Z s db s, where E Z 2 d <. Acually, we can show direcly ha E Z 2 d <, which is equivalen o E K 2 T <. Indeed, le v T be a sopping ime such ha E K2 v <.

17 718 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ We have v E K 2 v =2E K v K dk = 2E v v = 2E [ 2E E K v K dk v E Y Y v 2 sup Y + T [ 2 E 2 sup Y + T f s ds ) dk f s ds )K v ) 2 1/2 f s ds EK 2 v 1/2 Taking he limi as v T, he resul follows. We can now esablish he following heorem. Theorem 5.2. Under he above assumpions, in paricular i), ii), iii) and iv), he RBSDE wih v), vi), vii), viii) has a unique soluion Y Z K. Proof. Denoe by he space of progressively measurable Y Z T wih values in R R d which saisfy v) and vii). We define a mapping from ino iself as follows. Given U V, le Y Z = U V be he unique elemen of which is such ha, if we define he process K = Y Y f s U s V s ds + Z s db s T hen he riple Y Z K solves he BRP associaed wih f s =f s U s V s. In oher words, he pair Y Z is he unique soluion of he same BRP, in he sense of Remark 3.8. Le U V be anoher elemen of, and define Y Z = U V, U = U U V = V V Y = Y Y Z = Z Z I follows from argumens similar o hose in he proofs of Proposiions 3.5 and 3.6 ha for any β>, e β E Y 2 T +E e βs[ βy 2 s + Z s 2 ds = 2E 4K 2 E e βs Y s f s U s V s f s U s V s ds e βs Y 2 T s ds E e βs[ U 2 s + V s 2 ds

18 SOLUTIONS OF BACKWARD SDE S 719 so ha if we choose β = 4K 2 + 1, we deduce E e β[ Y 2 + Z 2 d 1 2 E e β[ U 2 + V 2 d Hence he mapping is a sric conracion on equipped wih he norm 1/2 Y Z β = E e β Y 2 + Z 2 d) and i has a unique fixed poin, which is he unique soluion of he RBSDE in he sense of Remark 3.8). 6. Exisence of a soluion of he RBSDE: approximaion via penalizaion. In his secion, we will give anoher proof of Theorem 5.2, based on approximaion via penalizaion. The resul of his secion will be useful in Secion 8. In he following, c will denoe a consan whose value can vary from line o line. For each n N, le Y n Z n T denoe he unique pair of progressively measurable processes wih values in R R d saisfying and 15) E Z n 2 d < Y n = ξ + f s Y n s Zn s ds + n Y n s S s ds Z n s db s where ξ and f saisfy he assumpions saed in Secion 2. We define K n = n Y n s S s ds T I follows from he heory of unconsrained) BSDE s ha for each n, E Y n 2) < sup T We now esablish a priori esimaes, uniform in n, on he sequence Y n Z n K n. E Y n 2 + E = E ξ 2 + 2E Z n s 2 ds f s Y n s Zn s Yn s ds + 2E Y n s dkn s E ξ 2 + 2E f s +K Y n s +K Zn s Yn s ds + 2E S s dk n s ) c 1 + E Y n s 2 ds E Z n s 2 ds + 1 [ α E sup S αe K n T Kn 2 T

19 72 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ bu K n T Kn = Yn ξ f s Y n s Zn s ds + Z n s db s Hence { E K n T Kn 2 c E Y n 2 +E ξ Choosing α = 1/3c, wehave 2 3 E Yn E Z n s 2 ds c 1 + E I hen follows from Gronwall s lemma ha } Y n s 2 + Z n s 2 ds ) Y n s 2 ds sup E Y n 2 +E T Z n 2 d + E K n T 2 c n N Using again equaion 15) and he Burkholder Davis Gundy inequaliy, we deduce ha ) 16) E sup Y n 2 + Z n 2 d + K n T 2 c n N T Noe ha if we define f n y z =f y z +n y S f n y z f n+1 y z and i follows from he comparison Theorem 4.1 in fac is version for nonrefleced BSDE s, from [19 or [13, is sufficien for our purpose) ha Y n Y n+1, T, a.s. Hence and from 16) and Faou s lemma, Y n Y T a.s. E sup T Y 2 ) c I hen follows by dominaed convergence ha 17) E Y Y n 2 d as n

20 SOLUTIONS OF BACKWARD SDE S 721 Now i follows from Iô s formula ha E Y n Yp 2 +E Z n s Zp s 2 ds = 2E f Y n s Zn s f Yp s Zp s Yn s Yp s ds + 2E Y n s Yp s d Kn s Kp s 2KE Y n s Yp s 2 + Y n s Yp s Zn s Zp s ds + 2E Y n s S s dk p s + 2E Y p s S s dk n s from which one deduces he exisence of a consan c such ha E Z n s Zp s 2 ds ce Y n s Yp s 2 ds + 4E Y n s S s dk p s 18) + 4E Y p s S s dk n s Le us admi for a momen he following lemma. Lemma 6.1. E sup T Y n S 2) as n We can now conclude. Indeed, 16) and Lemma 6.1 imply ha E Y n S dk p + E hence from 17) and 18): Y p S dk n as n p Moreover, E Y n Yp 2 + Z n Zp 2 d as n p Y n Yp 2 + = Z n s Zp s 2 ds f Y n s Zn s f Yp s Zp s Yn s Yp s ds Y n s Yp s d Kn s Kp s Y n s Yp s Zn s Zp s db s

21 722 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ and sup Y n Yp 2 2 T f Y n s Zn s f Yp s Zp s Yn s Yp s ds + 2 Y n s S s dk p s + 2 Y p s S s dk n s + 2 sup Y n s Yp s Zn s Zp s db s T and from he Burkholder Davis Gundy inequaliy, E Y n Yp 2) ce Y n Y p 2 + Z n Zp 2) ds sup T + 2E E Y n S dk p + 2E sup T Y n Yp 2) + ce Y p S dk n Z n Zp 2 ds Hence E sup Y n Y p 2, as n and p, and consequenly from 15), 19) E K n Kp 2) as n p sup T Consequenly here exiss a pair Z K of progressively measurable processes wih values in R d R such ha ) E Z Z n 2 d + sup K K n 2 T as n, and v) and vi) are saisfied by he riple Y Z K ; vii) follows from Lemma 6.1. I remains o check viii). Clearly, K is increasing. Moreover, we have jus seen ha Y n K n ends o Y K uniformly in in probabiliy. Then he measure dk n ends o dk weakly in probabiliy, Y n S dk n Y S dk in probabiliy, as n. We deduce from he same argumen and Lemma 6.1 ha On he oher hand, Hence Y S dk Y n S dk n Y S dk = n N and we have proved ha Y Z K solves he RBSDE. We finally urn o he proof. a.s.

22 SOLUTIONS OF BACKWARD SDE S 723 Proof of Lemma 6.1. Since Y n Y, we can w.l.o.g. replace S by S Y ; ha is, we may assume ha E sup T S 2 <. We firs wan o compare a.s. Y and S for all T, while we do no know ye ha Y is a.s. coninuous. From he comparison heorem for BSDE s, we have ha a.s. Y n Ỹ n T n N, where Ỹn Z n T is he unique soluion of he BSDE Ỹ n = ξ + f Y n s Zn s ds + n S s Ỹn s ds Z n s db s Le v be a sopping ime such ha v T. Then [ Ỹ n v = E v e n T v ξ + e n s v f Y n s Zn s ds + n I is easily seen ha e n T v ξ + n v v e n s v S s ds ξ1 v=t + S v 1 v<t v e n s v S s ds a.s. and in L 2, and he condiional expecaion converges also in L 2. Moreover, T e n s v f Y n s Zn s ds 1 ) 1/2 f 2 Y n s Zn s ds 2n v hence E v T v e n s v f Y n s Zn s ds inl2, asn. Consequenly Ỹn v ξ1 v=t + S v 1 v<t in mean square, and Y v S v a.s. From his and he secion heorem in Dellacherie and Meyer [7, page 22, i follows ha a.s. Y S T Hence Y n S, T, a.s., and from Dini s heorem he convergence is uniform in. The resul finally follows by dominaed convergence, since Y n S S Y + S + Y. 7. Refleced backward sochasic differenial equaion and opimal sopping ime conrol problems. I is clear from Proposiion 5.1 ha in he case where f is a given sochasic process, he soluion Y T of he RBSDE which we called BRP in ha paricular case) is he value funcion of an opimal sopping ime problem. We shall now see how his fac can be generalized, firs o he case where f y z is a linear funcion of y z, and second o he case where f is a concave or convex) funcion of y z. In he laer case, Y T will be he value funcion of a mixure of an opimal sopping ime problem and a classical opimal sochasic conrol problem. We shall inerpre hose resuls in he Markovian case. Noe ha in ha case we shall make explici he corresponding Hamilon Jacobi Bellman equaion in he nex secion. We sar wih a proposiion.

23 724 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Proposiion 7.1. Suppose ha f is affine in y z; ha is, i akes he form f y z =δ + β y + γ z where δ β γ T are progressively measurable processes wih values in R R R d, such ha E O δ2 d <, β + γ C a.s., T. Le Ɣ T denoe he R-valued soluion of he linear SDE dɣ = Ɣ β d + γ db Ɣ = 1 Then he unique soluion Y Z K T of he BSDE wih coefficien f saisfies, for each T, [ u Ɣ Y = ess sup E Ɣ v ξ1 v=t + Ɣ v S v 1 v<t + Ɣ s δ s ds v Proof. I follows from Iô s formula ha Y Ɣ = ξɣ T + Ɣ s δ s ds + Ɣ s dk s Ɣ s Z s + Y s γ s db s Le Y Z K = Y Ɣ Ɣ Z + Y γ Ɣ s dk s, T. This riple solves he BRP wih final condiion ξɣ T and coefficien Ɣ δ T, wihou condiion v). Also we only have ha [ E ξɣ T 2 ε + δ Ɣ 2 ε d < for each ε>, and no for ε = ; he argumen leading o 2) in Proposiion 2.3 is sill valid here. Hence [ v Y Ɣ = ess sup E Ɣ T ξ1 v=t + Ɣ v S v 1 v<t + Ɣ s δ s ds v from which he resul follows. We now suppose ha for each fixed ω, f y z is a concave funcion of y z R R d. We define he conjugae funcion F β γ as follows. For each ω β γ T R R d, F ω β γ =sup f ω y z βy γ z y z D F ω = β γ R Rd F ω β γ < I follows from well-known resuls see, e.g., [13) ha f y z = inf F β γ +βy + γ z β γ D F he infimum is achieved, and he se D F is a.s. bounded.

24 SOLUTIONS OF BACKWARD SDE S 725 Le us now denoe by he se of bounded progressively measurable R R d - valued processes β γ T which are such ha E F β γ 2 d < To each β γ we associae he unique soluion Y β γ Z β γ K β γ T of he RBSDE wih he affine coefficien f β γ y z =F β γ + β y + γ z. We shall denoe Y Z K T he unique soluion of he RBSDE wih coefficien f y z. I follows from a secion heorem in [7, page 22, ha here exiss β γ such ha Hence f Y Z =F β γ +β Y + γ Z d dp Y Z K = Y β γ as value func- We can now deduce an inerpreaion of Y β γ ions of opimizaion problems. Theorem 7.2. For each β γ, where Y β γ a.e. Z β γ K β γ T a.s. and Y = Y β γ = ess sup v E v β γ v β γ =Ɣ β γ v S v 1 v<t + ξ1 v=t + v Ɣ β γ s F s β s γ s ds and for each T, Ɣ β γ s s T is he unique soluion of he linear SDE Moreover, dɣ s = Ɣ s β s ds + γ s db s Y = ess inf β γ Y β γ Ɣ = 1 = ess inf ess sup E v β γ β γ v = ess sup inf E v β γ ess v β γ In oher words, Y is he value funcion of a minimax conrol problem, and he riple β γ D, where D = inf s T Y s = S s is opimal. Proof. The firs par of he saemen follows from Proposiion 7.1. Moreover, from he comparison Theorem 4.1, Y Y β γ β γ

25 726 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ On he oher hand, and consequenly Y = ess Y = Y β γ inf β γ inf β γ Y β γ ess sup v E v β γ and he fac ha D is opimal follows from an argumen given in he proof of Proposiion 5.1. We finally prove ha ess inf and ess sup can be inerchanged. We cerainly have On he oher hand, where D β γ Y = ess inf β γ ess sup Y = ess ess inf v β γ inf β γ ess sup ess inf v β γ = inf s T Y β γ s ess sup v E v β γ E v β γ E D β γ β γ = S s. E v β γ We finally noe ha one has a similar represenaion of Y in case f is a convex funcion of y z, wih ess inf β γ ess sup v replaced by ess sup β γ ess sup v. 8. Relaion beween a RBSDE and an obsacle problem for a nonlinear parabolic PDE. In his secion, we will show ha he refleced BSDE sudied in he previous secions allows us o give a probabilisic represenaion of soluions of some obsacle problems for PDE s. For ha purpose, we will pu he RBSDE in a Markovian framework. Le b T R d R d and σ = T R d R d d be coninuous mappings, which are Lipschiz wih respec o heir second variable, uniformly wih respec o T. For each x T R d, le X x s s T be he unique R d -valued soluion of he SDE: X x s = x + s b r X x r dr + s σ r X x r db r We suppose now ha he daa ξ f S of he RBSDE ake he following form: ξ = g X x T f s y z =f s X x s y z S s = h s X x s

26 SOLUTIONS OF BACKWARD SDE S 727 where g, f and h are as follows. Firs, g C R d and has a mos polynomial growh a infiniy. Second, f T R d R R d R is joinly coninuous and for some K>, p N, saisfies 2) f x K 1 + x p 21) f x y z f x y z K y y + z z for T, x z z R d y y R Finally, h T R d R is joinly coninuous in and x and saisfies 22) h x K 1 + x p T x R d We assume moreover ha h T x g x x R d For each >, we denoe by s s T he naural filraion of he Brownian moion B s B s T, argumened by he P-null ses of. I follows from he resuls of he above secions ha for each x, here exiss a unique riple Y x Z x K x of s progressively measurable processes, which solves he following RBSDE: 23) i) ii) iii) iv) E Y x s Y x s Y x s = g X x T 2 + Z x 2 ds < s + Z x s s f r X x r Y r x Z r x r db r s T h s X x s <s T K s x is increasing and coninuous, and Y x s h s X s x dk s x = dr + K x T K x s We now consider he relaed obsacle problem for a parabolic PDE. Roughly speaking, a soluion of he obsacle problem is a funcion u T R d R which saisfies: min u x h x u ) 24) x L u x f x u x uσ x = u T x =g x x R d x T R d

27 728 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ where L = 1 2 d σσ 2 d x i j + x i x j i j=1 i=1 b i x x i More precisely, we shall consider soluions of 24) in he viscosiy sense. I will be convenien for he sequel o define he noion of viscosiy soluion in he language of sub- and super-jes; see [4. Below, S d will denoe he se of d d symmeric nonnegaive marices. Definiion 8.1. Le u C T R d and x T R d.wedenoe by 2 + u x [he parabolic superje of u a x he se of riples p q X R R d S d which are such ha u s y u x +p s + q y x X y x y x +o s + y x 2 Similarly, we denoe by 2 u x [he parabolic subje of u a x he se of riples p q X R R d S d which are such ha u s y u x +p s + q y x X y x y x +o s + y x 2 Example 8.2. Suppose ha ϕ C 1 2 T R d.ifu ϕhas a local maximum a x, hen ) ϕ x xϕ x 2 xϕ x 2 + u x If u ϕ has a local minimum a x, hen ) ϕ x xϕ x 2 xϕ x 2 u x We can now give he definiion of a viscosiy soluion of he parabolic obsacle problem 24). Definiion 8.3. a) I can be said ha u C T R d is a viscosiy subsoluion of 24) if u T x g x, x R d, and a any poin x T R d, for any p q X 2 + u x, min u x h x p 1 2 Tr ax b q f x u x qσ x ) In oher words a any poin x where u x >h x, p 1 Tr ax b q f x u x qσ x 2

28 SOLUTIONS OF BACKWARD SDE S 729 b) I can be said ha u C T R d is a viscosiy supersoluion of 24) if u T x g x, x R d, and a any poin x T R d, for any p q X 2 u x, min u x h x p 1 2 Tr ax b q f x u x qσ x ) In he oher words, a each poin, we have boh u x h x and p 1 Tr ax b q f x u x qσ x 2 c) u C T R d is said o be a viscosiy soluion of 24) if i is boh a viscosiy sub- and supersoluion. We now define 25) u x = Y x x T R d which is a deerminisic quaniy. Lemma 8.4. u C T R d Proof. We define Y s x for all s T by choosing Y s x s. I suffices o show ha whenever n x n x, 26) E Y n x n s Y s x 2) Indeed, his will show ha sup s T = Y x for s x Y s x is mean-square coninuous, and so is x Y x Bu Y x is deerminisic, since i is measurable. Now 26) is a consequence of Proposiion 3.6 and he following convergences as n : E E sup s T 1 T s f s X s x E g X x T g X n x n T 2 h s X s x h s X n x n s 2) 1 n T s f s X n x n s Y s x Z s x Y x s Z s x 2 ds which follow from he coninuiy assumpions, 2), 21), 22) and he polynomial growh of f, g and h. Defined by 25), u is a viscosiy soluion of he obsacle prob- Theorem 8.5. lem 24).

29 73 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Proof. We are going o use he approximaion of he RBSDE 23) by penalizaion, which was sudied in Secion 6. For each x T R d, n N, le n Y s x n Z s x s T denoe he soluion of he BSDE n Y s x = g X x T + f r X r x n Y r x n Z r x dr + n s s n Y x r I is known from [18 ha h r X r x s n Z r x db r s T u n x = n Y x T x R d is he viscosiy soluion of he parabolic PDE u n x +L u n x +f n x u n x u n σ x = T x R d u T x =g x x R d where f n x r pσ x = f x r pσ x + n r h x However, from he resuls of he previous secion, for each T, x R d, u n x u x as n Since u n and u are coninuous, i follows from Dini s heorem ha he above convergence is uniform on compacs. We now show ha u is a subsoluion of 24). Le x be a poin a which u x >h x, and le p q X 2 + u x. From Lemma 6.1 in [4, here exiss sequences n j + j x j x p j q j X j 2 + u nj j x j such ha p j q j X j p q X Bu for any j, p j 1 2 Tr ax j b q j f j x j u nj j x j q j σ j x j n j u nj j x j h j x j From he assumpion ha u x >h x and he uniform convergence of u n, i follows ha for j large enough u nj j x j >h j x j ; hence, aking he limi as j in he above inequaliy yields: p 1 Tr ax b q f x u x qσ x 2 and we have proved ha u is a subsoluion of 24).

30 SOLUTIONS OF BACKWARD SDE S 731 We conclude by showing ha u is a supersoluion of 24). Le x be an arbirary poin in T R d, and p q X 2 u x. We already know ha u x h x. By he same argumen as above, here exis sequences: such ha Bu for any j, Hence n j j x j x p j q j X j 2 u nj j x j p j q j X j p q X p j 1 2 Tr ax j b q j f j x j u nj j x j q j σ j x j n j u nj j x j h j x j p j 1 2 Tr ax j b q j f j x j u nj j x j q j σ x and aking he limi as j, we conclude ha: p 1 Tr ax b q f x u x qσ x 2 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 + such ha m R = and 27) f x r p f y r p m R x y 1 + p for all T, x y R, r R, p R d. Theorem 8.6. Under he above assumpion, including condiion 27), he obsacle problem 24) has a mos one viscosiy soluion in he class of coninuous funcions which grow a mos polynomially a infiniy. Proof. I suffices o show ha if u v C T R d have a mos polynomial growh a infiniy, saisfy u T x =v T x =g x, x R d, and are, respecively, a sub- and a supersoluion of he obsacle problem 24), hen u v. For some λ> o be chosen below, le ũ x =u x e λ ξ 1 x ṽ x =v x e λ ξ 1 x h x =h x e λ ξ 1 x g x =g x e λt ξ 1 x

31 732 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ where ξ x = 1+ x 2 k/2, and k N is choosen such ha ũ and ṽ are bounded. We noe ha η x =ξ 1 x Dξ x = k 1 + x 2 1 x κ x =ξ 1 x D 2 ξ x =k 1 + x 2 1 I k k x 2 2 x x where Dξ denoes he gradien of ξ, and D 2 ξ he marix of second order parial derivaives of ξ. Then ũ resp. ṽ ) is a bounded viscosiy subsoluion resp. supersoluion) of he obsacle problem: min ũ x h x where ũ ) x Lũ x f x ũ x ũσ x = ũ T x = g x Lϕ = Lϕ + aη Dϕ + [ 1 2 Tr aκ + b η λ ϕ f x ũ x ũσ x = e λ ξ 1 x f x e λ ξ x ũ x e λ ξ x Dũσ x +e λ Dξ x σũ x ) We rewrie he above problem as min ũ x h x ũ ) x +F x ũ x Dũ x D2 ũ x = We choose λ large enough so ha ũ T x = g x r F x r q X is sricly increasing for any x q X T R d R d S d, which is possible since aκ and b η are bounded. Hence F is proper in he erminology of [4, and i also saisfies 27), since in paricular aη is Lipschiz. From now on, we drop he ildes, and we make a las modificaion. Namely we replace v x by v x + ε/, wih ε>. Since ε is arbirary, if we prove ha for he ransformed funcions, u v saisfy u v, we will have proved he same inequaliy for he old funcions u and v. Moreover, since F is proper, and he old v was a supersoluion, we have ha 28) v + F x v x Dv x D2 v x ε 2 and moreover v x + as, uniformly in x.

32 SOLUTIONS OF BACKWARD SDE S 733 For any R>, le B R = x R d x <R. We need only show ha for any R>, sup u v + sup u v + T B R T B R since he righ-hand side ends o zero as R. Le us suppose ha for some R>, here exiss x T B R such ha 29) δ = u x v x > sup u v + T B R and we will find a conradicion. For each α>, le ˆ ˆx ŷ be a poin in he compac se T B R B R where he coninuous funcion α x y =u x v y α x y 2 2 achieves is maximum. Le us admi for a momen he following lemma. Lemma 8.7. i) For α large enough, ˆ x ŷ T B R B R. ii) α x ŷ 2 and x ŷ 2, asα. iii) u ˆ x v ˆ ŷ +δ. Theorem 8.3 from [4 ells us ha here exiss such ha p X Y R d d p α x ŷ X 2 + u ˆ x p α x ŷ Y 2 v ˆ x and 3) ) X I I 3α Y I I ) Now from Lemma 8.7iii), u ˆ x h ˆ ŷ +δ, since v is a supersoluion. Then since h is uniformly coninuous on compacs, for α large enough, u ˆ x > h ˆ x. Hence since u is a subsoluion, and from 28) Nex from Lemma 8.7iii), p + F ˆ x u ˆ x α x ŷ X p + F ˆ ŷ v ˆ ŷ α x ŷ Y ε/ 2 u ˆ x v ˆ ŷ

33 734 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Hence, since F is proper, and consequenly Define G by p + F ˆ x v ˆ ŷ α x ŷ X ε/ 2 F ˆ ŷ v ˆ ŷ α x ŷ Y F ˆ x v ˆ ŷ α x ŷ X F x r q X = 1 Tr ax +G x r q 2 We have ε/ 2 Tr a ˆ x X a ˆ ŷ Y + G ˆ ŷ v ˆ ŷ α x ŷ G ˆ x v ˆ ŷ α x ŷ ε/ 2 Tr a ˆ x X a ˆ ŷ Y + Kα x ŷ 2 + m R x ŷ +α x ŷ 2 where R = R sup x T BR v x, since G saisfies he same condiion as f in 27). However, from 3), q q R d, and Tr a ˆ x X a ˆ ŷ Y Xq q Yq q 3α q q 2 = Tr σ ˆ x Xσ ˆ x σ ˆ ŷ Yσ ˆ ŷ d = Xσ ˆ x e i σ ˆ x e i Yσ ˆ ŷ e i σ ˆ ŷ e i i=1 3αdK 2 x ŷ 2 Finally, we deduce ha ε/ 2 c x ŷ 2 + α x ŷ 2 +m R x ŷ +α x ŷ 2 which conradics Lemma 8.7. We proceed o he proof. Proof of Lemma 8.7. Le us firs prove ii). We have ha u ˆ ŷ v ˆ ŷ sup α y y y sup α x y x y = u ˆ x v ˆ ŷ α 2 x ŷ 2 Hence α 2 x ŷ 2 u ˆ x u ˆ ŷ

34 SOLUTIONS OF BACKWARD SDE S 735 and consequenly α x ŷ 2 is bounded, and as α, x ŷ. Since u is uniformly coninuous on T B R, ii) is esablished. We now prove iii). From 29), δ sup α x x x sup α x y x y u ˆ x v ˆ ŷ We finally prove i). Since u T x = g x = v T x T/ε, from ii), he uniform coninuiy of u T and v T on B R, and iii), ˆ <T. Since u and v ε/ are bounded, C ε α 2 x y 2 α x y C ε Taking he sup over x y in he lef inequaliy yields: C ε T α ˆ x ŷ C εˆ hence ˆ 2C + T 1 ε 1 ε> Moreover, from ii), iii) and he uniform coninuiy of u and v on T B R, for any <δ <δ, here exiss M such ha α M implies ha u ˆ x v ˆ x δ, u ˆ ŷ v ˆ ŷ δ. In view of 29), if δ is chosen close enough o δ, hese inequaliies imply ha x ŷ B R. In order o conclude from Theorems 8.5 and 8.6 ha u x = Y x is he unique viscosiy soluion of he obsacle problems 24), i remains o show ha i grows a mos polynomially a infiniy. A careful analysis of he esimaes leading o he inequaliy preceding 16) shows ha here exiss a universal consan c, independen of he daa, such ha for each n N, ) sup E Y n 2 ce T ξ 2 + f 2 s ds + sup S +2 T From Faou s lemma, he same inequaliy holds for Y = lim n Y n. Hence, wih he noaion of he presen secion, we have in paricular ha Y x 2 ce g X x T 2 + f 2 s X s x ds + sup h s X s x ) 2 s T The resul now follows from 2), 22), he same assumpion for g and he sandard esimae sup s T E X s x 2 c T 1 + x 2

35 736 EL KAROUI, KAPOUDJIAN, PARDOUX, PENG AND QUENEZ Remark 8.8. Suppose now ha for each x, f x y z is a concave funcion of y z. We hen associae he conjugae funcion by he formula F T R d R R d R F x β γ =sup f x y z βy γ z y z Define moreover Ɣ β γ s as in Theorem 7.2 and for v T, x v β γ =Ɣ β γ v + [ h v X x v Ɣ β γ v 1 v<t + g X x T s F s X x β s γ s ds s 1 v=t I follows from Theorem 7.2 ha Y x = ess inf ess sup E x v β γ / β γ A v and 24) is he Hamilon Jacobi Bellman equaion of he corresponding minimax conrol problem. REFERENCES [1 Barles, G. 1994). Soluions de Viscosié des Équaions de Hamilon Jacobi du Premier Ordre e Applicaions. Springer, New York. [2 Barles, G. and Burdeau, J. 1995). The Dirichle problem for semilinear second order degenerae ellipic equaions and applicaions o sochasic exi ime problems. Comm. Parial Differ. Equaions [3 Bensoussan, A. and Lions, J. L. 1978). Applicaions des Inéquaions Variionelles en Conrôle Sochasique. Dunod, Paris. [4 Crandall, M., Ishii, H. and Lions, P. L. 1992). User s guide o he viscosiy soluions of second order parial differenial equaions. Bull. Amer. Mah. Soc [5 Darling, R. 1995). Consrucing gamma maringales wih prescribed limis, using backward SDEs. Ann. Probab [6 Davis, M. and Karazas, I. 1994). A deerminisic approach o opimal sopping. In Probabiliy, Saisics and Opimizaion F. P. Kelly, ed.) Wiley, New York. [7 Dellacherie, C. and Meyer, P. A. 1975). Probabiliés e Poeniel. I IV. Hermann, Paris. [8 Dellacherie, C. and Meyer, P. A. 198). Probabiliés e Poeniel. V VIII. Hermann, Paris. [9 Duffie, D. and Epsein, L. 1992). Sochasic differenial uiliy. Economerica [1 El Karoui, N. 1981). Les aspecs probabilises du conrôle sochasique. In Ecole d Eé de Sain Flour Lecure Noes in Mah Springer, Berlin. [11 El Karoui, N. and Chaleya-Maurel, M. 1978). Un problème de réflexion e ses applicaions au emps local e aux equaions differenielles sochasiques sur R. Cas coninu. In Temps Locaux. Asérisque Soc. Mah. France, Paris. [12 El Karoui, N. and Jeanblanc-Picqué, M. 1993). Opimizaion of consumpion wih labor income. Unpublished manuscrip. [13 El Karoui, N., Peng, S. and Quenez, M. C. 1994). Backward sochasic differenial equaions in finance. Mah. Finance. To appear. [14 Hamadene, S. and Lepelier, J. P. 1995). Zero-sum sochasic differenial games and backward equaions. Sysems Conrol Le [15 Jacka, S. 1993). Local imes, opimal sopping and semimaringales. Ann. Probab

36 SOLUTIONS OF BACKWARD SDE S 737 [16 Karazas, I. and Shreve, S. 1996). Mahemaical Finance. To appear. [17 Pardoux, E. and Peng, S. 199). Adaped soluion of a backward sochasic differenial equaion. Sysems Conrol Le [18 Pardoux, E. and Peng, S. 1992). Backward SDEs and quasilinear PDEs. In Sochasic Parial Differenial Equaions and Their Applicaions B. L. Rozovskii and R. B. Sowers, eds.). Lecure Noes in Conrol and Inform. Sci Springer, Berlin. [19 Pardoux, E. and Peng, S. 1996). Some backward SDEs wih non-lipschiz coefficiens. Proc. Conf. Mez. To appear. [2 Revuz, D. and Yor, M. 1994). Coninuous Maringales and Brownian Moion. Springer, New York. N. El Karoui Laboraoire de Probabiliés, URA CNRS 224 Universié Pierre emarie Curie 4, Place Jussieu Paris Cedex 5 France ne@ccr.jussieu.fr E. Pardoux LATP, URA CNRS 225 Cenre de Mahémaiques e d Informaique Universié de Provence 39, rue F. JolioCurie F13453 Marseille cedex 13 France pardoux@gypis.univ-mrs.fr C. Kapoudjian Ecole Normale Supérieure 46, Allée d Ialie Lyon cedex France S. Peng Insiue of Mahemaics Shandong Universiy Jinan, 251 China pengsg@shandong.ihep.ac.cn M. C. Quenez Equipe de Mahémaiques Universié de Marne la Vallée 2 rue de la Bue Vere Noisy-Le-Grand France quenez@mah.univ-mlv.fr