A NUMERICAL SCHEME FOR BSDES. BY JIANFENG ZHANG University of Southern California, Los Angeles

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1 The Annals of Appled Probably 24, Vol. 14, No. 1, Insue of Mahemacal Sascs, 24 A NUMERICAL SCHEME FOR BSDES BY JIANFENG ZHANG Unversy of Souhern Calforna, Los Angeles In hs paper we propose a numercal scheme for a class of backward sochasc dfferenal equaons (BSDEs) wh possble pah-dependen ermnal values. We prove ha our scheme converges n he srong L 2 sense and derve s rae of convergence. As an nermedae sep we prove an L 2 -ype regulary of he soluon o such BSDEs. Such a noon of regulary, whch can be hough of as he modulus of connuy of he pahs n an L 2 sense, s new. Some oher feaures of our scheme nclude he followng: () boh componens of he soluon are approxmaed by sep processes (.e., pecewse consan processes); () he regulary requremens on he coeffcens are praccally mnmum ; () he dmenson of he negrals nvolved n he approxmaon s ndependen of he paron sze. 1. Inroducon. In hs paper we are neresed n he followng backward sochasc dfferenal equaon (BSDE, for shor): (1.1) Y = ξ + f(r,y r,z r )dr Z r dw r, where W s a Brownan moon defned on some complee, flered probably space (, F,P;F T ) and ξ F T. The BSDEs of hs knd, naed by Bsmu [3] and laer developed by Pardoux Peng [2], have been suded exensvely n he pas decade. We refer he readers o he books of El Karou Mazlak [9], Ma Yong [16] and he survey paper of El Karou Peng Quenez [1] for more nformaon on boh heory and applcaon, especally n mahemacal fnance and sochasc conrol, for such equaons. A long-sandng problem n he heory of BSDEs s o fnd an mplemenable numercal mehod. Many effors have been made n hs drecon as well. For example, n he Markovan case, Douglas, Ma and Proer [8] esablshed a numercal mehod for a class of forward backward SDEs, a more general verson of he BSDEs (1.1), based on a four sep scheme developed by Ma Proer Yong [15]. In hs Ph.D. hess, Chevance [6] proposed a numercal mehod for BSDEs by usng bnomal approach o approxmae he process Y. We should pon ou ha, besdes he Markovan requremen [.e., he ermnal value ξ has o be of he form g(x T ),wherex s some forward dffuson], boh mehods requre raher hgh regulary of he coeffcens. Receved Aprl 22; revsed February 23. AMS 2 subjec classfcaons. Prmary 6H1; secondary 65C3. Key words and phrases. Backward SDEs, L -Lpschz funconals, sep processes, L 2 -regulary. 459

2 46 J. ZHANG In he non-markovan case where he ermnal value ξ s allowed o depend on he hsory of a forward dffuson X, Bally [2] presened a dscrezaon scheme and obaned s rae of convergence. The man dea heren was o use a random me paron o overcome some dffculy n approxmang he marngale negrand he process Z n (1.1). However, hs scheme requres some exra approxmaons n order o gve an acual mplemenaon, and nvolves compung mulple negrals whose dmenson s proporonal o he paron sze, whch s que undesrable n mplemenaon. Recenly, Brand Delyon Mémn [5] and Ma Proer San Marn Torres [14] proposed some numercal mehods for BSDEs wh pah-dependen ermnal values. In hese works only weak convergence resuls are obaned. Fnally, n [23] Zhang and Zheng suggesed anoher mehod va PDE approach, whch also requres hgh regulary of coeffcens. We noe ha he man dffculy n a numercal scheme for BSDEs usually les n he approxmaon of he marngale negrand Z. In fac, n a sense he problem ofen comes down o he pah regulary of Z. To our bes knowledge, mos exsng mehods eher requre hgh regulary condons (e.g., [6, 15]) so as o guaranee he pah regulary of he process Z or oherwse lack a good rae of convergence (e.g., [2, 14]). In hs paper we ry o fnd some mddle ground among he exsng mehods. We shall consder a class of BSDEs whose ermnal value ξ akes he form (X), where X s a dffuson process and ( ) s a so-called L -Lpschz funconal (see Secon 2 for precse defnon). Wh he help of some resuls n our prevous work [18], we frs prove ha he marngale negrand Z sasfes a new ype of pah regulary, called he L 2 -regulary n hs paper, whch n essence characerzes he modulus of connuy n a mean-square sense. Wh such a regulary resul, and n lgh of an underlyng dscrezaon scheme, we hen desgn a numercal scheme and sudy s rae of convergence. Our man resuls are he followng: f s an L -Lpschz funconal, hen he asympoc rae of convergence s (log n)/n;f s a so-called L 1 -Lpschz funconal, or s of he form g(x T ), hen he rae of convergence wll be 1/ n. We should noe ha, whle he laer rae s more or less sandard (cf. [8]), he former rae of convergence, o our bes knowledge, s new. In fac, can be shown ha such rae s ndeed sharp (see Remark 4.3). There are several oher feaures of our numercal scheme. Frs, our approxmang soluons are all sep processes (.e., pecewse consan processes), whch s raher convenen n mplemenaons for obvous reasons. Second, besdes he L Lpschz propery of, our mehod vrually requres only Lpschz condons, whch s needed for he well posedness of he problem. Thrd, he hgh dmensonaly caused by he non-markovan naure of he BSDE s overcome sgnfcanly. We noe ha o mplemen a scheme whch nvolves calculang a ceran number of negrals, one needs o deal wh wo ypes of hgh dmensonaly: One s he dmenson of each negral and he oher s he number of negrals nvolved.

3 NUMERICAL SCHEME FOR BSDES 461 In our scheme, he former dmenson s equal o ha of W and he laer number, whch s exponenal o he paron sze n general, s lnear o ha sze f he non- Markovan problem can be convered o a Markovan one by addng one (or some) sae varable. We should noe ha hs ype of non-markovan problem has los of applcaons n fnance heory (e.g., lookback opons and Asan opons) and he dea of converng o a Markovan one has been exploed by many auhors (see, e.g., [4, 7]). The res of he paper s organzed as follows. In Secon 2 we presen some prelmnares. In Secon 3 we esablsh he L 2 -ype regulary for he process Z,as well as ha for X and Y. In Secon 4 we revew he Euler scheme used o dscreze he forward dffuson X; hen n Secon 5 we dscreze he BSDE and prove he rae of convergence. In Secon 6 he numercal scheme s presened explcly, wh specal aenon o he BSDEs whch can be convered o Markovan ones. 2. Prelmnares. Throughouhs paperwele T>beafxed ermnal me and (, F, F,P) be a complee, flered probably space on whch s defned a sandard Brownan moon W, such ha F =F T s he naural flraon of W, augmened by all he P-null ses. The followng spaces wll be frequenly used n he sequel: Le E denoe a generc Eucldean space wh nner produc, and norm. D s he space of all càdlàg funcons defned on [,T]. Cb m ([,T] E) s he space of all connuous funcons ϕ : [,T] E R,such ha ϕ has unformly bounded dervaves wh respec o he spaal varables up o order m. We ofen denoe Cb m = Cm b ([,T] E) for smplcy, when he conex s clear. C 1/2,1 ([,T] E) s he space of all connuous funcons ϕ : [,T] E R, such ha ϕ s unformly 1 2-Hölder connuous n and unformly Lpschz connuous n he spaal varables. Agan, we ofen denoe C 1/2,1 = C 1/2,1 ([,T] E) for smplcy, when he conex s clear. For 1 p<, L p (F T ) s he space of all F T -measurable and L p -negrable random varables; L p (F) s he space of all F-adaped processes ξ sasfyng = E ξ p d <. (2.1) ξ p p,t We consder he followng (decoupled) forward backward SDE: X = x + Y = (X) + b(s,x s )ds+ σ(s,x s )dw s, f(s,x s,y s,z s )ds Z s dw s, where b, σ and f are deermnsc funcons and s a deermnsc funconal. To smplfy presenaons, n wha follows we assume ha X R d,andw, Y and Z are all one-dmensonal (nong ha X and W may have dfferen

4 462 J. ZHANG dmensons). Bu he resuls can be exended o cases wh hgher-dmensonal W, Y and Z whou sgnfcan dffcules. For smplcy we also denoe he soluon o (2.1) by = (X,Y,Z). DEFINITION 2.1. A funconal : D d R s called L -Lpschz, f here exss a consan K such ha (2.2) (x 1 ) (x 2 ) K sup x 1 () x 2 () x 1, x 2 D d ; T and s called L 1 -Lpschz, f sasfes he followng esmae: (2.3) (x 1 ) (x 2 ) K x 1 () x 2 () d x 1, x 2 D d. Two ypcal examples of L -Lpschz and L 1 -Lpschz connuous funconals are (x) = max T x() and (x) = x() d, movaed by lookback opons and Asan opons, respecvely. The followng approxmaon resul, due o Ma Zhang [18], for L -Lpschz funconal wll be useful n he sequel. LEMMA 2.2. Suppose ha s an L -Lpschz funconal sasfyng he condon (2.2). Le =π be a famly of parons of [,T]. Then here exss a famly of dscree funconals g π : π such ha () for each π, assumng π : = < < n = T, we have ha g π C b (Rd(n+1) ), and sasfes (2.4) n x g π (x) K x = (x,...,x n ) R d(n+1), = where K s he same consan as ha n (2.2). () for any x D d, holds ha lm ( (2.5) g π x( ),...,x( n ) ) (x) =. π We shall make use of he followng Sandng assumpons: ASSUMPTION 2.3. The funcons b,σ,f C 1/2,1 ;and s an L -Lpschz funconal. We use a common consan K>odenoe all he Lpschz consans and assume ha b(,) + σ(,) + f(,,, ) + () K, sup T where s he consan funcon akng value on [,T].

5 NUMERICAL SCHEME FOR BSDES 463 The followng lemma collecs some sandard resuls n SDE and BSDE leraure. We ls hem for ready references. LEMMA 2.4. Suppose ha b, σ : [,T] R d R d and f : [,T] R R R are F-adaped random felds, such ha: (a) hey are unformly Lpschz connuous wh respec o x R d, y R and z R, wh he common Lpschz consan K>; (b) b(, ), σ(,), f(,, ) L 2 (F). For any ξ L 2 (F T ), denoe = (X, Y, Z) be he soluon o followng FBSDE: (2.6) (2.7) X = x + Y = ξ + b(s, X s )ds+ f(s,y s,z s )ds σ(s,x s )dw s, Z s dw s. Then, we have he followng esmaes: () for any p 2, here exss a consan C p >, dependng only on T, K and p, such ha E X p (2.8) (2.9) E sup T C p E x p + sup T ( Y p + Z 2 d [ b(, ) p + σ(,) p] d, ) p/2 C p E ξ p + f(,, ) p d, E X X s p (2.1) C p E x p + sup b(,) p + sup σ(,) p s p/2, T T E Y Y s p [ (2.11) C p E ξ p + sup ] f(,, ) p s p 1 T p/2 + Z r 2 dr. s () (Sably) Le ε = (X ε,y ε,z ε ) be he soluon o he perurbed FBSDE (2.6) and (2.7) n whch he coeffcens are replaced by b ε, σ ε, f ε, wh nal

6 464 J. ZHANG sae x ε and ermnal value ξ ε. Assume ha he assumpons (a) and (b) hold for all coeffcens b ε, σ ε and f ε and assume ha lm ε x ε = x, and for fxed (x,y,z), lm E b ε (, x) b(,x) 2 + σ ε (, x) σ(,x) 2 =, ε Then, we have lm E sup ε lm E ξ ε ξ 2 + ε T f ε (, y, z) f(,y,z) 2 d =. X ε X 2 + sup Y ε Y 2 + Z ε Z 2 d =. T The nex lemma conans some deeper resuls on he srucure of he soluon (Y, Z) o he BSDE (2.7). The proof of hese resuls can be found n [17, 18]. LEMMA 2.5. Assume ha Assumpon 2.3 holds. () Suppose ha dm(x) = dm(w), σσ T δi d for some consan δ> and ha sasfes he L -Lpschz condon (2.2), hen he process Z adms a càdlàg verson. () Suppose ha akes he specal form: (X) = g(x,...,x n ) and ha b,σ,f,g Cb 1. Denoe = ( X, Y, Z), =,...,n, o be he soluon o he followng varaonal equaons : (2.12) X = I d + Y = j j g X j + x b(r) X r dr + f (r) r dr x σ(r) X r dw r, Z r dw r, where I d s he d d deny marx, x b(r) s a d d marx whose jh column s j b(r,x r ), σ and f are defned n a smlar manner, and f (r) r = x f(r) X r + y f(r) Y r + z f(r) Z r. Then, for [,T], holds ha (2.13) where (2.14) Z = Y [ X ] 1 σ(,x ), n Y = Y 1 [ 1, )() + n Y T 1 T (). =1 REMARK 2.6. () Pars () and () n Lemma 2.4 are sandard for SDEs and BSDEs (see, e.g., [1, 11]). The par () can be found n [16]. () Lemma 2.5 gves suffcen condons for he pah regulary of Z. By Remark of [22], Z s also càdlàg f, n addon o Assumpon 2.3, σ s unformly Lpschz wh respec o.

7 NUMERICAL SCHEME FOR BSDES 465 To end hs secon, we gve a useful represenaon formula for he marngale negrand Z n he smples BSDE case. LEMMA 2.7. Assume ha g s a Lpschz connuous funcon and ha g(w T ) = Eg(W T )+ η dw for some predcable process η. Then η s a marngale and η = T 1 Eg(W T )W T. PROOF. Snce g s Lpschz, by [19] we know ha here exss ζ L 2 (F T ) such ha η = Eζ F, hus η s a marngale. The formula for η s a resul of [17]. 3. L 2 -regulary. In hs secon we esablsh he frs man resul of hs paper, whch we shall call he L 2 -regulary of he marngale negrand Z. Sucha regulary, combned wh he esmae for X and Y (3.2) below, plays a key role for dervng he rae of convergence of our numercal scheme n Secon 5. To begn wh, le π := < < n = T be a paron of [,T]. We would lke o esmae Z Z 1 2 n erms of he paron sze π = max, where =. We noe ha n general one canno expec an esmae of E Z Z s 2 as srong as ha for X (2.1) or ha for Y (2.11). In lgh of he norm for he process Z n (2.9), we wll ry o derve an esmae n he space of n=1 L 2 ([, ] ). Our man resul s he followng heorem. THEOREM 3.1. Assume ha Assumpon 2.3 holds rue and ha Z s càdlàg. Le π be any paron of [,T], hen he followng esmae holds: n [ E Z Z Z Z 2 ] d C(1 + x 2 ) π, =1 where C> s a consan dependng only on T and K, bu ndependen of he paron π. We noe agan ha Lemma 2.5 and Remark 2.6 gve some suffcen condons for he pah regulary of Z. The proof of Theorem 3.1 s que lenghy, we spl no several lemmas. The frs resul s neresng n s own rgh. LEMMA 3.2. Assume Assumpon 2.3 holds rue. Then for p 2, here exss a consan C p >, dependng only on T, K and p, such ha, (3.1) Z p C p (1 + x ) a.e. [,T]. Moreover, for any paron π, we have he followng esmae: (3.2) max sup 1 n (, ] E X X Y Y 1 2 C(1 + x 2 ) π,

8 466 J. ZHANG where C> s a consan dependng only on T and K, bu ndependen of he paron π. PROOF. Frs we assume ha all he condons n Lemma 2.5() hold rue (recallng ha a funcon n C 1/2,1 s only Lpschz connuous, bu no necessarly dfferenable on x,y,z!), and ha g sasfes (2.4). For p 2, by (2.8) we have Esup T X p C p and by (2.9) we ge, for =,...,n, E Y p sup T C p E C p E j [( n p T j g X j + x f(r) X r p dr p j g ) + 1 j= sup T ] sup X p T C p. Snce X s he soluon o he lnear SDE (2.12), one can easly check ha [ X] 1 also sasfes a lnear SDE and holds ha E [ X ] 1 p C p. Now recallng (2.13) and (2.14) and applyng Hölder s nequaly, we have Z p Y 3p [ X ] 1 3p σ(,x ) 3p C p (1 + X 3p ) C p (1 + x ). For he general case, le π be a sequence of parons of [,T], such ha π. By Lemma 2.2, we may choose smooh funcons g π sasfyng (2.4) and (2.5). Now le b π,σ π,f π be smooh molfers of b,σ,f, respecvely. For each π, by he above argumens we know Z π p C p (1+ x ),wherec p > s ndependen of π. Moreover, applyng Lemma 2.4() we ge ha (3.3) lm E Z π Z 2 d =. π Therefore, for a.e. [,T], here exss a subsequence of π such ha lm π Z π = Z, n probably. Applyng Faou s lemma, we hen oban ha Z p C p (1 + x ). To fnsh he proof we noe ha n (3.2) he esmae for X s a drec consequence of (2.1), whle by applyng (3.1) and (2.11) we can easly ge E Y Y 1 2 C(1 + x 2 ). The proof s now complee. The followng echncal lemma s he buldng block of he proof of Theorem 3.1.

9 ξ j LEMMA 3.3. NUMERICAL SCHEME FOR BSDES 467 Assume L 2 (F) and η j L 2 (F), j = 1,...,n. Denoe = α j + η j r dw r, where α j are some consans. Then E n =1 +1 where n+1 = n and 2 ( ) 2 η j n r dr 1 CE sup ξ j T, j T j=1 = sup s s. PROOF. By Iô s formula we have +1 E η j 1 r ηj 2 r dr F 1 = E ξ j 1 +1 ξ j 2 +1 ξ j 1 ξ j 2 F 1. Noe ha snce F 1, holds obvously ha +1 E η j 1 r ηj 2 r dr = E ( ξ j 1 +1 ξ j 2 +1 ξ j 1 ξ j ) 2 1. Denoe n+1 = T and 1 =. Snce s ncreasng, by some smple calculaon and applyng he Abel ransformaon one can show ha n +1 2 E η j r dr 1 (3.4) =1 E = E = E = E n j +1 =1 n =1 j 1,j 2 n =1 j 1,j 2 n [ j 1,j 2 =1 2 η j r dr 1 j +1 η j 1 r ηj 2 r dr ( ξ j 1 +1 ξ j 2 +1 ξ j 1 ξ j 2 ) 1 ξ j1 +1 ξ j 2 1 j 1 j 2 +1 ( 1 +1 ) + ξ j 1 j1 j 2 ξ j 2 j1 j 2 j1 j 2 + ξ j 1 j1 j 2 +1 ξ j 2 j1 j 2 +1 j1 j 2 +1 ξ j 1 ξ j 2 ξ j 1 1 ξ j ].

10 468 J. ZHANG Noe ha n E ξ j 1 j1 ξ j 2 j 2 j1 j 2 j1 j 2 j 1,j 2 =1 n n 2E ξ j1 j1 ξ j 2 j1 j1 j 1 =1 j 2 =j 1 n n = 2E ξ j1 j1 ξ j 2 T j1 j 1 =1 j 2 =j 1 n n = 2E E ξ j 1 T j1 F j1 ξ j 2 T j1 j 1 =1 j 2 =j 1 n 2E E ξ j 1 T n T F j1 ξ j2 T T j 1 =1 j 2 =j 1 n 2E E ξ j 1 T n T F j1 ξ j T T j 1 =1 j=1 n n = 2E ξ j 1 T T E ξ j T T F j1 j 1 =1 j=1 n n 2E ξ j 1 T T sup E ξ j T T F j 1 =1 T j=1 ( n ) 2E sup E ξ j 2 T T F T j=1 ( n ) 2 CE ξ j T T j=1 ( n ) 2 = CE ξ j T T, j=1 where he las nequaly s hanks o Doob s nequaly. Analogously we have n ( n ) 2 E ξ j 1 ξ j 2 j1 j 2 +1 j1 j 2 +1 j1 CE ξ j j 2 +1 T T. j 1,j 2 =1 j=1

11 Then (3.4) leads o ha NUMERICAL SCHEME FOR BSDES 469 n +1 2 E η j =1 r dr 1 j n ( n ) 2( E ξ j ) =1 j=1 ( n ) 2 ( + C n ) 2 ( T ξ j T ξ j n ) 2 ξ j 1 1 j=1 j=1 j=1 ( n ) 2 [ n ] E sup ξ j ( +1 ) + C T T j=1 = ( n ) 2 CE sup T. T ξ j j=1 Ths proves he lemma. Le ξ n = ξ and ξ j = forj = 1,...,n 1. Then he followng resul s a drec consequence of Lemma 3.3. COROLLARY 3.4. n E =1 If ξ = α + η r dw r and η, L 2 (F), hen +1 η r 2 dr 1 CE sup T ξ 2 T We now urn o he proof of Theorem 3.1. We would lke o use (3.3) and ake advanage of he represenaon formula (2.13) of Z π. Bu one should be careful ha (3.3) does no mply lm π E Z π Z 1 2 =.. PROOF OF THEOREM 3.1. We fx a paron π := < < n = T and wll prove he heorem for π. Le π : = s < <s m = T be any paron of [,T] fner han π and whou loss of generaly, we assume = s l for = 1,...,n.Forϕ = b,σ,f, le ϕ π Cb 1 be smooh molfers of ϕ such ha he dervaves of ϕ are bounded by K and lm π ϕ π = ϕ. Moreover, snce sasfes he L -Lpschz condon (2.2), by vrue of Lemma 2.2 one can fnd g π C 1 (R d(m+1) ) sasfyng (2.4) and (2.5). Le π = (X π,y π,z π ) denoe he adaped soluon o he

12 47 J. ZHANG followng FBSDE: (3.5) X π = x + Y π b π (r, Xr π )dr + σ π (r, Xr π )dw r, = g π ( Xs π,...,xs π ) T m + f π (r, π r )dr Zr π dw r. Now by (2.5), applyng Lemma 2.4() we know ha lm E [ (3.6) sup X π X 2 + Y π Y 2] + π T Z π Z 2 d =. By (3.6) here exss a subsequence of π such ha lm π E Z π Z 2 =, for d-a.s.. From now on we always assume π s n ha subsequence. Obvously we may fnd a sequence r k such ha lm E Z π π +r k Z +r 2 (3.7) k =, k. Whou loss of generaly, we assume + r k (, +1 ),forall and k. Noe ha, for [, ), E Z Z Z Z 2 CE Z Z π 2 + Z π Z π Z π 1 Z π 2 1 +r k (3.8) + Z π 1 +r k Z 1 +r 2 k + Z 1 +r k Z Z π Z π +r k 2 + Z π +r k Z +r k 2 + Z +r k Z 2. By (3.6), (3.7) and he rgh connuy of Z, o prove he heorem suffces o esmae E Z π Z π 1 2 for (, +1 ). To hs end we recall Lemma 2.5 wh he coeffcens ϕ replaced by ϕ π and denoe X π, Y π as he erms correspondng o X and Y, respecvely. For he convenence of applcaon, we shall rewre Y π n some oher form. Noe ha for each (2.12) s lnear. Le (γ,ζ ) and (γ j,ζ j ), j = 1,...,m,behe adaped soluons o he BSDEs (3.9) γ = γ j = j g π X π j + [ f π x (r) Xr π + f y π (r)γ r + f z π (r)ζ r ] T dr ζ r dw r, [ f π y (r)γr j + f z π (r)ζ r j ] T dr ζr j dw r, respecvely, hen we have he followng decomposon: (3.1) Y π = γ m + j= γ j, [s 1,s ).

13 NUMERICAL SCHEME FOR BSDES 471 We may smplfy (3.1) furher. Le us defne (3.11) = exp M = exp f π y (r) dr, f π z (r) dw r 1 2 fz π (r) 2 dr. Snce fz π s unformly bounded, by Grsanov s heorem (see, e.g., [12]) we know ha M s a P -marngale on [,T], and W = W fz π (r) dr, [,T], san F-Brownan moon on he new probably space (, F, P),where P s defned by d P dp = M T. Moreover, nong ha fy π and f z π are unformly bounded, one can deduce easly from (3.11) ha, for p 1, here exss a consan C p dependng only on T,K and p, such ha [ sup p + 1 p] C p, T [ E sup T M p + M 1 p] C p, (3.12) Now we defne (3.13) ξ = γ s p + 1 E M M s p + M 1 1 s p C p s p, M 1 s p C p s p/2. fx π (r) Xπ r 1 r dr, ζ = ζ 1, = γ 1 + fx π (r) Xπ r 1 r dr, ξ = g π Xs π 1 T, ζ = ζ 1, γ = γ 1. Then by (3.9) we have, for =,...,m, γ = ξ T ζ r d W r, [,T]. Therefore, by he Bayes rule (see, e.g., [12], Lemma 3.5.3) we have, for [,T] and = 1,...,m, (3.14) γ = γ = Ẽ ξ F = E M T ξ F M 1 γ = γ = ξ M 1 = ξ M 1, fx π (r) Xπ r 1 dr r f π x (r) Xπ r 1 r dr,

14 472 J. ZHANG where, for =, 1,...,m, (3.15) ξ = E M T ξ F = E MT ξ + ηr dw r. Noe ha f π z s bounded, hus M T L p (F T ) and X π L p (F) for all p 2. Therefore for each p 1, (2.4) leads o (3.16) n E M ξ j p T C p E M T p sup X π p C p. j=1 T In parcular, for each j, M T ξ j L 2 (F T ), hus (3.15) makes sense. Now we fx.for [s 1,s ), by applyng Lemma 2.5, (3.1) and (3.14) mply ha Z π = [( ξ + ξ j j ) M 1 f x (r) Xr π 1 r dr ] [ X π ] 1 σ π (, X π ). Therefore, (3.17) Z π Z π I I 2 + I 3, where (recallng ha 1 = s l 1!) I 1 I 2 I 3 [ = ξ + ξ j j ] M = ξ + j = 1 ξ j [ ξ + ξ 1 j 1] j l 1+1 [ X π ] 1σ π ( 1 1,X π ) 1, M 1 [ X π ] 1 σ π (, X π ) M 1 [ 1 1 X π ] 1σ π ( 1 1,X π ) 1, f π x (r) Xπ r 1 r dr [ X π ] 1 σ π (, X π ) fx π [ (r) Xπ r 1 r dr 1 X π ] 1σ π ( 1 1,X π ) 1. We assume ( 1, +1). Recallng (3.12) and applyng Lemma 2.4 one can

15 easly show ha NUMERICAL SCHEME FOR BSDES 473 (3.18) E I 3 2 C(1 + x 2 ) π. Recallng (3.15), (3.13) and (3.12) we have m m ξ j E M ξ j T F (3.19) j= j= E M T + M T 1 T f π x (r) Xπ r 1 r dr m g j π Xπ F s j CE M T sup Xs π F, s T where he las nequaly s due o (2.4) and he fac ha 1 and fx π are unformly bounded. Then by applyng (3.16), (3.12) and Lemma 2.4, one can show ha (3.2) E I 2 2 C(1 + x 2 ) π. I remans o esmae I 1. To hs end we denoe Ɣ = sup 1 + X π s + [ Xs π ] 1 + Ms 1, s Ɣ = sup 1 + Xs π. s Nong ha s bounded and ha Ɣ, 1 Ɣ F 1, and recallng ha 1 1 = s l 1, by (3.15) we have E [ I 1 2 CE Ɣ 4 Ɣ 2 1 ξ 1 + ] ξ j j CE Ɣ 4 1 j=1 [ ξ + 2 ξ 1 j 1] j l 1+1 Ɣ 2 1[ ξ ξ 2 1 (3.21) + l 1<j< ξ j 2 + j>l 1 ( j ξ ξ j ) 2] 1

16 474 J. ZHANG = CE Ɣ 4 Ɣ E + Ɣ 4 1 ξ j 2 T F l 1<j< Ɣ 2 1 E ξ ξ j>l 1 CE Ɣ 4 Ɣ 2 1 1[ ξ j 2 T l 1<j< 2 + η j r 1 j>l 1 CE Ɣ 4 1 Ɣ 2 1 [( l +1 j=l 1 ( j ξ ξ j ) 2 F ηr 2 dr 1 dr ] ) 2 ξ j T +1 + ηr 2 dr η j r 1 j>l 1 Applyng Lemma 3.3 and Corollary 3.4, (3.21) leads o ha n E I 1 [ 2 d + I 1 2 +r k + I 1 2 ] +r k =1 ( ) 2 n C E ƔT 4 Ɣ T 2 ξ j T =1 l 1 +1 j l +1 dr ]. (3.22) C π E Ɣ 4 T Ɣ 2 T [ + Ɣ 4 1 Ɣ ηr 2 dr [ n ( = l 1 +1 j l +1 ξ j T + sup ξ 2 + T ( ( ) 2 ] η r j dr k l k 1 +1 j l k ) 2 sup T n k=1 ξ j l k 1 +1 j l k ) 2 ]

17 NUMERICAL SCHEME FOR BSDES 475 ( ) 2 C π E ƔT 4 Ɣ m T 2 sup ξ j T j= ( 2 C π ƔT 4 3 Ɣ T 2 m 3 sup ξ j ) T j= 3 C(1 + x 2 m ) π sup ξ j 2. T j= 6 Now by (3.19) and applyng Doob s nequaly one has E ( sup T m j= ) 6 ξ j CE sup T M E T sup s T CE MT 6 sup X π 6 T C, X π s F 6 whch, ogeher wh (3.22), mples ha n E I 1 [ 2 d + I 1 2 +r k + I 1 2 (3.23) ] +r k C(1 + x 2 ) π. =1 Combnng (3.23), (3.2) and (3.18), we deduce from (3.17) ha n E Z π Z π 2 [ d + Z π 1 +r k Z π 2 + Z π +r k Z π 2 ] =1 C(1 + x 2 ) π, whch, combned wh (3.8), leads o n [ E Z Z Z Z 2 ] d (3.24) =1 T n CE Z π Z 2 d + Z π +r k Z 1 +r 2 k =1 n + C E Z 1 +r k Z Z +r k Z 2 =1 + C(1 + x 2 ) π.

18 476 J. ZHANG Recallng (3.6) and (3.7) and leng π, (3.24) mples ha (3.25) n =1 [ E Z Z Z Z 2 ] d n C E Z 1 +r k Z Z +r k Z 2 =1 + C(1 + x 2 ) π. Fnally, snce Z s càdlàg, applyng Lemma 3.2 and Faou s lemma we ge ha E Z p C p (1 + x ) for all [,T], whch mples ha Z T s unformly negrable. Now le r k, agan by he assumpon ha Z s càdlàg, we have lm E Z 1 +r k Z Z +r k Z 2 =, k whch, combned wh (3.25), proves he heorem. 4. Dscrezaon of he FSDE. In hs secon, we brefly revew he Euler scheme for he forward dffuson X. Leπ : = < 1 < < n = T be a paron of [,T]. Defneπ() =,for [, ).LeX π be he soluon of he followng SDE: X π = x + b ( π(r),xπ(r) π ) (4.1) dr + σ ( π(r),xπ(r)) π dwr, and we defne a sep process ˆX π as follows. (4.2) ˆX π = X π π(), [,T]. The followng esmaes are well known (see, e.g., [13]). LEMMA 4.1. Assume ha b and σ sasfy he condons n Assumpon 2.3. Then for X and X π defned as n (1.1) and (4.1), respecvely, here exss a consan C, dependng only on T and K, such ha E E sup T sup T X π 4 C(1 + x 4 ), X X π 2 C(1 + x 2 ) π. We now urn our aenon o he esmaes nvolvng ˆX π. The followng heorem s more or less sandard. For compleeness we shall skech a proof.

19 NUMERICAL SCHEME FOR BSDES 477 THEOREM 4.2. Assume ha b and σ sasfy he condons n Assumpon 2.3. Then here exss a consan C>, dependng only on T and K, such ha he followng esmaes hold: sup E X ˆX π 2 C(1 + x 2 ) π ; T E sup T X ˆX π 2 C(1 + x 2 ) π log 1 π. PROOF. Frs, for [, ), applyng Lemmas 3.2 and 4.1 we have E X ˆX π 2 = E X X π 2 1 2E X X X 1 X π 2 C(1 + x 2 ) π. Nex, recallng (4.1) one can easly ge (4.3) sup X ˆX π sup X X π + max sup X π X π T T 1 n 1 < sup X X π [ + max b (,X π ) ] T 1 n 1 [ + max σ ( ],X π ) 1 n 1 sup W W 1. < Now usng Lemmas 2.8 and 4.1, we nfer from (4.3) ha E sup X ˆX π 2 T CE sup X X π 2 + π 2 max b (,X π ) 2 T 1 n 1 + max σ (,X π ) 2 1 n 1 max sup W W 1 2 (4.4) 1 n < [ C(1 + x 2 ] ) π + E max sup W W n < [ ] C(1 + x 2 ) π + E max ( ) 2 N 4, 1 n

20 478 J. ZHANG where N, = 1, 2,...,n, are..d. random varables wh sandard Normal dsrbuon. Denoe C ε = 2ε log(1/ε),henwehavec π / 2log(1/ π ) and E max ( ) 2 N 4 1 n = E max ( ) 2 N n max N 2 C π (4.5) + E max ( ) 2 N n max N 2 C π n C π 2 + ( ) 2 E N 4 1 N 2 C π / =1 C 2 π + T π E N N 2 1 2log(1/ π ). Moreover, by drec calculaon we have, for a 1, E N1 4 1 ( N1 2 C a >a 3 exp a ) (4.6). 2 Seng a = 2log(1/ π ) and assumng ha he paron π s fne enough so ha a 1, we oban from (4.6) and (4.5) ha ( E max ( ) 2 N 4 C 2 π + C π log 1 ) 3/2 exp( log 1 ) n 1 π π ( C π 2 log 1 ) 2. π Ths, ogeher wh (4.4), proves he heorem. REMARK 4.3. We noe ha π log(1/ π ) s ndeed sharp. In fac, le X = W and π be a paron wh equal sze, one can show ha [ π log(1/ π )] 1 sup T X ˆX π 2 2 n dsrbuon (see, e.g., [1], Proposon 1). The followng resul s a drec consequence of Theorem 4.2. COROLLARY 4.4. Assume all he condons n Theorem 4.2 hold. If : D d R sasfes he L -Lpschz condon (2.2), hen E (X) ( ˆX π ) 2 C(1 + x 2 ) π log 1 π. Moreover, f sasfes he L 1 -Lpschz condon (2.3) or akes he form (X) = g(x T ), hen E (X) ( ˆX π ) 2 C(1 + x 2 ) π.

21 NUMERICAL SCHEME FOR BSDES Dscrezaon of he BSDE. In hs secon, we consruc an approxmang soluon (Y, Z) by usng he sep processes. Le π := < < n = T be any paron on [,T] and defne he approxmang pars (Y π,z π ) recursvely (n a backward manner), such ha (5.1) Y π n = ξ π, Z π,1 n =, Y π where ξ π L 2 (F T ), π,1 (5.2) = Y π + f (, π,1 ) = (X π,y π,z π,1 Z π, = E Zr π dr F +1 ) and Z π r dw r, [, ), = n, n 1,...,1,, =,...,n 1. We should pon ou here ha he famly (Y π,z π ) s dfferen from ha n (3.5). Also, we noe ha n (5.1) he ermnal value of he BSDE s Y π + f(, π,1 ). REMARK 5.1. As we shall see n (6.4) below, f one chooses ξ π appropraely, hen Z π,1 = Z π. Therefore he compuaon of he negral and he condonal expecaon n (5.2) becomes redundan and can be omed. Ths wll be sgnfcan n mplemenaon. To prove he convergence of (5.1), le us nroduce a new noon regardng he paron π. DEFINITION 5.2. Le K > be a consan. A paron π s called K-unform f π /K for = 1,...,n. THEOREM 5.3. Assume ha all he condons n Theorem 3.1 hold rue and suppose ha he paron π s K-unform. Then he followng esmae holds: max E Y Y π 2 n +E Z r Zr π 2 dr C [ (1+ x 2 ) π +E ξ ξ π 2], where ξ = (X) and C depends only on T and K. Before we prove he heorem, le us frs gve a dscree verson of he Gronwall nequaly for convenence. LEMMA 5.4. Assume π : = < < n = T. If a,b, =, 1,...,n, sasfy ha a n, b and ha a 1 (1 + C )a + b for = 1,...,n.

22 48 J. ZHANG Then we have max n a e CT [a n + n =1 b ]. PROOF. Noe ha 1 + C e C. By nducon one can easly show ha [ n a e C(T ) a T + b j ], whch clearly proves he lemma. j=+1 PROOF OF THEOREM 5.3. We frs denoe, for = 1,...,n, I 1 = E Y 1 Y π Z r Zr π 2 dr. By (2.1) and (5.1) we have Y 1 Y π + (Z r Z π r )dw r = Y Y π + f(r, r )dr f(, π,1 ). Squarng boh sdes and hen akng expecaons, also nong ha Y 1 Y π 1 s uncorrelaed wh (Z r Zr π)dw r,wege [ (Y I 1 = E Y π ) ( + f(r, r ) f ( ], π,1 )) 2 dr E Y Y π [ + C r + X r X π + Y r Y π + Z r Z π,1 ] 2 dr. Noe ha for any ε>, one has ( (a + b) 2 = a 2 + b 2 + 2ab 1 + ε ) ( a ε ) b 2. Thus we have ( I 1 E 1 + C ) Y ε Y π 2 ( + C 1 + ε ) ( [ π + X r X π + Y r Y + ) Z r Z π,1 ] 2 (5.3) dr

23 ( E ( E 1 + C ε + C(ε + ) 1 + C ε NUMERICAL SCHEME FOR BSDES 481 ) Y Y π 2 [ π + X r X 2 + X X π 2 + Y r Y 2 + Z r Z 2 + Z Z π,1 2 ] dr ) Y Y π 2 + C Z r Z 2 dr + C(ε + ) Z Z π,1 2 + C π 2 hanks o Lemmas 3.2 and 4.1. Snce π s K-unform, we have K +1. Therefore, E Z Z π,1 2 CE +1 Z Z π,1 2 = C [ +1 ( E E Z π ) ] 2 F r Z dr CE Zr π Z 2 dr +1 [ C Z π r Z r 2 + Z r Z 2 ] dr. Thus (5.3) leads o ( I 1 E 1 + C ) Y ε Y π C1 (ε + ) Zr π Z r 2 dr +1 + C Z r Z 2 dr + C π 2. Now choosng ε = 1/(4C 1 ), hen for π 1/(4C 1 ),wehave I 1 E (1 + C ) Y π +1 Y Zr π Z r 2 dr +1 + C Z r Z 2 dr + C 2. Therefore, (5.4) I E +1 Zr π Z r 2 dr +1 (1 + C )I + CE Z r Z 2 dr + π 2.,

24 482 J. ZHANG Applyng Lemma 5.4 and recallng Theorem 3.1, we have n max I CE ξ ξ π 2 [ + Z r Z Z r Z 2 ] dr n (5.5) =1 C [ (1 + x 2 ) π +E ξ ξ π 2]. + (1 + x 2 ) π Moreover, summng boh sdes of (5.4) over from o n 1, we have n 2 I E Zr π Z r 2 dr Therefore, (5.6) = n 1 = (1 + C )I + C [ (1 + x 2 ) π +E ξ ξ π 2]. E Zr π Z r 2 dr 2(1 + C n 1 )I n 1 + C [ n 1 = I + (1 + x 2 ) π +E ξ ξ π 2]. Pluggng (5.5) no (5.6), one can easly prove ha E Z r Zr π 2 dr CE ξ ξ π 2 + C(1 + x 2 ) π, whch, combned wh (5.5), proves he heorem. REMARK 5.5. If f s ndependen of z, hen he K-unform assumpon of π s no necessary. In fac, n such a case (5.3) does no nvolve he process Z. Thus, one may apply Lemma 5.4 drecly on (5.3) and oban a smplfed proof. Now le us defne he followng wo sep processes: Ŷ π = Y π, Ẑπ Then we have he followng heorem. = Z π,1 for [, ). THEOREM 5.6. Assume ha all he condons n Theorem 3.1 hold rue and le K> be gven. Then for any K-unform paron π, he followng esmae

25 NUMERICAL SCHEME FOR BSDES 483 holds: sup E Y Ŷ π 2 + E Z r Ẑπ r 2 dr T C [ (1 + x 2 ) π +E ξ ξ π 2]. PROOF. Frs, combnng Lemma 3.2, Theorem 5.3 and Lemma 2.4 we have, for [, ), E Y Ŷ π 2 2E Y Y Y 1 Y π 2 1 (5.7) C [ (1 + x 2 + E ξ 2 ) π +E ξ ξ π 2] C [ (1 + x 2 ) π +E ξ ξ π 2]. To esmae Z Ẑ π, we recall ha a condonal expecaon mnmzes he condonal mean square error. By (5.2) one can easly show ha E Zr π Zπ,1 2 dr E Zr π Z 2 1 dr. Therefore, by Theorems 3.1 and 5.3, we have n E Z Ẑπ 2 d 2E Z Z π 2 d + Z π Z π,1 2 d =1 n 2E Z Z π 2 d + Z π Z 1 2 d =1 n CE Z Z π 2 d + Z Z 1 2 d =1 C [ (1 + x 2 ) π +E ξ ξ π 2]. Ths, combned wh (5.7), proves he heorem. 6. Numercal schemes. In hs secon we propose a numercal scheme based on he resuls of prevous secons. To presen our scheme explcly, we frs recall he process ˆX π defned by (4.2) and defne he dscree funconal ξ π = ( ˆX π ) n (5.1). Nex, noe ha we can rewre (4.1) as X π ( = X 1,X π ) (6.1) 1, where, for s< T and x R, (6.2) X (s, x) = x + b(s, x)( s) + σ (s, x)(w W s ). Furher, for each n, we denoe x () = (x,...,x ). Our scheme s as follows.

26 484 J. ZHANG THEOREM 6.1. Le ξ = (X) and ξ π = ( ˆX π ), (Y, Z) and (Y π,z π ) are soluons o BSDEs (2.1) and (5.1), respecvely. Assume ha all he condons n Theorem 3.1 hold rue. Defne u π,vπ : R d(+1) R, =,...,n, as follows: u π ( n x (n) ) ( n ) = x 1 1 [ 1, )( ) + x n 1 T ( ), v π ( n x (n) ) =, =1 U π(x( 1),ω) = u π ( x ( 1),X (,x 1 ) ) ( + f,x (,x 1 ), u π ( x ( 1),X (,x 1 ) ), (6.3) ( x ( 1),X (,x 1 ) )), u π 1( x ( 1) ) = E U π (x ( 1),ω), v 1 π ( x ( 1) ) 1 = E U π ( x ( 1),ω )[ ] W W 1. Then we have Ŷ π = Y π = u π ( X π,...,x π ), Ẑπ = Z π,1 = Z π = v π ( X π,...,x π ) (6.4). If f s ndependen of z, or f π s K-unform, hen holds ha sup E Y Ŷ π 2 (6.5) + E Z Ẑπ 2 d C(1 + x 2 ) π log 1 T π. Moreover, f sasfes he L 1 -Lpschz condon (2.3) or akes he form (X) = g(x T ), hen we have (6.6) sup E Y Ŷ π 2 + E T v π Z Ẑπ 2 d C(1 + x 2 ) π. PROOF. By Theorem 5.3 and Corollary 4.4, suffces o prove (6.4). We would also show smulaneously ha, for each, u π and v π are Lpschz. (In fac can be shown ha u π s unformly Lpschz, unformly n π and.) We proceed boh by nducon. For = n obvously (6.4) holds rue and u π n and vπ n are Lpschz. Assume ha for boh (6.4) and he Lpschz connuy hold rue. Frs, by (5.1) obvously we have Y π 1 = E u π ( X π,...,x π ) + f (,X π,u π ( X π,...,x π ),v π ( X π,...,x π )) F 1, whch, combned wh (6.1), mples ha Y π = u π 1 (Xπ,...,X π ). Nex, by he Lpschz connuy assumpon, one may apply Lemma 2.7. So Z π r s a marngale on [, ), herefore Z π,1 = Z π. Moreover, followng he

27 NUMERICAL SCHEME FOR BSDES 485 formula of η n Lemma 2.7, we ge Z π 1 = v 1 π (Xπ,...,X π 1 ). Ths proves (6.4) for 1. Fnally, by (6.2) and (6.3) one can easly check ha u π 1 and vπ 1 are also Lpschz. Ths fnshes he nducon and hus proves he heorem. REMARK 6.2. By (6.2) and (6.3), o calculae (u π 1,vπ 1 ) from (uπ,vπ ) one needs only approxmae an expecaon nvolvng W W 1, ha s, a onedmensonal negral. We noe ha n Bally s mehod [2] one has o approxmae negrals whose dmenson s proporonal o he paron sze n. There sll remans, however, anoher ype of hgh dmensonaly: n (6.1) one has o compue he values of ( ˆX π ) for all possble choces of he hghdmensonal vecor (X π,...,x π n ). Tha s, f we dscreze he real lne R no M pars, hen he number of values nvolved n he scheme s of he order M d(n+1), whch would cause sgnfcan echncal dffcules n mplemenaon. We herefore mpose he followng resrcon on, movaed by applcaons n fnance heory, so as o make our scheme mplemenable. DEFINITION 6.3. Le E 1 and E 2 be wo generc Eucldean spaces. A funconal : D E1 [,T] E 2, s called consrucble wh consrucon ϕ ffor each s< T here exss a funcon : D E1 [,] E 2 and ϕ s, : E 2 D E1 [s,] E 2, such ha for s< T : () T = and (x) = (x 1 [,) ); () (x) = ϕ s, ( s (x), x 1 [s,) ); () all ϕ s, sasfy he L -Lpschz condon (2.2) wh a unformly Lpschz consan. We should pon ou here ha snce X s Markovan, by () one can easly see ha ( (X), X ) s Markovan. Thus hs defnon essenally amouns o sayng ha we may add a sae varable (X) such ha ( (X), X,Y,Z ) T s Markovan. Moreover, s farly easy o see ha also sasfes he L -Lpschz condon (2.2). [In fac, under assumpons () and (), sasfes (2.2) f and only f () holds rue.] The followng are some easy examples of consrucble funconals. EXAMPLE 6.4. () The funconal : x x() d, x D d [,T], s consrucble. Indeed, f we defne (x) = x(r) dr, ϕ s, (a, x) = a + hen s consrucble wh consrucon ϕ. s x(r) dr,

28 486 J. ZHANG () The funconal : x sup T x(), x D 1 [, 1], s also consrucble, wh (x) = sup x(r), r ϕ s, (a, x) = a sup x(r). s r In he sequel, by a slgh abuse of noaons we denoe ϕ s, (a, x) = ( ) ϕ s, a,x1[s,) (a, x) R k R d. Now we can smplfy he numercal scheme (6.1) as follows. THEOREM 6.5. Assume ha n BSDEs (2.1) and (5.1) ξ = g( (X),X T ) and ξ π = g( ( ˆX π ), ˆX T π), respecvely, and denoe (Y, Z) and (Y π,z π ) o be he correspondng soluons. Assume also ha all he condons n Theorem 3.1 hold rue and assume furher ha g s Lpschz connuous, and : D d [,T] R k s consrucble wh consrucon ϕ. Defne, for (a, x) R d+k, (6.7) u π n (a, x) = g(a,x), vn π(a, x) =, U π(a, x, ω) = u π ( ϕ 1, (a, x), X (,x) ) ( + f,x (,x),u π ( ϕ 1, (a, x), X (,x) ), ( ϕ 1, (a, x), X (,x) )), v π u π 1 (a, x) = EU π(a,x,ω), v 1 π (a, x) = 1 E U π (a, x, ω)[ ] W W 1. Then all he resuls n Theorem 6.1 hold rue. We noe ha f we dscreze he real lne R no M pars, hen for each, he number of grd pons for a s M k and ha for x s M d. Therefore, he oal number of values nvolved n scheme (6.7) s of he order M d+k (n+1), raher han M d(n+1) as n scheme (6.4). REMARK 6.6. In Theorem 6.5, f we assume ha ξ = g(x T ),henwemay consder as a consan funconal. In hs case, ϕ s, are also consan funconals, hus (6.7) becomes (6.8) u π n (x) = g(x), vn π(x) =, U π(x, ω) = u π ( X (,x) ) ( + f,x (,x),u π ( X (,x) ),v π ( X (,x) )), u π 1 (x) = EU π(x, ω), vπ 1 (x) = 1 E U π (x, ω)[ ] W W 1.

29 We sll have Ŷ π = Y π NUMERICAL SCHEME FOR BSDES 487 = u π ( ˆX π ), Ẑπ = Z π,1 = Z π = v π ( ˆX π ). Moreover, snce E g(x T ) g( ˆX π T ) 2 C(1+ x 2 ) π, we conclude ha he rae of convergence becomes C(1 + x 2 ) π, whch concdes wh he resul of [16]. Acknowledgmens. The auhor hanks J. Ma for los of useful dscusson and he referee for hs/her careful readng of he manuscrp and many helpful suggesons. REFERENCES [1] ASMUSSEN, S., GLYNN, P. and PITMAN, J. (1995). Dscrezaon error n smulaon of onedmensonal reflecng Brownan moon. Ann. Appl. Probab [2] BALLY, V. (1995). An approxmaon scheme for BSDEs and applcaons o conrol and nonlnear PDEs. Prepublcaon du Laboraore de Sasque e Processus de l Unverse du Mane. [3] BISMUT, J. M. (1973). Théore probablse du conrôle des dffusons. Mem. Amer. Mah. Soc [4] BJÖRK, T. (1998). Arbrage Theory n Connuous Tme. Oxford Unv. Press. [5] BRIAND, P., DELYON,B. and MÉMIN, J. (21). Donsker-ype heorem for BSDEs. Elecron. Comm. Probab [6] CHEVANCE, D. (1996). Ph.D. hess, Unv. Provence. [7] DAI,M.andJIANG, L. (2). On pah-dependen opons. In Mahemacal Fnance Theory and Applcaons (J. Yong and R. Con, eds.) Hgher Educaon Press, Chna. [8] DOUGLAS, J., Jr., MA, J. and PROTTER, P. (1996). Numercal mehods for forward backward sochasc dfferenal equaons. Ann. Appl. Probab [9] EL KAROUI, N. andmazliak, L. (1997). Backward Sochasc Dfferenal Equaons. Longman, Harlow. [1] EL KAROUI, N., PENG, S. and QUENEZ, M. C. (1997). Backward sochasc dfferenal equaons n fnance. Mah. Fnance [11] GIKHMAN, I. I. andskorohod, A. V. (1972). Sochasc Dfferenal Equaons. Sprnger, New York. [12] KARATZAS, I. andshreve, S. E. (1987). Brownan Moon and Sochasc Calculus. Sprnger, New York. [13] KLOEDEN, P.E.andPLATEN, E. (1992). Numercal Soluon of Sochasc Dfferenal Equaons. Sprnger, New York. [14] MA, J., PROTTER, P., SAN MARTÍN, J. and TORRES, S. (22). Numercal mehod for backward SDEs. Ann. Appl. Probab [15] MA, J., PROTTER, P. and YONG, J. (1994). Solvng forward backward sochasc dfferenal equaons explcly a four sep scheme. Probab. Theory Relaed Felds [16] MA, J. andyong, J. (1999). Forward Backward Sochasc Dfferenal Equaons and Ther Applcaons. Lecure Noes n Mah Sprnger, Berln. [17] MA,J.andZHANG, J. (22). A represenaon heorem for backward SDE s wh non-smooh coeffcens. Ann. Appl. Probab [18] MA, J. and ZHANG, J. (22). Pah regulary for soluons of backward SDE s. Probab. Theory Relaed Felds [19] NUALART, D. (1995). The Mallavn Calculus and Relaed Topcs. Sprnger, New York.

30 488 J. ZHANG [2] PARDOUX, E. and PENG, S. (199). Adaped soluons of backward sochasc equaons. Sysem and Conrol Le [21] PARDOUX, E. and PENG, S. (1992). Backward sochasc dfferenal equaons and quaslnear parabolc paral dfferenal equaons. Lecure Noes n Conrol and Inform. Sc Sprnger, Berln. [22] ZHANG, J. (21). Some fne properes of backward sochasc dfferenal equaons. Ph.D. dsseraon, Purdue Unv. [23] ZHANG, Y. and ZHENG, W. (21). Dscrezng a backward sochasc dfferenal equaons. Preprn. DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTHERN CALIFORNIA LOS ANGELES,CALIFORNIA 989 USA janfenz@usc.edu