MALLIAVIN CALCULUS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATION TO NUMERICAL SOLUTIONS

Transcription

1 The Annal of Applied Probabiliy 211, Vol. 21, No. 6, DOI: /11-AAP762 Iniue of Mahemaical Saiic, 211 MALLIAVIN CALCULUS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATION TO NUMERICAL SOLUTIONS BY YAOZHONG HU 1,DAVID NUALART 2 AND XIAOMING SONG Univeriy of Kana In hi paper we udy bacward ochaic differenial equaion wih general erminal value and general random generaor. In paricular, we do no require he erminal value be given by a forward diffuion equaion. The randomne of he generaor doe no need o be from a forward equaion, eiher. Moivaed from applicaion o numerical imulaion, fir we obain he L p -Hölder coninuiy of he oluion. Then we conruc everal numerical approximaion cheme for bacward ochaic differenial equaion and obain he rae of convergence of he cheme baed on he obained L p - Hölder coninuiy reul. The main ool i he Malliavin calculu. 1. Inroducion. The bacward ochaic differenial equaion BSDE, for hor we hall conider in hi paper ae he following form: 1.1 Y = ξ + fr,y r,z r dr Z r dw r, T, where W ={W } T i a andard Brownian moion, ξ i he given erminal value and f i he given random generaor. To olve hi equaion i o find a pair of adaped procee Y ={Y } T and Z ={Z } T aifying he above equaion 1.1. Linear bacward ochaic differenial equaion were fir udied by Bimu [3] in an aemp o olve ome opimal ochaic conrol problem hrough he mehod of maximum principle. The general nonlinear bacward ochaic differenial equaion were fir udied by Pardoux and Peng [15]. Since hen here have been exenive udie of hi equaion. We refer o he review paper by El Karoui, Peng and Quenez [7], o he boo of El Karoui and Mazlia [6] and of Ma and Yong [12] and he reference herein for more comprehenive preenaion of he heory. A curren imporan opic in he applicaion of BSDE i he numerical approximaion cheme. In mo wor on numerical imulaion, a cerain forward Received March 21; revied Ocober Suppored by he NSF Gran DMS Suppored by he NSF gran DMS MSC21 ubjec claificaion. 6H7, 6H1, 6H35, 65C3, 91G6. Key word and phrae. Bacward ochaic differenial equaion, Malliavin calculu, explici cheme, implici cheme, Clar Ocone Hauman formula, rae of convergence, Hölder coninuiy of he oluion. 2379

2 238 Y. HU, D. NUALART AND X. SONG ochaic differenial equaion of he following form: 1.2 X = X + br,x r,y r dr + σr,x r dw r i needed. Uually i i aumed ha he generaor f in 1.1 depend on X r a he ime r: fr,y r,z r = fr,x r,y r,z r,wherefr,x,y,z i a deerminiic funcion of r,x,y,z,andf i global Lipchiz in x,y,z. If in addiion he erminal value ξ i of he form ξ = hx T,whereh i a deerminiic funcion, a o-called four-ep numerical cheme ha been developed by Ma, Proer and Yong in [11]. A baic ingredien in hi paper i ha he oluion {Y } T o he BSDE i of he form Y = u, X,whereu, x i deermined by a quai-linear parial differenial equaion of parabolic ype. Recenly, Bouchard and Touzi [4] propoe a Mone-Carlo approach which may be more uiable for high-dimenional problem. Again in hi forward bacward eing, if he generaor f ha a quadraic growh in Z, a numerical approximaion i developed by Imeller and Do Rei [9] in which a runcaion procedure i applied. In he cae where he erminal value ξ i a funcional of he pah of he forward diffuion X, namely, ξ = gx, differen approache o conruc numerical mehod have been propoed. We refer o Bally [1] for a cheme wih a random ime pariion. In he wor by Zhang [16], he L 2 -regulariy of Z i obained, which allow one o ue deerminiic ime pariion a well a o obain he rae eimae ee Bender and Den [2], Gobe, Lemor and Warin [8]andZhang[16] for differen algorihm. We hould alo menion he wor by Briand, Delyon and Mémin [5] and Ma e al. [1], where he Brownian moion i replaced by a caled random wal. The purpoe of he preen paper i o conruc numerical cheme for he general BSDE 1.1, wihou auming any paricular form for he erminal value ξ and generaor f.thimeanhaξ can be an arbirary random variable, and fr,y,z can be an arbirary F r -meaurable random variable ee Aumpion 2.2 in Secion 2 for precie condiion on ξ and f. The naural ool ha we hall ue i he Malliavin calculu. We emphaize ha he main difficuly in conrucing a numerical cheme for BSDE i uually he approximaion of he proce Z. I i neceary o obain ome regulariy properie for he rajecorie of hi proce Z.The Malliavin calculu urn ou o be a uiable ool o handle hee problem becaue he random variable Z can be expreed in erm of he race of he Malliavin derivaive of Y, namely, Z = D Y. Thi relaionhip wa proved in he paper by El Karoui, Peng and Quenez [7] and wa ued by hee auhor o obain eimae for he momen of Z. We hall furher exploi hi ideniy o obain he L p -Hölder coninuiy of he proce Z, which i he criical ingredien for he rae eimae of our numerical cheme. Our fir numerical cheme wa inpired by he paper of Zhang [16], where he auhor conider a cla of BSDE whoe erminal value ξ ae he form gx,

3 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2381 where X i a forward diffuion of he form 1.2, and g aifie a Lipchiz condiion wih repec o he L or L 1 norm imilar aumpion for f. The dicreizaion cheme i baed on he regulariy of he proce Z in he mean quare ene; ha i, for any pariion π ={ = < 1 < < n = T }, one obain 1.3 n 1 i= i+1 E [ Z Z i 2 + Z Z i+1 2 ] d K π, i where π =max i n 1 i+1 i,andk i a conan independen of he pariion π. We conider he cae of a general erminal value ξ which i wice differeniable in he ene of Malliavin calculu, and he fir and econd derivaive aify ome inegrabiliy condiion; we alo made imilar aumpion for he generaor f ee Aumpion 2.2 in Secion 2 for deail. In hi ene our framewor exend ha of [13] and i alo naural. In hi framewor, we are able o obain an eimae of he form 1.4 E Z Z p K p/2, where K i a conan independen of and. Clearly, 1.4 wih p = 2 implie 1.3. Moreover, 1.4 implie he exience of a γ -Hölder coninuou verion of he proce Z for any γ< p. Noice ha, up o now he pah regulariy of Z ha been udied only when he erminal value and he generaor are funcional of a forward diffuion. Afer eablihing he regulariy of Z, we conider differen ype of numerical cheme. Fir we analyze a cheme imilar o he one propoed in [16][ee3.2]. In hi cae we obain a rae of convergence of he following ype: E up Y Y π 2 + T E Z Z π 2 d K π +E ξ ξ π 2. Noice ha hi reul i ronger han ha in [16] which can be aed a when ξ π = ξ up E Y Y π 2 + E Z Z π 2 d K π. T We alo propoe and udy an implici numerical cheme [ee 4.1 in Secion 4 for he deail]. For hi cheme we obain a much beer reul on he rae of convergence, p/2 E up Y Y π p + E Z Z d π 2 K π p/2 + E ξ ξ π p, T where p>1 depend on he aumpion impoed on he erminal value and he coefficien.

4 2382 Y. HU, D. NUALART AND X. SONG In boh cheme, he inegral of he proce Z i ued in each ieraion, and for hi reaon hey are no compleely dicree cheme. In order o implemen he cheme on compuer, one mu replace an inegral of he form i+1 i Z π d by dicree um, and hen he convergence of he obained cheme i hardly guaraneed. To avoid hi dicreizaion we propoe a ruly dicree numerical cheme uing our repreenaion of Z a he race of he Malliavin derivaive of Y ee Secion 5 for deail. For hi new cheme, we obain a rae of convergence reul of he form E max { Y i Y π i n i p + Z i Z π i p } K π p/2 ε for any ε>. In fac, we have a lighly beer rae of convergence ee Theorem 5.2, E max { Y i Y π i n i p + Z i Z π i p } K π p/2 p/2log1/ π log 1 p/2. π However, hi ype of reul on he rae of convergence applie only o ome clae of BSDE, and hu hi cheme remain o be furher inveigaed. In he compuer realizaion of our cheme or any oher cheme, an exremely imporan procedure i o compue he condiional expecaion of form EY F i. In hi paper we hall no dicu hi iue bu only menion he paper [2, 4] and [8]. The paper i organized a follow. In Secion 2 we obain a repreenaion of he maringale inegrand Z in erm of he race of he Malliavin derivaive of Y, and hen we ge he L p -Hölder coninuiy of Z by uing hi repreenaion. The condiion ha we aume on he erminal value ξ and he generaor f are alo pecified in hi ecion. Some example of applicaion are preened o explain he validiy of he condiion. Secion 3 i devoed o he analyi of he approximaion cheme imilar o he one inroduced in [16]. Under ome differeniabiliy and inegrabiliy condiion in he ene of Malliavin calculu on ξ and he nonlinear coefficien f, we eablih a beer rae of convergence for hi cheme. In Secion 4, we inroduce an implici cheme and obain he rae of convergence in he L p norm. A compleely dicree cheme i propoed and analyzed in Secion 5. Throughou he paper for impliciy we conider only calar BSDE. The reul obained in hi paper can be eaily exended o muli-dimenional BSDE. 2. The Malliavin calculu for BSDE Noaion and preliminarie. Le W ={W } T be a one-dimenional andard Brownian moion defined on ome complee filered probabiliy pace, F,P,{F } T. We aume ha {F } T i he filraion generaed by he Brownian moion and he P -null e, and F = F T. We denoe by P he progreive σ -field on he produc pace [,T]. For any p 1 we conider he following clae of procee:

5 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2383 M 2,p,foranyp 2, denoe he cla of quare inegrable random variable F wih a ochaic inegral repreenaion of he form F = EF + u dw, where u i a progreively meaurable proce aifying up T E u p <. H p F [,T] denoe he Banach pace of all progreively meaurable procee ϕ : [,T],P R, B wih norm p/2 1/p ϕ H p = E ϕ d 2 <. S p F [,T] denoe he Banach pace of all he RCLL righ coninuou wih lef limi adaped procee ϕ : [,T],P R, B wih norm ϕ S p = E up ϕ p 1/p <. T Nex, we preen ome preliminarie on Malliavin calculu, and we refer he reader o he boo by Nualar [14] for more deail. Le H = L 2 [,T] be he eparable Hilber pace of all quare inegrable realvalued funcion on he inerval [,T] wih calar produc denoed by, H.The norm of an elemen h H will be denoed by h H.Foranyh H we pu Wh = h dw. We denoe by Cp Rn he e of all infiniely coninuouly differeniable funcion g : R n R uch ha g and all of i parial derivaive have polynomial growh. We mae ue of he noaion i g = g x i whenever g C 1 R n. Le S denoe he cla of mooh random variable uch ha a random variable F S ha he form 2.1 F = gwh 1,..., Wh n, where g belong o Cp Rn, h 1,...,h n are in H and n 1. The Malliavin derivaive of a mooh random variable F of he form 2.1ihe H-valued random variable given by n DF = i gwh 1,..., Wh n h i. i=1 For any p 1 we will denoe he domain of D in L p by D 1,p, meaning ha D 1,p i he cloure of he cla of mooh random variable S wih repec o he norm F 1,p = E F p + E DF p H 1/p. We can define he ieraion of he operaor D in uch a way ha for a mooh random variable F, he ieraed derivaive D F i a random variable wih value

6 2384 Y. HU, D. NUALART AND X. SONG in H. Then for every p 1 and any naural number 1 we inroduce he eminorm on S defined by 1/p F,p = E F p + E D j F p H j. j=1 We will denoe by D,p he compleion of he family of mooh random variable S wih repec o he norm,p. Le μ be he Lebegue meaure on [,T]. Forany 1andF D,p,he derivaive D F ={D 1,..., F, i [,T],i = 1,...,} i a meaurable funcion on he produc pace [,T], which i defined a.e. wih repec o he meaure μ P. We ue L 1,p a o denoe he e of real-valued progreively meaurable procee u ={u } T uch ha: i For almo all [,T],u D 1,p. ii E u 2 d p/2 + D θ u 2 dθ d p/2 <. Noice ha we can chooe a progreively meaurable verion of he H-valued proce {Du } T Eimae on he oluion of BSDE. The generaor f in he BSDE 1.1 i a meaurable funcion f : [,T] R R, P B B R, B, andhe erminal value ξ i an F T -meaurable random variable. DEFINITION 2.1. A oluion o he BSDE 1.1 i a pair of progreively meaurable procee Y, Z uch ha Z 2 d <, f,y,z d <,a.. and Y = ξ + fr,y r,z r dr Z r dw r, T. The nex lemma provide a ueful eimae on he oluion o he BSDE 1.1. LEMMA 2.2. Fix q 2. Suppoe ha ξ L q, f,, H q F [,T] and f i uniformly Lipchiz in y, z; namely, here exi a poiive number L uch ha μ P a.e. f,y 1,z 1 f,y 2,z 2 L y 1 y 2 + z 1 z 2 for all y 1,y 2 R and z 1,z 2 R. Then here exi a unique oluion pair Y, Z S q F [,T] H q F [,T] o 1.1. Moreover, we have he following eimae for he

7 oluion: 2.2 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2385 q/2 E up Y q + E Z 2 d T q/2 K E ξ q + E f,, 2 d, where K i a conan depending only on L, q and T. PROOF. The proof of he exience and uniquene of he oluion Y, Z can be found in [7], Theorem 5.1, wih he local maringale M, ince he filraion here i he filraion generaed by he Brownian moion W. Eimae 2.2 can be eaily obained from Propoiion 5.1 in [7] wih f 1,ξ 1 = f, ξ and f 2,ξ 2 =,. A we will ee laer, for a given BSDE he proce Z will be expreed in erm of he Malliavin derivaive of he oluion Y, which will aify a linear BSDE wih random coefficien. To udy he properie of Z we need o analyze a cla of linear BSDE. Le {α } T and {β } T be wo progreively meaurable procee. We will mae ue of he following inegrabiliy condiion: ASSUMPTION 2.1. H1 For any λ>, C λ := E exp λ H2 For any p 1, α +β 2 d <. K p := up E α p + β p <. T Under condiion H1, we denoe by {ρ } T he oluion of he linear ochaic differenial equaion { dρ = α ρ d + β ρ dw, T, 2.3 ρ = 1. The following heorem i a criical ool for he proof of he main heorem in hi ecion, and i ha alo i own inere. THEOREM 2.3. Le q>p 2 and le ξ L q and f H q F [,T]. Aume ha {α } T and {β } T are wo progreively meaurable procee aifying condiion H1 and H2 in Aumpion 2.1. Suppoe ha he random

8 2386 Y. HU, D. NUALART AND X. SONG variable ξρ T and ρ f d belong o M 2,q, where {ρ } T i he oluion o 2.3. Then he following linear BSDE, 2.4 Y = ξ + [α r Y r + β r Z r + f r ] dr Z r dw r, T, ha a unique oluion pair Y, Z, and here i a conan K>uch ha 2.5 E Y Y p K p/2 for all, [,T]. We need he following lemma o prove he above reul. LEMMA 2.4. Le {α } T and {β } T be wo progreively meaurable procee aifying condiion H1 in Aumpion 2.1, and {ρ } T be he oluion of 2.3. Then, for any r R we have 2.6 E up ρ r <. T PROOF. Le [,T]. The oluion o 2.3 can be wrien a { ρ = exp α β2 d + β dw }. 2 For any real number r, wehave { exp r α β2 2 } d + r β dw } β 2 d E up ρ r = E up T T { E exp r α d r +r2 { up exp r β dw r2 β }. 2 T 2 d Then, fixing any p>1 and uing Hölder inequaliy, we obain { E up ρ r C E up exp rp β dw pr2 } 1/p 2.7 β 2 T T 2 d, where { C = E exp q r α d + q } 1/q 2 r +r2 β 2 d and p 1 + q 1 = 1.

9 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2387 Se M = exp{r β dw r2 2 β 2 d}. Then{M } T i a maringale due o H1. We can rewrie 2.7 ino E up ρ r C E up M p 1/p. 2.8 T T By Doob maximal inequaliy, we have 2.9 E up M p c p EM p T T for ome conan c p > depending only on p. Finally, chooing any γ>1,λ>1 uch ha γ 1 + λ 1 = 1 and applying again he Hölder inequaliy yield { EM p T exp = E rp β dw γ } 2 p2 r 2 β 2 d { γp 1 T } exp pr 2 β 2 d { E exp rpγ 2 β dw 1 } 1/γ 2 γ 2 p 2 r 2 β 2 d { λγp 1 T } 1/λ E exp pr 2 β 2 2 d { λγp 1 T } 1/λ = E exp pr 2 β 2 2 d <. Combining hi inequaliy wih 2.8 and2.9 we complee he proof. PROOF OF THEOREM 2.3. The exience and uniquene i well nown. We are going o prove 2.5. Le [,T]. Denoe γ = ρ 1,where{ρ } T i he oluion o 2.3. Then {γ } T aifie he following linear ochaic differenial equaion: { dγ = α + β 2γ d β γ dw, T, γ = 1. For any T and any poiive number r 1, we have, uing H2, he Hölder inequaliy, he Burholder Davi Gundy inequaliy and Lemma 2.4 applied o he proce {γ } T, E γ γ r = E α u + βu 2 r γ u du β u γ u dw u [E r 1 C r/2, α u +β 2 u γ u du r +C r E β 2 u γ 2 u du r/2]

10 2388 Y. HU, D. NUALART AND X. SONG where C r i a conan depending only on r, andc i a conan depending on T, r and he conan appearing in condiion H1 and H2. From 2.3, 2.4 and by Iô formula, we obain A a conequence, 2.11 Y = ρ 1 E ξρ T + dy ρ = ρ f d + β ρ Y + ρ Z dw. ρ r f r drf = E ξρ,t + wherewewrieρ,r = ρ 1 ρ r = γ ρ r for any r T. Now, fix T.Wehave E Y Y p T = E ξρ E,T + ρ,r f r drf E ξρ,t + 2 p 1 [ E Eξρ,T F Eξρ,T F p ρ,r f r drf, p ρ,r f r drf + E E T p] ρ,r f r drf E ρ,r f r drf = 2 p 1 I 1 + I 2. Fir we eimae I 1.Wehave I 1 = E Eξρ,T F Eξρ,T F p = E Eξρ,T F Eξρ,T F + Eξρ,T F Eξρ,T F p 2 p 1[ E Eξρ,T F Eξρ,T F p + E Eξρ,T F Eξρ,T F p] 2 p 1[ E ξρ,t ρ,t p + E Eξρ,T F Eξρ,T F p] = 2 p 1 I 3 + I 4. Uing he Hölder inequaliy, Lemma 2.4 and he eimae 2.1 wih r = 2pq q p,he erm I 3 can be eimaed a follow: I 3 E ξ q p/q E ρ,t ρ,t pq/q p q p/q E ξ q p/q E γ γ 2pq/q p q p/2q Eρ 2pq/q p T C p/2, q p/2q where C i a conan depending only on p,q,t, E ξ q and he conan appearing in condiion H1 and H2. In order o eimae he erm I 4 we will mae ue of he condiion ξρ T M 2,q. Thi condiion implie ha ξρ T = Eξρ T + u r dw r,

11 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2389 where u i a progreively meaurable proce aifying up T E u q <. Therefore, by he Burholder Davi Gundy inequaliy, we have E Eξρ T F Eξρ T F q q = E u r dw r C q E u 2 q/2 r dr C q q 2/2 E u r q dr C q q/2 up T E u q. A a conequence, from he definiion of I 4 we have I 4 = E γ [Eξρ T F Eξρ T F ] p Eγ pq/q p C p/2, q p/q E Eξρ T F Eξρ T F q p/q where C i a conan depending on p,q,t,up T E u q < and he conan appearing in condiion H1 and H2. The erm I 2 can be decompoed a follow: I 2 = E E T p ρ,r f r drf E ρ,r f r drf 3 p 1 [E E + E E + E E = 3 p 1 I 5 + I 6 + I 7. T p ρ,r f r drf E ρ,r f r drf ρ,r f r drf E ρ,r f r drf E p ρ,r f r drf p] ρ,r f r drf Le u fir eimae he erm I 5. Suppoe ha p<p <q. Then, uing 2.1 and he Hölder inequaliy, we can wrie I 5 = E E T p ρ,r f r drf E ρ,r f r drf E ρ,r ρ,r f r dr { E γ γ pp /p p } p p/p { E p T = E γ γ p p ρ r f r dr ρ r f r dr p } p/p

12 239 Y. HU, D. NUALART AND X. SONG C {E p/2 { q/2 } p/q E fr 2 dr ρ 2 r dr p q/2q p } pq p /p q Ĉ p/2 f p H q, where Ĉ i a conan depending on p,p, q, T and he conan appearing in condiion H1 and H2. Now we eimae I 6. Suppoe ha p<p <q. We have, a in he eimae of he erm I 5, I 6 = E E T p ρ,r f r drf E ρ,r f r drf p E ρ,r f r dr = E ρ p p ρ r f r dr { Eρ pp /p p } p p/p { p } p/p E ρ r f r dr { p } p/p = C E ρ r f r dr C p/2{ E up T ρ p q/q p } pq p /p q p f H q = Ĉ p/2, where Ĉ i a conan depending on p,p, q, T and he conan appearing in condiion H1 and H2. The fac ha ρ r f r dr belong o M 2,q implie ha ρ r f r dr = E ρ r f r dr + v r dw r, where {v } T i a progreively meaurable proce aifying up E v q <. T Then, by he Burholder Davi Gundy inequaliy we have E E T q ρ r f r drf E ρ r f r drf = E E T q ρ r f r drf E ρ r f r drf q = E v r dw r C q q/2 up E v q. T

13 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2391 Finally, we eimae I 7 a follow: I 7 = E E T p ρ,r f r drf E ρ,r f r drf = E ρ 1 T p E ρ r f r drf E ρ r f r drf { Eρ pq/q p } q p/p 2.12 { E E T q} p/q ρ r f r drf E ρ r f r drf { C E E T q} p/q ρ r f r drf E ρ r f r drf Ĉ p/2, where Ĉ i a conan depending on p, q, T,up T E v q and he conan appearing in condiion H1 and H2. A a conequence, we obain for all, [,T] E Y Y p K p/2, where K i a conan independen of and The Malliavin calculu for BSDE. We reurn o he udy of 1.1. The main aumpion we mae on he erminal value ξ and generaor f are he following: ASSUMPTION 2.2. Fix 2 p< q 2. i ξ D 2,q, and here exi L>, uch ha for all θ,θ [,T], and E D θ ξ D θ ξ p L θ θ p/2, up E D θ ξ q < θ T 2.15 up θ T up E D u D θ ξ q <. u T ii The generaor f,y,z ha coninuou and uniformly bounded firand econd-order parial derivaive wih repec o y and z, and f,, H q F [,T]. iii Aume ha ξ and f aify he above condiion i and ii. Le Y, Z be he unique oluion o 1.1 wih erminal value ξ and generaor f. For each

14 2392 Y. HU, D. NUALART AND X. SONG y, z R R, f,y,z, y f,y,z and z f,y,z belong o L 1,q a,andhe Malliavin derivaive Df,y,z, D y f,y,zand D z f,y,zaify q/2 up E D θ f,y,z d 2 <, θ T θ q/2 up E D θ y f,y,z d 2 <, θ T θ q/2 up E D θ z f,y,z d 2 <, θ T θ and here exi L> uch ha for any,t],andforany θ,θ T p/ E D θ fr,y r,z r D θ fr,y r,z r dr 2 L θ θ p/2. For each θ [,T], and each pair of y, z, D θ f,y,z L 1,q a and i ha coninuou parial derivaive wih repec o y,z, which are denoed by y D θ f,y,z and z D θ f,y,z, and he Malliavin derivaive D u D θ f,y,zaifie 2.2 up θ T q/2 up E D u D θ f,y,z d 2 <. u T θ u The following propery i eay o chec and we omi he proof. REMARK 2.5. Condiion 2.17 and2.18 imply q/2 up E y D θ f,y,z d 2 < θ T θ and q/2 up E z D θ f,y,z d 2 <, θ T θ repecively. The following i he main reul of hi ecion. THEOREM 2.6. Le Aumpion 2.2 be aified. a There exi a unique oluion pair {Y,Z } T o he BSDE 1.1, and Y,Z are in L 1,q a. A verion of he Malliavin derivaive {D θ Y,D θ Z } θ, T of

15 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2393 he oluion pair aifie he following linear BSDE: D θ Y = D θ ξ + [ y fr,y r,z r D θ Y r z fr,y r,z r D θ Z r + D θ fr,y r,z r ] dr D θ Z r dw r, θ T ; 2.22 D θ Y =, D θ Z =, <θ T. Moreover, {D Y } T defined by 2.21 give a verion of {Z } T, namely, μ P a.e Z = D Y. b There exi a conan K>, uch ha, for all, [,T], 2.24 E Z Z p K p/2. PROOF. Par a: The proof of he exience and uniquene of he oluion Y, Z, andy,z La 1,2 i imilar o ha of Propoiion 5.3 in [7], and alo he fac ha D θ Y,D θ Z i given by 2.21 and2.22. In Propoiion 5.3 in [7] he exponen q i equal o 4, and one aume ha D θ f,y,z 2 dθ <, H 2 which i a conequence of 2.16 and he fac ha Y,Z L 1,2 a. Furhermore, from condiion 2.14and2.16 and he eimae in Lemma 2.2, we obain { q/2 } 2.25 up E up D θ Y q + E D θ Z 2 d <. θ T θ T θ Hence, by Propoiion in [14], Y and Z belong o L 1,q a. Par b: Le T. In hi proof, C>will be a conan independen of and, and may vary from line o line. By repreenaion 2.23 wehave 2.26 Z Z = D Y D Y = D Y D Y + D Y D Y. From Lemma 2.2 and equaion 2.21forθ = and θ =, repecively, we obain, uing condiion 2.13 and2.19, p/2 E D Y D Y p + E D Z r D Z r 2 dr [ C E D ξ D ξ p 2.27 p/2 ] + E D fr,y r,z r D fr,y r,z r 2 dr C p/2.

16 2394 Y. HU, D. NUALART AND X. SONG Denoe α u = y fu,y u,z u and β u = z fu,y u,z u for all u [,T]. Then, by Aumpion 2.2ii, he procee α and β aify condiion H1 and H2 in Aumpion 2.1, and from 2.21 wehaveforr [,T ] D Y r = D ξ + r [α u D Y u + β u D Z u + D fu,y u,z u ] du r D Z u dw u. Nex, we are going o ue Theorem 2.3 o eimae E D Y D Y p.fixp wih p<p < q 2 noice ha p < q p 2 i equivalen o q p < 1. From condiion 2.14 and2.16, i i obviou ha D ξ L q L p and D f,y,z H q [,T] H p [,T] for any [,T]. We are going o how ha, for any [,T], ρ T D ξ and ρ u D fu,y u,z u duare elemen in M 2,p,where { r r ρ r = exp β u dw u + α u 1 } 2 β2 u du. For any θ r T, le u compue { r D θ ρ r = ρ r [ yz fu,y u,z u D θ Y u θ + zz fu,y u,z u D θ Z u + D θ z fu,y u,z u ] dw u + z fθ,y θ,z θ + + r θ r θ + yy fu,y u,z u yz fu,y u,z u β u Dθ Y u du yz fu,y u,z u zz fu,y u,z u β u Dθ Z u du r θ Dθ y fu,y u,z u β u D θ z fu,y u,z u } du. By he boundedne of he fir- and econd-order parial derivaive of f wih repec o y and z,2.17, 2.18, 2.25, Lemma2.4, he Hölder inequaliy and he Burholder Davi Gundy inequaliy, i i eay o how ha for any p <q, 2.28 up θ T E up θ r T D θ ρ r p <. By he Clar Ocone Hauman formula, we have ρ T D ξ = Eρ T D ξ+ = Eρ T D ξ+ = Eρ T D ξ+ ED θ ρ T D ξ F θ dw θ ED θ ρ T D ξ + ρ T D θ D ξ F θ dw θ u θ dw θ

17 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2395 and ρ r D fr,y r,z r dr = E + = E + ρ r D fr,y r,z r dr E D θ ρ r D fr,y r,z r dr ρ r D fr,y r,z r drf θ dw θ E [D θ ρ r D fr,y r,z r + ρ r y D fr,y r,z r D θ Y r + ρ r z D fr,y r,z r D θ Z r + ρ r D θ D fr,y r,z r ] drf θ dw θ = E ρ r D fr,y r,z r dr + vθ dw θ. We claim ha up θ T E u θ p < and up θ T E v θ p <. In fac, E u θ p = E ED θ ρ T D ξ + ρ T D θ D ξ F θ p 2 p 1 E D θ ρ T D ξ p + E ρ T D θ D ξ p 2 p 1 E D θ ρ T p q/q p q p /q E D ξ q p /q + Eρ p q/q p q p /q T E Dθ D ξ q p /q. By 2.14, 2.15, 2.28 and Lemma 2.4, wehaveup T up θ T E u θ p <. On he oher hand, E vθ T p = E E [D θ ρ r D fr,y r,z r + ρ r y D fr,y r,z r D θ Y r + ρ r z D fr,y r,z r D θ Z r 4 p 1 [J 1 + J 2 + J 3 + J 4 ], p + ρ r D θ D fr,y r,z r ] drf θ

18 2396 Y. HU, D. NUALART AND X. SONG where and J 1 = E J 2 = E J 3 = E For J 1,wehave J 1 E For J 2,wehave J 2 E up θ r T J 4 = E up θ r T D θ ρ r D fr,y r,z r dr p, ρ r y D fr,y r,z r D θ Y r dr ρ r z D fr,y r,z r D θ Z r dr ρ r D θ D fr,y r,z r dr p. p, T p D θ ρ r p D fr,y r,z r dr E up D θ ρ r p q/q p q p /q θ r T E D fr,y r,z r dr q p /q T p /2 E up D θ ρ r p q/q p q p /q θ r T q/2 p /q E D fr,y r,z r 2 dr. D θ Y r p p up ρ r y D fr,y r,z r dr r T E up D θ Y r q p /q θ r T E up r T ρ r E up D θ Y r q p /qe up θ r T r T p q/q p q p /q y D fr,y r,z r dr q p /q E y D fr,y r,z r dr p ρ p q/q 2p q 2p /q r

19 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2397 T p /2 E up D θ Y r q p /qe up θ r T r T q/2 p E y D fr,y r,z r 2 /q dr. Uing a imilar echnique a before, we obain ha and q/2 p /q J 3 T p /2 E D θ Z r 2 dr E up r T q/2 p E z D fr,y r,z r 2 /q dr J 4 T p /2 E up r T ρ p q/q p q p /q r ρ p q/q 2p q 2p /q r ρ p q/q 2p q 2p /q r q/2 p E D θ D fr,y r,z r 2 /q dr. By 2.16, , 2.28 and Lemma 2.4, we obain ha up T up E vθ p <. θ T Therefore, ρ T ξ and ρ u D fu,y u,z u dubelong o M 2,p. Thu by Theorem 2.3 wih p<p, here i a conan C >, uch ha E D Y D Y p C p/2 for all [,T ]. Furhermore, aing ino accoun he proof of he eimae I = 3, 4,...,7 in he proof of Theorem 2.3, we can how ha up T C =: C<. Thu we have 2.29 E D Y D Y p C p/2 for all, [,T]. Combining 2.29 wih 2.26 and2.27, we obain ha here i a conan K>independen of and, uch ha E Z Z p K p/2 for all, [,T]. COROLLARY 2.7. Under he aumpion in Theorem 2.2, le Y, Z S q F [,T] H q F [,T] be he unique oluion pair o 1.1. If up T E Z q <, hen here exi a conan C, uch ha, for any, [,T], 2.3 E Y Y q C q/2.

20 2398 Y. HU, D. NUALART AND X. SONG PROOF. Wihou lo of generaliy we aume T. C>i a conan independen of and, which may vary from line o line. Since Y = Y + fr,y r,z r dr Z r dw r, we have, by he Lipchiz condiion on f, E Y Y q q = E fr,y r,z r dr Z r dw r 2 E q 1 q q fr,y r,z r dr + E Z r dw r q/2 C q q/2 E fr,y r,z r dr 2 + E [ C { q/2 E C q/2. The proof i complee. q/2 Y r dr 2 + E Z r 2 dr q/2 Z r 2 dr q/2 q/2 ] + E fr,, 2 dr } + q/2 up E Z r q r T REMARK 2.8. From Theorem 2.6 we now ha {D θ Y,D θ Z } θ T aifie equaion 2.21andZ = D Y, μ P a.e. Moreover, ince 2.14and2.16 hold, we can apply he eimae 2.2 in Lemma 2.2 o he linear BSDE 2.21and deduce up T E Z q <. Therefore, by Lemma 2.7, he proce Y aifie he inequaliy 2.3. By Kolmogorov coninuiy crierion hi implie ha Y ha Hölder coninuou rajecorie of order γ for any γ< 1 2 q Example. In hi ecion we dicu hree paricular example where Aumpion 2.2 i aified. EXAMPLE 2.9. Conider equaion 1.1. Aume ha: a f,y,z: [,T] R R R i a deerminiic funcion ha ha uniformly bounded fir- and econd-order parial derivaive wih repec o y and z, and f,, 2 d <. b The erminal value ξ i a muliple ochaic inegral of he form 2.31 ξ = g 1,..., n dw 1 dw n, [,T ] n

21 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2399 where n 2 i an ineger and g 1,..., n i a ymmeric funcion in L 2 [,T] n, uch ha up g 1,..., n 1,u 2 d 1 d n 1 <, u T [,T ] n 1 g 1,..., n 2,u,v 2 d 1 d n 2 <, [,T ] n 2 up u,v T and here exi a conan L> uch ha for any u, v [,T] [,T ] n 1 g 1,..., n 1,u g 1,..., n 1,v 2 d 1 d n 1 <L u v. From 2.31, we now ha D u ξ = n g 1,..., n 1,udW 1 dw n 1. [,T ] n 1 The above aumpion implie Aumpion 2.2, and herefore, Z aifie he Hölder coninuiy propery EXAMPLE 2.1. Le = C [, 1] equipped wih he Borel σ -field and Wiener meaure. Then, i a Banach pace wih upremum norm,and W = ω i he canonical Wiener proce. Conider equaion 1.1 on he inerval [, 1]. Aume ha: g1 f,y,z: [, 1] R R R i a deerminiic funcion ha ha uniformly bounded fir- and econd-order parial derivaive wih repec o y and z, and 1 f,, 2 d <. g2 ξ = ϕw, whereϕ : R i wice Fréche differeniable, and he firand econd-order Fréche derivaive δϕ and δ 2 ϕ aify ϕω + δϕω + δ 2 ϕω C 1 exp {C 2 ω r } for all ω and ome conan C 1 >, C 2 > and<r<2, where denoe he operaor norm oal variaion norm. g3 If λ denoe he igned meaure on [, 1] aociaed wih δϕ, here exi a conan L> uch ha for all θ θ 1, for ome p 2. E λθ, θ ] p L θ θ p/2 I i eay o how ha D θ ξ = λθ, 1] and D u D θ ξ = νθ, 1] u, 1], whereν denoe he igned meaure on [, 1] [, 1] aociaed wih δ 2 ϕ. From he above aumpion and Fernique heorem, we can ge Aumpion 2.2, and herefore, he Hölder coninuiy propery 2.24 ofz.

22 24 Y. HU, D. NUALART AND X. SONG EXAMPLE Conider he following forward bacward ochaic differenial equaion FBSDE for hor: X = X + br,x r dr + σr,x r dw r, 2.32 Y = ϕ Xr 2 dr + fr,x r,y r,z r dr Z r dw r, where b,σ, ϕ and f are deerminiic funcion, and X R. We mae he following aumpion: h1 b and σ ha uniformly bounded fir- and econd-order parial derivaive wih repec o x, and here i a conan L>, uch ha, for any, [,T], x R, σ,x σ,x L 1/2. h2 up T { b, + σ, } <. h3 ϕ i wice differeniable, and here exi a conan C > and a poiive ineger n uch ha ϕ X 2 d T + ϕ X 2 d T + ϕ X 2 d C1 + X n, where x = up{ x, T } for any x C[,T]. h4 f,x,y,z ha coninuou and uniformly bounded fir- and econdorder parial derivaive wih repec o x,y and z and f,,, 2 d <. Noice ha in hi example, X = ϕ X 2 d i no necearily globally Lipchiz in X, and he reul of [16] canno be applied direcly. Under he above aumpion, h1 and h4, equaion 2.32 ha a unique oluion riple X,Y,Z, and we have he following claical reul: for any real number r>, here exi a conan C>uch ha E up X r <, E X X r C r/2 T for any, [,T]. Foranyfixedy, z R R, wehaved θ f,x,y,z= x f,x,y,zd θ X. Then, under all he aumpion in hi example, by Theorem and Lemma in [14] and he reul lied above, we can verify Aumpion 2.2. Therefore, Z ha he Hölder coninuiy propery Noe ha in he mulidimenional cae we do no require he marix σσ T o be inverible. 3. An explici cheme for BSDE. In he remaining par of hi paper, we le π ={ = < 1 < < n = T } be a pariion of he inerval [,T] and π = max i n 1 i+1 i. Denoe i = i+1 i, i n 1.

23 3.1 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 241 From 1.1, we now ha, when [ i, i+1 ], Y = Y i+1 + i+1 fr,y r,z r dr i+1 Z r dw r. Comparing wih he numerical cheme for forward ochaic differenial equaion, we could inroduce a numerical cheme of he form Y 1,π n = ξ π, Y 1,π i = Y 1,π i+1 + f i+1,y 1,π i+1,z 1,π i+1 i i+1 i Z 1,π r dw r, [ i, i+1, i = n 1,n 2,...,, where ξ π L 2 i an approximaion of he erminal condiion ξ. Thileado a bacward recurive formula for he equence {Y 1,π i,z 1,π i } i n. In fac, once Y 1,π i+1 and Z 1,π i+1 are defined, hen we can find Y 1,π i by Y 1,π i = E Y 1,π i+1 + f i+1,y 1,π i+1,z 1,π i+1 i F i, and {Zr 1,π } i r< i+1 i deermined by he ochaic inegral repreenaion of he random variable Y 1,π i Y 1,π i+1 f i+1,y 1,π i+1,z 1,π i+1 i. Alhough {Zr 1,π } i r< i+1 can be expreed explicily by Clar Ocone Hauman formula, i compuaion i a hard problem in pracice. On he oher hand, here are difficulie in udying he convergence of he above cheme. An alernaive cheme i inroduced in [16], where he approximaing pair Y π,z π are defined recurively by 3.2 Y π n = ξ π, Z π n =, Y π 1 = Y π i+1 + f i+1,y π i+1, E i+1 i+2 i+1 Zr π dr Fi+1 i i+1 Zr π dw r, [ i, i+1, i = n 1,n 2,...,, where, by convenion, E 1 i+2 i+1 i+1 Zr π dr F i+1 = wheni = n 1. In [16] he following rae of convergence i proved for hi approximaion cheme, auming ha he erminal value ξ and he generaor f are funcional of a forward diffuion aociaed wih he BSDE, 3.3 max E Y i Y π i n i 2 + E Z Z π 2 d K π. The main reul of hi ecion i he following, which on one hand improve he above rae of convergence, and on he oher hand exend erminal value ξ and generaor f o more general iuaion.

24 242 Y. HU, D. NUALART AND X. SONG THEOREM 3.1. Conider he approximaion cheme 3.2. Le Aumpion 2.2 be aified, and le he pariion π aify max i n 1 i / i+1 L 1, where L 1 i a conan. Aume ha a conan L 2 > exi uch ha 3.4 f 2,y,z f 1,y,z L /2 for all 1, 2 [,T] and y,z R. Then here are poiive conan K and δ, independen of he pariion π, uch ha, if π <δ, hen 3.5 E up Y Y π 2 + E T Z Z π 2 d K π +E ξ ξ π 2. PROOF. In hi proof, C> will denoe a conan independen of he pariion π, which may vary from line o line. Inequaliy 2.24 in Theorem 2.6b yield he following eimae Theorem 3.1 in [16] wih p = 2: n 1 i+1 E Z Z i 2 + Z Z i+1 2 d C π. i= i Uing hi eimae and following he ame argumen a he proof of Theorem 5.3 in [16], we can obain he following reul: 3.6 max E Y i Y π i n i 2 + E Denoe Z π 3.7 i = 1 i+1 E i Z Z π 2 d C π +E ξ ξ π 2., if i = n; Zr π dr Fi i, if i = n 1,n 2,...,. If i < i+1, i = n 1,n 2,...,, hen, by ieraion, we have 3.8 Therefore, Y π Y π i+1 = Y π i+1 + f i+1,y π i+1, Z π i+1 i Zr π dw r n = ξ π + f,y π, Z π 1 Zr π dw r. = E ξ π + n We rewrie he BSDE 1.1 a follow: 3.9 Y = ξ + = ξ + n f,y π, Z π F 1, [ i, i+1. fr,y r,z r dr Z r dw r f,y,z 1 Z r dw r + R π,

25 where MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 243 R π = fr,y r,z r dr = n n n f,y,z 1 1 [fr,y r,z r f,y,z ] dr 1 fr,y r,z r f,y,z dr + By Lemma 2.2 and he Lipchiz condiion on f,wehave p/2 E fr,y r,z r dr 2 <, and hence, 3.1 E max i n 1 i+1 i p fr,y r,z r dr i i+1 p/2 π p/2 E fr,y r,z r dr 2. Define a funcion {r} r T by { T, if r = T, r= i+1, if i r< i+1, i = n 1,...,. i fr,y r,z r dr fr,y r,z r dr. By he Hölder inequaliy, he boundedne of he fir-order parial derivaive of f,3.4, 2.24, Remar 2.8 and 3.1, i i eay o ee ha E up R π p 2 [E p 1 fr,y r,z r f p r,y r,z r dr T i+1 p ] + E max fr,y r,z r dr i n 1 i T p 1 E + 2 p 1 π p/2 E C π p/2, fr,y r,z r f r,y r,z r p dr p/2 fr,y r,z r 2 dr where, by convenion, R T =. In paricular, we obain 3.12 E up R π 2 C π. T

26 244 Y. HU, D. NUALART AND X. SONG To implify he noaion we denoe and δy π = Y Y π, δz π = Z Z π for all [,T] Ẑ π i = Z i Z π i for i = n, n 1,...,. Then, when i < i+1,by3.8 and3.9 we can wrie δy π n = [f,y,z f,y π, Z π ] 1 δz π r dw r + R π + δξ π, where δξ π = ξ ξ π. Therefore, we obain n 3.13 = E δy π [f,y,z f,y π, Z π ] 1 + R π + δξ π F Denoe f π = f,y,z f,y π, Z π. From equaliy 3.13 for j < j+1,wherei j n 1, and aing ino accoun ha δyt π = δy π n = δξ π,we obain up δy π n up E f π 1 + up Rr π + δξπ F. i T i T r T The above condiional expecaion i a maringale if i i conidered a a proce indexed by [ i,t]. Thu, uing Doob maximal inequaliy, we obain [ E up δy π 2 E up i T i T From 3.12, we deduce n E f π 1 + up r T ] 2 Rr π + δξπ F n 2 CE f π 1 + up Rr π + δξπ r T { n 2 C E f π 1 + E up Rr π 2 + E δξ π }. 2 r T { n 2 } E up δy π 2 C E f π 1 + E δξ π 2 + π. i T.

27 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 245 Uing he Lipchiz condiion on f, we obain { 3.14 E up δy π 2 C i T T i 2 E up δy π 2 i+1 n + E n 1 + CE δξ π 2 + π. Ẑπ E Ẑ n 2 2 n 1 Noice ha n 1 2 n 1 E Ẑπ 1 = E Z EZu π F du 1 n E E Z Zu π F du n 1 L 2 1 E 2 +1 E Z Zu π F du 3.15 { n 1 2L E E Z Z u F du n 1 2 } +1 + E E Z u Zu π F du = 2L 2 1 I 1 + I 2. Now he Minowi and he Hölder inequaliie yield } 3.16 I 1 E n 1 { +1 T i T i CT i n 1 n 1 n 1 E Z Z u F } 1/2 2 du 1/ E E Z Z u F 2 du E Z Z u 2 du u du C π. 2

28 246 Y. HU, D. NUALART AND X. SONG In a imilar way and by 3.6, we obain 3.17 On he oher hand, I 2 T i n 1 +1 E Z u Z π u 2 du = T i E δzu π 2 du C π. i EẐπ n n 1 2 = E Z n 2 n 1 2 C π 2. From , we have E up δy π 2 C 1 T i 2 E up δy π 2 i T i+1 n C 2 E δξ π 2 + π, where C 1 and C 2 are wo poiive conan independen of he pariion π. We can find a conan δ>independen of he pariion π, uch ha C 1 3δ 2 < 1 2 and T>2δ. Denoe l =[T 2δ ] [x] mean he greae ineger no larger han x. Then l 1 i an ineger independen of he pariion π. If π <δ, hen for he pariion π we can chooe n 1 >i 1 >i 2 > >i l, uch ha, T 2δ i1 1, i1 ], T 4δ i2 1, i2 ],...,T 2δl [, il ] wih 1 =. For impliciy, we denoe i = T and il+1 =. Each inerval [ ij+1, ij ],j =, 1,...,l, ha lengh le han 3δ, hai, ij ij+1 < 3δ. On each inerval [ ij+1, ij ],j =, 1,...,l, we conider he recurive formula 3.2, and 3.19 become E up δy π 2 C 1 ij ij+1 2 E up δy π 2 ij+1 ij i j+1 +1 i j C 2 E δy π ij 2 + π. Uing 3.2, we can obain inducively E up δy π 2 ij+1 ij 3.21 C 1 ij ij+1 2 E up δy π 2 + C 2 E δy π ij 2 + π i j+1 +1 i j C 1 ij ij+1 2 C 1 ij ij 1 2 E δy π ij 2 + C 2 E δy π ij 2 + π 1 + C 1 ij ij C 1 ij ij+1 2 C 1 ij ij C 1 ij ij+1 2 C 1 ij ij C 1 ij ij 1 2

29 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 247 C 1 3δ 2 i j i j+1 E δy π ij 2 + C 2 E δy π ij 2 + π 1 + C 1 3δ 2 + C 1 3δ C 1 3δ 2 i j i j+1 E δy π ij 2 C C 1 3δ 2 E δy π ij 2 + π E δy π ij 2 + 2C 2 E δy π ij 2 + π = 2C 2 + 1E δy π ij 2 + 2C 2 π. By recurrence, we obain 3.22 E up δy π ij+1 ij 2 2C j+1 E δξ π 2 + C 2 π 1 + 2C C j 2C l+1 E δξ π 2 + C 2 π 1 + 2C C l 32C l+1 E δξ π 2 + π. 2 Therefore, aing C = 32C 2+1 l+1 2, we obain E up δy π 2 max T j l E up δy π 2 C π +E ξ ξ π 2. ij+1 ij Combining he above eimae wih 3.6, we now ha here exi a conan K> independen of he pariion π, uch ha E up Y Y π 2 + E Z Z π 2 d K π +E ξ ξ π 2. T REMARK 3.2. The numerical cheme inroduced before, a oher imilar cheme, involve he compuaion of condiional expecaion wih repec o he σ -field F i+1. To implemen hi cheme in pracice we need o approximae hee condiional expecaion. Some wor ha been done o olve hi problem, and we refer he reader o he reference [2, 4] and[8]. 4. An implici cheme for BSDE. In hi ecion, we propoe an implici numerical cheme for he BSDE 1.1. Define he approximaing pair Y π,z π

30 248 Y. HU, D. NUALART AND X. SONG recurively by 4.1 Y π n = ξ π, Y π = Y π i+1 + f i+1,y π i+1, 1 i+1 i+1 Zr π dr i Zr π dw r, i i [ i, i+1, i = n 1,n 2,...,, where he pariion π and i, i = n 1,...,, are defined in Secion 3, andξ π i an approximaion of he erminal value ξ. In hi recurive formula 4.1, on each ubinerval [ i, i+1, i = n 1,...,, he nonlinear generaor f conain he informaion of Z π on he ame inerval. In hi ene, hi formula i differen from formula 3.2, and 4.1 i an equaion for {Y π,z π} i < i+1.when π i ufficienly mall, he exience and uniquene of he oluion o he above equaion can be eablihed. In fac, equaion 4.1 i of he following form: b b 4.2 Y = ξ + g Z r dr Z r dw r, [a,b] and a<b T. a For he BSDE 4.2, we have he following heorem. THEOREM 4.1. Le a<b T and p 2. Le ξ be F b -meaurable and ξ L p. If here exi a conan L>uch ha g : R, F b B R, B aifie gz 1 gz 2 L z 1 z 2 for all z 1,z 2 R and g L p, hen here i a conan δp,l >, uch ha, when b a<δp,l, equaion 4.2 ha a unique oluion Y, Z S p F [a,b] H p F [a,b]. PROOF. We hall ue he fixed poin heorem for he mapping from H p F [a,b] ino H p F [a,b] which map z o Z, wherey, Z i he oluion of he following BSDE: b b 4.3 Y = ξ + g z r dr Z r dw r, [a,b]. a In fac, by he maringale repreenaion heorem, here exi a progreively meaurable proce Z ={Z } a b uch ha E b a Z 2 d < and b b Fa b ξ + g z r dr = E ξ + g z r dr + Z dw. a a a By he inegrabiliy properie of ξ,g and z, one can how ha Z H p F [a,b]. Define Y = Eξ + g b a z r dr F, [a,b].theny, Z aifie equaion 4.3.

31 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 249 Noice ha Y i a maringale. Then by he Lipchiz condiion on g, he inegrabiliy of ξ,g and z, and Doob maximal inequaliy, we can prove ha Y S p F [a,b]. Le z 1,z 2 be wo elemen in he Banach pace H p F [a,b], andley 1,Z 1, Y 2,Z 2 be he aociaed oluion, ha i, Y i = ξ + g b b zr i dr Zr i dw r, [a,b],i = 1, 2. a Denoe Ȳ = Y 1 Y 2, Z = Z 1 Z 2, z = z 1 z 2. Then b b b 4.4 Ȳ = g zr 1 dr g zr 2 dr Z r dw r a a for all [a,b].so b b F Ȳ = E g zr 1 dr g zr 2 dr a a for all [a,b]. Thu by Doob maximal inequaliy, we have E up Ȳ p b b F = E up g E zr 1 dr g zr 2 p dr a b a b a a b b CE g zr 1 dr g zr 2 p dr a a 4.5 b b CE zr 1 dr zr 2 p dr a Cb a p/2 E a b a z r 2 dr p/2, where C> i a generic conan depending on L and p, which may vary from line o line. From 4.4, i i eay o ee Ȳ = Ȳ a + a Z r dw r for all [a,b]. Therefore, by he Burholder Davi Gundy inequaliy and 4.5, we have b p/2 E Z r dr 2 p CE up Z r dw r a a b a [ 4.6 C E Ȳ a p + E up Ȳ p] a b b p/2 Cb a p/2 E z r dr 2, a

32 241 Y. HU, D. NUALART AND X. SONG ha i, Z H p C 1 b a 1/2 z H p, where C 1 i a poiive conan depending only on L and p. Tae δp,l = 1/C1 2. I i obviou ha he mapping i a conracion when b a<δp,l, and hence here exi a unique oluion Y, Z S p F [a,b] [a,b] o he BSDE 4.2. H p F Now we begin o udy he convergence of he cheme 4.1. THEOREM 4.2. Le Aumpion 2.2 be aified, and le π be any pariion. Aume ha ξ π L p and here exi a conan L 1 > uch ha, for all 1, 2 [,T], f 2,y,z f 1,y,z L /2. Then, here are wo poiive conan δ and K independen of he pariion π, uch ha, when π <δ, we have p/2 E up Y Y π p + E Z Z d π 2 K π p/2 + E ξ ξ π p. T PROOF. If π <δp,l,whereδp,l i he conan in Theorem 4.1, hen Theorem 4.1 guaranee he exience and uniquene of Y π,z π. Denoe, for i = n 1,n 2,...,, Z π 1 i+1 i+1 = Zr π i+1 dr. i Noice ha { Z π i, } i=n 1,n 2,..., here i differen from ha in Secion 3. Then Y π i = Y π i+1 + f i+1,y π i+1, Z π i+1 i Recurively, we obain Denoe and Y π δξ π = ξ ξ π, i+1 i = ξ π + i i Z π r dw r, i = n 1,n 2,...,. n f,y π, Z π 1 i Zr π dw r, i = n 1,n 2,...,. δy π = Y Y π, δz π = Z Z π, [,T], Ẑ π i = Z i Z π i, i = n 1,...,.

33 4.7 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2411 If [ i, i+1, i = n 1,n 2,...,, hen by ieraion, we have δy π = δξ π + n [f,y,z f,y π, Z π ] 1 δzr π dw r + R π, i where R π i exacly he ame a ha in Secion 3. Denoe f π = f,y,z f,y π, Z π. Then for [ i, i+1, i = n 1,n 2,...,, we have n δy π = E δξ π + f π 1 + R π 4.8 F. From equaliy 4.8for j < j+1,wherei j n 1, and aing ino accoun ha δyt π = δy π n = δξ π, we obain n up δy π up E i T i T f π 1 + up r T Rr π + δξπ F. The above condiional expecaion i a maringale if i i conidered a a proce indexed by for [ i,t]. Uing Doob maximal inequaliy, 3.11, and he Lipchiz condiion on f,wehave E up δy π p i T [ E up E i T n f π 1 + up r T ] p Rr π + δξπ F n p CE f π 1 + up Rr π + δξπ r T { n p } C E f π 1 + E up Rr π p + E δξ π p r T { n n C C { E p δy π 1 + E T i p E up δy π p i+1 n n p + E Ẑπ 1 + π p/2 + E δξ π }, p p } Ẑπ 1 + π p/2 + E δξ π p

34 2412 Y. HU, D. NUALART AND X. SONG where, and in he following, C> denoe a generic conan independen of he pariion π and may vary from line o line. On he oher hand, we have, by he Hölder coninuiy of Z givenby2.24, n p E Ẑπ 1 n = E Z 1 p Zr π dr n n p E Z Z r dr + Z r Zr π dr 1 1 p C π p/2 + 2 p 1 E Z r Zr π dr Hence, we obain i C π p/2 + 2 p 1 T i p/2 E Z r Zr π 2 dr i p/2 = C π p/2 + 2 p 1 T i p/2 E δzr π 2 dr. i E up δy π p i T p/2 4.9 C 1 {T i p E up δy p i+1 n p/2 + T i p/2 E δzr π 2 dr i + π p/2 + E δξ π p }, where C 1 i a conan independen of he pariion π. By he Burholder Davi Gundy inequaliy, we have p/2 E δzr π dr c p E i From 4.7, we obain i δzr π dw p r i δz π r dw r = δξ π + n f π 1 + R π i δy π i.

35 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2413 Thu, from 4.1 and4.11, we obain p/2 E δzr π 2 dr i n p } C p {E f π 1 + E δξ π p + E R π i p + E δy π i p. Similar o 4.9, we have E δzr π 2 dr i p/2 C 2 {T i p E up δy p i+1 n p/2 } + T i p/2 E δzr π 2 dr + π p/2 + E δξ π p, i where C 2 i a conan independen of he pariion π. If C 2 T i p/2 < 1 2,henwehave 4.12 p/2 E δzr π 2 dr 2C 2 T i p E up δy p i i+1 n Subiuing 4.12 ino4.9, we have 4.13 E up δy π p i T C C2 T i p/2 T i p E + 2C 2 π p/2 + E δξ π p. up i+1 n + C C2 T i p/2 π p/2 + E δξ π p 2C 1 T i p E up i+1 n δy p δy p + 2C 1 π p/2 + E δξ π p. We can find a poiive conan δ<δp,lindependen of he pariion π, uch ha, 4.14 C 2 3δ p/2 < 1 2, C13δ p < 1 2 and T>2δ. Denoe l =[ 2δ T ].Thenl 1 i an ineger independen of he pariion π. If π <δ, hen for he pariion π we can chooe n 1 >i 1 >i 2 > > i l, uch ha, T 2δ i1 1, i1 ], T 4δ i2 1, i2 ],...,T 2δl [, il ] wih 1 =. For impliciy, we denoe i = T and il+1 =. Each inerval

36 2414 Y. HU, D. NUALART AND X. SONG [ ij+1, ij ],j =, 1,...,l, ha lengh le han 3δ, hai, ij ij+1 < 3δ. On [ ij+1, ij ], we conider he recurive formula 4.1. Then yield E up δy π ij+1 ij p C 1 ij ij+1 p E up δy p + 2C 1 π p/2 + E δy π ij p i j+1 +1 i j 2C 1 3δ p E up δy p + 2C 1 π p/2 + E δy π ij p i j+1 +1 i j 1 2 up δy p + 2C 1 π p/2 + E δy π ij p. i j+1 +1 i j A in he proof of 3.21 and3.22, we have E up δy π p 4C 1 + 1E δy π ij p + 4C 1 π p/2 ij+1 ij and E up δy π p 34C l+1 E δξ π 2 + π p/2. ij+1 ij 2 Therefore, we obain 4.17 E up δy π p max T j l E up δy π ij+1 ij p 34C l+1 E δξ π p + π p/2. 2 On [ ij+1, ij ],j =, 1,...,l, baed on he recurive formula 4.1 and4.17, inequaliy 4.12 become ij p/2 E δzr π 2 dr ij+1 2C 2 ij ij+1 p E up δy p + 2C 2 π p/2 + E δξ π p i j+1 +1 i j 2C 2 3δ p E up δy p + 2C 2 π p/2 + E δξ π p i j+1 +1 i j 1 2 E up δy p + 2C 2 π p/2 + E δξ π p i j+1 +1 i j 34C1 + 1 l+1 + 2C 2 π p/2 + E δξ π p. 4

37 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2415 Thu 4.18 p/2 E δz π 2 d l ij = E δz π 2 d ij+1 j= l + 1 p/2 1 l j= p/2 ij p/2 E δz π 2 d ij+1 l + 1 p/2 34C1 + 1 l+1 + 2C 2 π p/2 + E δξ π p. 4 Combining 4.17 and4.18, we now ha here exi a conan K = l + 1 p/2 34C1 + 1 l+1 + 4C 2 2 independen of he pariion π, uch ha p/2 E up Y Y π p + E Z Z π 2 d T K π p/2 + E ξ ξ π p. REMARK 4.3. The advanage of hi implici numerical cheme are: i we can obain he rae of convergence in L p ene; ii he pariion π can be arbirary π hould be mall enough wihou auming max i n 1 i / i+1 L A new dicree cheme. For all he numerical cheme conidered in Secion 3 and 4, one need o evaluae procee {Z π} T wih coninuou index. In hi ecion, we ue he repreenaion of Z in erm of he Malliavin derivaive of Y o derive a compleely dicree cheme. From 2.21, {D θ Y } θ T can be repreened a 5.1 D θ Y = E ρ,t D θ ξ + ρ,r D θ fr,y r,z r drf, where 5.2 { r ρ,r = exp β dw + r wih α = y f,y,z and β = z f,y,z. α 1 } 2 β2 d

38 2416 Y. HU, D. NUALART AND X. SONG Uing ha Z = D Y, μ P a.e., from 1.1, 5.1 and5.2, we propoe he following numerical cheme. We define recurively 5.3 Y π n = ξ, Z π n = D T ξ, Y π i = E Y π i+1 + f i+1,y π i+1,z π i+1 i F i, Z π i = E n 1 ρ π i+1, n D i ξ + =i ρ π i+1, +1 D i f +1,Y π +1,Z π +1 Fi i = n 1,n 2,...,,, where ρ π i, i = 1,i =, 1,...,n,andfor i<j n, 5.4 ρ π i, j = exp { j 1 +1 =i j =i z fr,y π,z π dw r y fr,y π,z π 1 } 2 [ zfr,y π,z π ] 2 dr. An alernaive expreion for ρ π i, j i given by he following formula: 5.5 ρ π i, j = exp { j 1 =i z f,y π,z π W +1 W j 1 + y f,y π,z π 1 2 [ zf,y π,z π ] 2 }. =i However, we will only conider he cheme 5.3 wih ρ π i, j givenby5.4. We mae he following aumpion: G1 f,y,zi deerminiic, which implie D θ f,y,z=. G2 f,y,z i linear wih repec o y and z; namely, here are hree funcion g, h and f 1 uch ha f,y,z= gy + hz + f 1. Aume ha g, h are bounded and f 1 L 2 [,T]. Moreover, here exi a conan L 2 >, uch ha, for all 1, 2 [,T], g 2 g 1 + h 2 h 1 + f 1 2 f 1 1 L 2 1 1/2. G3 E up θ T D θ ξ r <, for all r 1.

39 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 2417 Noice ha G1 and G2 imply ii and iii in Aumpion 2.2. REMARK 5.1. We propoe condiion G1 in order o implify {Z π i } i=n 1,..., in formula 5.3. In fac, here are ome difficulie in generalizing he condiion G, epecially G1, o a forward bacward ochaic differenial equaion FBSDE, for hor cae. If we conider a FBSDE X = X + br,x r dr + σr,x r dw r, Y = ξ + fr,x r,y r,z r dr Z r dw r, where X R, and he funcion b,σ,f are deerminiic, hen under ome appropriae condiion [e.g., h1 h4 in Example 2.11] Z π i for i = n 1,...,in5.3 i of he form Z π i = E ρ π i+1, n D i ξ n 1 + =i ρ π i+1, +1 x f +1,X π +1,Y π +1,Z π +1 D i X π +1 Fi where X π,y π,z π i a cerain numerical cheme for X,Y,Z.Iihardo guaranee he exience and he convergence of Malliavin derivaive of X π,and herefore, he convergence of Z π i difficul o derive., THEOREM 5.2. Le Aumpion 2.2i and aumpion G1 G3 be aified. Then here are poiive conan K and δ independen of he pariion π, uch ha, when π <δwe have E max { Y i Y π i n i p + Z i Z π i p } K π p/2 p/2log1/ π log 1 p/2. π PROOF. In he proof, C> will denoe a conan independen of he pariion π, which may vary from line o line. Under he aumpion G1, we can ee ha 5.6 Z π i = Eρ π i+1, n D i ξ F i, i = n 1,n 2,...,. Denoe, for i = n 1,n 2,...,, δz π i = Z i Z π i, δy π i = Y i Y π i.

40 2418 Y. HU, D. NUALART AND X. SONG Since e x e y e x + e y x y, we deduce, for all i = n 1,n 2,...,, δz π i = Eρ i, n D i ξ F i Eρ π i+1, n D i ξ F i E ρ i, n ρ π i+1, n D i ξ F i E D i ξ ρ i, n + ρ π i+1, n hr dw r + grdr 1 i i 2 n 1 +1 hr dw r E D i ξ ρ i, n + ρ π i+1, n [ i+1 i+1 hr dw r + i i θ T [ i i+1 i T i+1 + n 1 n 1 hr 2 dr i gr dr ] Fi. hr 2 dr grdr hr 2 dr F i From G2, we have D i ξ ρ π i+1, n { n 1 +1 D i ξ exp hr dw r + grdr 1 } T hr 2 dr i+1 2 i+1 C 1 up D θ ξ { up exp hr dw r }, θ T T where C 1 > i a conan independen of he pariion π. In he ame way, we obain D i ξ ρ i, n <C 1 up D θ ξ { up exp hr dw r }. θ T T Thu for i = n 1,n 2,...,, δz π i 2C 1 E up D θ ξ { } up exp hr dw r i hr dw r i gr dr i+1 i ] Fi hr 2 dr

41 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE C 1 E up D θ ξ up θ T T [ +1 up n 1 { exp hr dw r hr dw r } +1 + up n up 2 n 1 gr dr ] Fi hr 2 dr. The righ-hand ide of he above inequaliy i a maringale a a proce indexed by i = n 1,n 2,...,. Le η = exp{ hu dw u }. Then, η aifie he following linear ochaic differenial equaion: { dη = hη dw h2 η d, η = 1. By G1, G2, he Hölder inequaliy and Lemma 2.4, i i eay o how ha, for any r, { } r E up exp hu dw u T { } { } r = E exp hu dw u up exp hu dw u T { } 1/2 5.7 E exp 2r hu dw u E up T { exp 2r = exp {r 2 hu dr} 2 E up T hu dw u } 1/2 η 2r 1/2 <. For any p p, q 2, by Doob maximal inequaliy and he Hölder inequaliy, G3 and 5.7, we have E up δz π i p i n CE up p D θ ξ up T θ T [ +1 up n up n 1 hr dw r { exp gr dr hr dw r } p +1 ] p up hr 2 dr n 1

42 242 Y. HU, D. NUALART AND X. SONG [ pp C E /p p up D θ ξ θ T [ E [ C E [ E up T +1 up n 1 up θ T up T [ +1 E up n 1 { } pp /p p] p p/p exp hr dw r hr dw r up n 1 2pp /p p] p /2p p D θ ξ +1 gr dr +1 p ] p/p up hr 2 dr n 1 { } 2pp /p p] p /2p p exp hr dw r hr dw r p + E up n 1 = C[I 1 + I 2 + I 3 ] p/p. For any r>1, by he Hölder inequaliy we can obain +1 p { I 1 = E up hr dw r E up n 1 { n 1 +1 E = hr dw r p r } 1/r. +1 p gr dr +1 p ] p/p + E up hr 2 dr n 1 n 1 +1 hr dw r p r} 1/r For any cenered Gauian variable X,andanyγ 1, we now ha E X γ C γ γ γ/2 E X 2 γ/2, where C i a conan independen of γ. Thu, we can ee ha I 1 C p r p r p r/2 n 1 i+1 p hr 2 r/2 1/r dr Cr p /2 π p /2 1/r. i= Tae r = 2log1/ π p. Aume π i mall enough; hen we have I 1 C π p /2 p /2log1/ π log 1 p /2. π i