A numerical scheme for backward doubly stochastic differential equations

Transcription

1 Bernoull 191, 213, DOI: 1.315/11-BEJ391 A numercal scheme for backward doubly sochasc dfferenal equaons AUGUSTE AMAN UFR Mahémaques e Informaque, Unversé de Cocody, BP 582 Abdjan 22, Côe d Ivore. E-mal: auguseaman5@yahoo.fr; auguse.aman@unv-cocody.c In hs paper we propose a numercal scheme for he class of backward doubly sochasc dfferenal equaons BDSDEs wh possble pah-dependen ermnal values. We prove ha our scheme converges n he srong L 2 -sense and derves s rae of convergence. As an nermedae sep we derve an L 2 -ype regulary of he soluon o such BDSDEs. 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. Keywords: backward doubly SDEs; L -Lpschz funconals; L 2 -regulary; numercal scheme; regresson esmaon 1. Inroducon In hs paper we are neresed n he followng backward doubly sochasc dfferenal equaons BDSDEs, n shor: Y = ξ + fs,x s,y s,z s ds + gs,x s,y s db s Z s dw s, 1.1 where W and B are wo ndependen Brownan moons defned on, F, P, a complee probably space. Ths knd of equaon has wo dfferen drecons of sochasc negrals: sandard forward sochasc negrals, drven by W, and backward sochasc ones, drven by B. Inaed by Pardoux and Peng 18, BDSDEs are conneced o quas-lnear sochasc paral dfferenal equaons SPDEs, n shor n order o derve he Feynman Kac formula for SPDEs. In hs seng, BDSDEs have been exensvely suded n he pas decade. We refer he readers o he papers of Buckdahn and Ma 7,8, Aman e al. 2, Aman 1, Bahlal and Gherbal 3 and references heren for more nformaon on boh heory and applcaons, especally n mahemacal fnance and sochasc conrol, for such equaons. In conras, here was lle progress made n he drecon of he numercal mplemenaon of BDSDEs. In specal case of BDSDEs g called BSDEs, many effors have been made n hs drecon as well. Up o now bascally wo ypes of schemes have been consdered. Based on he heorecal four-sep scheme from Ma e al. 15, he frs ype of numercal algorhms for BSDEs have been developed by Douglas e al. 1 and more recenly by Mlsen and Trekyakov 17. The man focus of hese algorhms s he numercal soluon of parabolc PDEs whch s relaed o BSDEs ISI/BS

2 94 A. Aman A second ype of algorhm works backward hrough me and res o ackle he sochasc problem drecly. Bally 4 and Chevance 9 were he frs o sudy hs ype of algorhm wh a hardly mplemenable random me paron under srong regulary assumpons. The works of Ma e al. 14 and Brand e al. 6 are n he same spr, replacng, however, he Brownan moon by a bnary random walk. Recenly, Zhang 2 proved a new noon of L 2 -regulary on he conrol par Z of he soluon whch allowed proof of convergence of hs backward approach wh deermnsc parons under raher weak regulary assumpons see Zhang 2, Bouchard and Touz 5 and Lemor e al. 13 for dfferen algorhms. All numercal schemes provde alernaves o consruc algorhms for PDEs. To he bes of our knowledge, o dae, here has been no dscusson n he leraure concernng numercal algorhms n he spr of he las hree works ced above n he general case, ha s, g. Ths consues an nsuffcency when we know ha almos all he deermnsc problems n hese appled felds PDEs have her sochasc counerpars SPDEs. In hs paper, o fll hs vod, our goal s o buld a numercal scheme followng he dea used by Bouchard and Touz 5 and sudy s convergence. These resuls are mporan from a purely mahemacal pon of vew, and also n applcaon o he world of fnance. Parcularly, hs numercal scheme clears he way for a possble algorhm for deermnng he prce of opons on fnancal asses whose dynamcs are a soluon of SPDEs. Smlarly o he specal case g, he man dffculy 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. However, n case g, hs regulary becomes a naural queson o ask. Therefore, he frs man resul n hs paper s o derve he pah regulary called L 2 -regulary for BDSDEs wh he ermnal value ξ s he form X, where X and are, respecvely, a dffuson process and an L -Lpschz funconal see Secon 3 for precse defnon. The proof s heavly relaed o Grsanov s Transformaon whch exss n he BDSDEs case, only f g does no depend on z. The above L 2 -regulary resul allows us o provde he rae of convergence of our numercal scheme whch s dfferen from he one consruced n Bouchard and Touz 5. Indeed, snce BDSDEs have wo drecons of negrals, our numercal scheme needs, a each sep, he condonal expecaon wh respec o he flraon F π defned by F π = σx π j,j σb j,j. However, we oban he same convergence rae. The res of hs paper s organzed as follows. In Secon 2, we nroduce some fundamenal knowledge and assumpons of BDSDEs. Secon 3 s devoed o L 2 -regulary resuls. In Secon 4, we bul our numercal scheme and prove he rae of convergence. Fnally n Secon 5, we focus some deas for he regresson approxmaon and gve a convergence rae. 2. Prelmnares Le, F, P be a complee probably space and T> be fxed hroughou hs paper. Le {W, T } and {B, T } be wo muually ndependen sandard Brownan moon processes, wh values, respecvely, n R d and R l, defned on, F, P.LeN denoe he class of P-null ses of F. For each,t, we defne F = F W F B,T,

3 A numercal scheme for backward doubly SDEs 95 where for any process η s : s T,F η s, = σ {η r η s,s r } N, F η = F η,. We noe ha snce he collecon F s neher ncreasng nor decreasng, does no consue a flraon. Therefore, we defne he flraon F by F = F W F B T, whch conans F, and play a key role n he buldng of our numercal scheme. For any real p 2 and k N,leS p R k denoe he se of jonly measurable processes {X },T, akng values n R k, whch sasfy: X S p = Esup T X p 1/p < + ; X s F -measurable, for any,t. We denoe smlarly by M p R k he se of classes of dp d a.e. equal k-dmensonal jonly measurable processes whch sasfy: X M p = E X 2 d p/2 1/p < + ; X s F -measurable, for a.e.,t. We denoe by: W 1, R k he space of all measurable funcons ψ : R k R, such ha for some consan K>holds ha ψx ψy K x y, x,y R k ; D he space of all càdàg funcons defned on,t; Cb m,t Rk he space of all connuous funcons no necessarly bounded ψ :,T R k R, such ha ψ has unformly bounded dervaves wh respec o he spaal varables up o order m. We ofen denoe Cb m = Cm b,t Rk for smplcy, when he conex s clear. Le b :,T R d R d, σ :,T R d R d d, f :,T R d R R d R, g :,T R d R R l be he funcons sasfyng he followng assumpons: here exss consan K>such ha for all s,s,t,x,x R d,y,y R,z,z R d, H1 bs,x bs,x + σs,x σs,x K x x. H2 fs,x,y,z fs,x,y,z 2 K s s 2 + x x 2 + y y 2 + z z 2, gs,x,y gs,x,y 2 K s s 2 + x x 2 + y y 2. H3 sup T { b, + σ, + f,,, + g,, } K. Gven ξ L 2, F T, P; R d, denoe X,Y,Zbe he soluon o he followng FBDSDE: X = x + bs,x s ds + σs,x s dw s, 2.1

4 96 A. Aman Y = ξ + fs,x s,y s,z s ds + gs,x s,y s db s Z s dw s. 2.2 Le us now recall some sandard resuls ha appear n he SDEs and BDSDEs leraure. Proposon 2.1 Karazas and Shreve 11. Assume H1 holds. Then for any nal condon x R d, FSDE 2.1 has a unque soluon X T ha belongs o S p R d. Moreover, for any p 2, here exss a consan C p >, dependng only on T,K and p, such ha and E sup T X p C p x p + b, p + σ, p d E X X s p C p E x p + sup b, p + sup σ, p s p/2. T T Proposon 2.2 Pardoux and Peng 18. Under assumpon H2, BDSDE 2.2 has a unque soluon Y,Z T n S p R M p R d. Moreover, for any p 2, here exss a consan C p >, dependng only on T,K and p, such ha E sup Y p + T and p/2 Z s 2 ds C p E ξ p + f,,, p + g,, p d E Y Y s p C p E{ ξ p + sup f,,, p + sup g,, p s p 1 T T + Z s 2 ds s p/2 }. Remark 2.1. In Proposon 2.2, he exsence and unqueness resul needs only a Lpschz condon on f and g, wh respec o varables y and z, unformly n and x. Proposon 2.3 Sably. Le X ε,y ε,z ε be he soluon o he perurbed FBDSDE 2.1 and 2.2 n whch he coeffcens are replaced by b ε,σ ε,f ε,g ε, wh nal sae x ε an ermnal value ξ ε. Assume ha he assumpon H1 and H2 hold for all coeffcens b ε,σ ε,f ε,g ε and assume ha lm ε x ε = x, and for fxed x,y,z, { lm E ξ ε ξ 2 + ε { } lm E b ε, x b,x 2 + σ ε, x σ,x 2 d =, ε } f ε,x,y,z f,x,y,z 2 + g ε,x,y g,x,y 2 d =.

5 A numercal scheme for backward doubly SDEs 97 Then, we have { lm E sup ε T X ε X 2 + sup Y ε Y 2 + sup Z ε Z 2} =. T T 3. L 2 -regulary resul for BDSDEs In hs secon we esablsh he frs man resul of hs paper, whch s L 2 -regulary of he marngale negrand Z, and whch can be hough of as he modulus of connuy of he pahs n an L 2 sense. Such a regulary, combned wh he esmae for X and Y, plays a key role for dervng he rae of convergence of our numercal scheme n Secon 4. We shall consder a class of BDSDEs wh ermnal values whch are pah dependen, ha s, of he form ξ = X, where a deermnsc funconal : D R sasfes: H4 L -Lpschz condon. There exss a consan K such ha X 1 X 2 K sup X 1 X 2 X 1,X 2 D. 3.1 T H5 s bounded by K, where denoes he consan funcon akng value on,t. Ths approxmaon, due o Ma and Zhang 16, for L -Lpschz funconal, wll be useful n he sequel. Lemma 3.1. Suppose H4 and H5 hold. Le ={π} be a famly of parons of,t. Then here exss a famly of dscree funconals {h π : π } such ha: for each π, assumng π: = < 1 < 2 < < n = T, we have ha h π Cb 1Rdn+1, and sasfes n x h π x K x = x,x 1,...,x n R dn+1, 3.2 =1 where K s he same consan as ha n 3.1; for any X D, holds ha where π =max 1 n 1. lm π hπ X,X 1,...,X n X =, 3.3 Our man resul n hs secon s he followng heorem. Theorem 3.1. Assume H1 H5. Le π : s <...,s m be any paron of,t, and for each 1 m, le us defne Z π 1 s s 1 = E Z s ds F s s s 1 s 1

6 98 A. Aman Then here exss a consan C dependng only on T and K, such ha m s E max sup Y Y s Z s Z s π 1 m 1 2 ds C π. 3.5 s 1 s s 1 =1 In he sequel, le π: =,..., n = T be any paron of,t, fner han, and, whou loss of generaly, we assume s = l for = 1,...,m. Snce sasfes he L -Lpschz condon 3.1, by vrue of Lemma 3.1 one can fnd h π C 1 R dn+1 sasfyng 3.2 and 3.3. Le Y π,z π be he soluon o he followng BDSDE: Y π = h π X,...,X n + + gs,x s,ys π db s fs,x s,y π s,zπ s ds 3.6 Z π s dw s. Moreover, seng π = π,z π, wh π = X, Y π,le X, Y π, Z π be he unque soluon of he followng FBDSDE: We denoe where X = I d + Y π = n j + + b x r, X r X r dr + h π x j X,...,X n X j σ x r, X r X r dw r, f x π r X r + f y π r Yr π + f z π r Zr π dr 3.7 g x π r X r + g y π r Y π r db r Z π r dw r,,t, =,...,n 1. ξ = f x π r X rnr 1 dr + g x π r X rnr 1 db r ; ξ = h π X,...,X n X N 1 T, = 1,...,n, N = exp f y π r dr + { M = exp f z π r dw r 1 2 g y π r db r 1 2 f z π r 2 dr }. g y π r 2 dr, 3.8

7 A numercal scheme for backward doubly SDEs 99 The followng echncal lemma s he buldng block of he proof of Theorem 3.1. Lemma 3.2. Le us consder he paron π defned above and h π gven by Lemma 3.1, and assume σ,b,f,g, Cb 1. Then for all = 1,...,n Y π = ξ + ξ j M 1 N f x π r X rnr 1 drn g x π r X rn 1 r db r N, j where ξ j = EM T ξ j F, j =,...,n. Proof. For each n, we recall X, Y π, Z π, he soluon of he lnear FBDSDE 3.7. Le γ,ζ and γ j,ζ j, j = 1,...,n, be he soluon of he BDSDEs γ = γ j + f x π r X r + f y π r γ r + f z π r ζ r dr g x π r X r + g y π r γ r db r = hπ X,...,X n X j + x j + g y π r γ j r db r ζ r dw r; f y π r γ j r + f z π r ζ j r dr ζ j r dw r, respecvely; hen we have he followng decomposon: 3.9 n Ys π = γs + γs j, s 1,. 3.1 j= Recall 3.8, and, snce f y,f z and g y are unformly bounded, by Grsanov s Theorem see, e.g., Karazas and Shreve 11, we know ha M s a P-marngale on,t, and W = W f z π r dr,,t,sanf -Brownan moon on he new probably space, F, P, where P s defned by d P dp = M T. Now for n, defne γ = γ N 1, ζ = ζ N 1,,T. Then, usng negraon by pars and equaon 3.9 wehave γ = ξ ζ j r d W r,,t. Therefore, by he Bayes rule see, e.g., Karazas and Shreve 11, Lemma we have, for,t, γ = γ N = EM T ξ F M 1 N = ξ M 1 N,

8 1 A. Aman where for =,...,n, ξ = E{M T ξ F }=EM T ξ + η s dw s Recallng 3.18, M T L p F T and X L p F for all p 2. Therefore for each p 1, 3.2 leads o { n p { E M T ξ } j C p E M T p sup X p} T j=1 In parcular, for each j,m T ξ j L 2 F T, hus 3.11 makes sense. Fnally he resul follows by 3.1. Proof of Theorem 3.1. Le us recall he paron π defned above, and consder π, s mesh, defned by Applyng Proposon 2.2, we ge π = max m s s 1. E Y Y s 1 2 C π, s 1,s, = 1,...,m, whch, ogeher wh Burkölder Davs Gundy nequaly, mples E max sup Y Y s 1 2 C π m s 1 s The esmae for Z s a lle nvolved. Ths par wll be dvde n wo seps. Sep 1. Frs we assume ha b,σ,f, g Cb 1. I follows from Lemma 3.1, ogeher wh Proposon 2.3, ha { } lm E sup Y π Y 2 + Z π Z 2 d = π T On he oher hand, accordng o 3.4, Z π s 1 L 2, F s 1. Then snce Zs π 1 L 2, F s 1, follows from Lemma of Zang 19, page 71, ha m s m s E Z s Z π s 1 2 ds E Z s Zs π 1 2 ds s 1 s 1 =1 =1 m s 2E Z s Zs π 2 + Zs π Zπ s 1 2 ds =1 s 1 m s C π +E Zs π Zπ s 1 2 ds. s 1 =1 3.15

9 A numercal scheme for backward doubly SDEs 11 By 3.14 and 3.15, remans o prove ha m =1 s E Zs π Zπ s 1 2 ds C π, 3.16 s 1 where C s ndependen of π or π.nowwefx.for s 1,s, follows from Proposon 2.3 of Pardoux and Peng 18, ogeher wh Lemma 3.2, Therefore, Z π = ξ + ξ j M 1 j N X 1 σx. where recallng ha s 1 = l 1 I 1 = I 2 = ξ + ξ j j ξ + j ξ j ξs + 1 M 1 f x π r X rnr 1 dr g x π r X rnr 1 db r Z π Z π s 1 I 1 + I 2 + I 3 + I 4, 3.17 j l 1+1 ξ j s 1 M 1 s 1 N s 1 X s 1 1 σx s 1, N X 1 σx M 1 s 1 N s 1 X s 1 1 σx s 1, I 3 = A 1, I 4 = A 2 wh A 1 = f x π r X rnr 1 dr N X 1 σx s 1 f x π r X rnr 1 dr N X s 1 s 1 1 σx s 1 and A 2 = g x π r X rnr 1 db r N X 1 σx s 1 g x π r X rnr 1 db r N X s 1 s 1 1 σx. s 1

10 12 A. Aman Recallng 3.8, and nong ha f y,f z and g y are unformly bounded, one can deduce ha, for all p 1, here exss a consan C p dependng only on T,K and p, such ha E sup T N p + N 1 p C p ; E sup T M p + M 1 p C p ; E N N s p + N 1 Ns 1 p C p s p/2 ; 3.18 E M M s p + M 1 Ms 1 p C p s p/2. Thus, applyng Proposon 2.1 and 2.2, one can show ha and Recallng 3.11 and 3.2 wehave ξ + j E I 3 2 C π, 3.19 E I 4 2 C π. 3.2 { CE sup X F }. ξ j T Thus by usng agan Proposons 2.1 and 2.2 ogeher wh 3.18, we ge As proved n Zhang 2 see he proof of Theorem 3.1, we have E I 2 2 C π E I 1 2 C π Combnng 3.19, 3.2, 3.21 and 3.22, we deduce from 3.17 ha 3.16 holds, whch ends he proof for he smooh case. Sep 2. Le us consder he general case, ha s, b,σ,f,g are only Lpschz. For ϕ = b,σ,f,g, no dffcul o consruc va a convoluon mehod, for any ε>, he funcon ϕ ε C 1 b be he smooh mollfers of ϕ such ha he dervaves of ϕ ε are unformly bounded by K and lm ε ϕ ε = ϕ. LeX ε,y ε,z ε and X ε,y π,ε,z π,ε denoe he soluon o correspondng FBDSDE replaced ϕ by ϕ ε, and se N ε = exp { M ε = exp fy ε π,ε r dr + f ε z π,ε r dw r 1 2 gy ε π,ε r db r 1 2 f ε z π,ε r 2 dr }. gy ε π,ε r 2 dr, Then one can derve, snce he funcon fy ε,fε z and gε y are unformly bounded by K, wh he sandard calculus abou BSDEs, ha, for all p 1, here exss a consan C p ndependen on ε

11 A numercal scheme for backward doubly SDEs 13 dependng only on T,K and p, such ha E sup T N ε p + N ε 1 p C p ; E E N ε Ns ε p + N ε 1 N ε 1 s p C p s p/2 ; E M ε Ms ε p + M ε 1 M ε 1 s p C p s p/2. Nex, defne Then follows from Sep 1 ha sup T Z ε,π 1 s s 1 = E Zs ε s s ds F s 1. 1 s 1 m =1 s s 1 Z ε s Z π,ε 1 2 ds C π. M ε p + M ε 1 p C p ; Therefore usng agan Lemma of Zang 19, page 71, we oban m =1 s E Zs π Z π s 1 2 ds s 1 m =1 s E Zs π Z π,ε s 1 2 ds s 1 m s E Z s Zs ε 2 + Zs ε Z π,ε 1 2 ds =1 s 1 E Z s Zs ε 2 ds + C π. Applyng Proposon 2.3,wehave lm E Z s Zs ε ε 2 ds =, whch, combned wh 3.23, proves he heorem Numercal scheme and rae of convergence In hs secon, we consder he BDSDE 2.2 n he specal case X = hx T, where h W 1, R d, such ha h s bounded by K. The goal of hs secon s o consruc an approxmaon of he soluon X,Y,Zby usng he sep processes. Le us recall π: < 1 < < n = T he paron of,t and π =max 1 n π, wh π = 1. We se also π W = W W 1, π B = B B 1 and for all n defne F π = σx j,j F B,

12 14 A. Aman he dscree-me flraon. Le us brefly revew he Euler scheme for he forward dffuson X. Defne π = 1,for 1,.LeX π be he soluon of he followng SDE: X π = x + b πs,xπs π ds + σ πs,xπs π dws, 4.1 and we defne a sep process ˆX π as follows: ˆX π = Xπ π,,t. 4.2 The followng esmae s well known see, e.g., Kloeden and Plaen 12. Proposon 4.1. Assume b and σ sasfy he assumpons H1 and H3. Then here exss a consan C dependng only on T and K, such ha max E sup 1 n T X π X 2 + sup 1 X X 1 2 C π. Moreover, we ge he followng esmae nvolvng he sep process ˆX π due o Zhang 2. Proposon 4.2. Assume b and σ sasfy he assumpons H1 and H3. Then here exss a consan C dependng only on T and K, such ha sup E ˆX π T E sup T X 2 C π ; ˆX π X 2 1 C π log. π The backward componen Y, Z wll be approxmaed by he followng numercal scheme: Y π n = hx π T, Zπ n =, Z π 1 = 1 π where E π = E F π and Ỹ π Remark 4.1. E π 1 Ỹ π π W, 4.3 Y π 1 = E π 1 Ỹ π +f 1,X π 1,Y π 1,Z π 1 π, 4.4 = Y π + g,x π,y π π B. Our approxmaon scheme dffers from he one appearng n Bouchard and Touz 5. Indeed, acually we use he condonal expecaon wh respec he enlarged flraon σx j,j F B, whch s necessary o exend Iô s represenaon heorem for backward doubly SDE see Pardoux and Peng 18.

13 A numercal scheme for backward doubly SDEs 15 The backward componen and he assocaed conrol Y, Z, whch solves he backward doubly SDE, can be expressed as a funcon of X and B, ha s, Y,Z = u, B,X, v, B,X, for some deermnsc funcons u and v. Then, he condonal expecaons, nvolved n he above dscrezaon scheme, reduce o he regresson of Ỹ π and Ỹ π W W 1 on he random varable X π 1,B 1. Nex, for all n, on can show ha Ỹ π belongs o L 2, F, hus an obvous exenson of Iô marngale represenaon heorem yelds he exsence of he F s s 1, -jonly measurable and square negrable process Z π sasfyng Ỹ π Therefore we defne he followng connuous verson: = EỸ π F 1 π + Z s π dw s Y π = Y π 1 1 f 1,X π 1,Y π 1,Z π 1 g,x π,y π B B Z π s dw s, 1 <. 4.6 Noe ha he process Z π s gven by he represenaon heorem; hus s useful and even necessary o fnd a relaonshp wh Z π, defned by 4.3. We have: Lemma 4.1. Assume b,σ,f and g sasfy he assumpons H1, H2 and H3 and le h W 1, R d such ha h s bounded by K. Then for all 1 n, we have Proof. Le us recall π Zπ 1 = 1 π Z π 1 = 1 π E π 1 1 Z s π ds. E π 1 Y π + g,x π,y π π B π W. Then follows from 4.5 ha Z π 1 = 1 π E π 1 π W 1 Z π s dw s. The resul follows by Iô s somery. We also need he followng, whch s he parcular case of Theorem 3.1.

14 16 A. Aman Lemma 4.2. Assume b,σ,f and g sasfy he assumpons H1, H2 and H3, and le h W 1, R d such ha h s bounded by K. Le defne, for each 1 n, Z π 1 = 1 π E π 1 Z s ds. 1 Then here exss a consan C dependng only on T and K, such ha n E max sup Y Y Z s Z π 1 n 1 2 ds C π =1 We are now ready o sae our man resul of hs secon, whch provdes he rae of convergence of he numercal scheme Theorem 4.1. Assume b,σ,f and g sasfy assumpons H1, H2 and H3, and le h W 1, R d such ha h s bounded by K. Then here exss a consan C dependng only on T and K, such ha sup E Y Y π 2 + E T Z s Ẑs π 2 ds C π. Proof. The proof follows he seps of he proof of Theorem 3.1 n Bouchard and Touz 5, so we wll only oulne. In he sequel, C> wll denoe he generc consan ndependen of and n, and may vary lne o lne. For {,...,n 1},wese δ π Y = Y Y π, δ π Z = Z Z π, δπ f= f,x,y,z f,x π,y π,z π and δ π g = g,x,y, g +1,X π +1,Y π +1,, +1. By Iô s formula, follows from a Lpschz condon on f, g and h, ogeher wh he nequaly ab βa 2 + b 2 /β, ha V = E δ π Y 2 + E = 2E C β β δ π Y s,δ π fs ds + δ π Z s 2 ds δ π Y δ π gs 2 ds E{ π 2 + X s X π 2 + Y s Y π 2 + Z s Z π 2 } ds 4.8 CE{ π 2 + X s X π Y s Y π +1 2 } ds E δ π Y s 2 ds,, +1.

15 A numercal scheme for backward doubly SDEs 17 Proposon 2.1, Proposon 2.2 and Lemma 4.1 yeld ha E X s X π 2 + E X s X π +1 2 C π, Pluggng 4.9no4.8, we ge V C β + C β E Y s Y π 2 2E Y s Y 2 + E δ π Y 2 C π +E δ π Y 2, E Y s Y π E Y s Y E δ π Y C π +E δ π Y +1 2, E Z s Z π 2 2 E Z s Z π π +1 E{ π + δ π Y 2 + Z s Z π 2 } ds + C E δ π Z s 2 ds + β +1 E δ π Y s 2 ds, +1 whch, along wh he defnons of V, provdes, for +1, +1 E δ π Z r 2 dr. E{ π + δ π Y +1 2 } ds where E δ π Y A = 1 + CπE δ π Y C β E δ π Z s 2 ds β +1 π 2 + π E Y π + E δ π Y s 2 ds + A, E Z s Z π 2 ds + C β +1 E δ π Z s 2 ds. Nex, by some calculus used n Gronwall s lemma, we have E δ π Y E δ π Z s 2 ds 1 + Cβ π A ; 4.11 hence for = and β suffcenly larger han C, such ha C β < 1, we oban for small π. E δ π Y C +1 E δ π Z s 2 ds β { 1 + C π E δ π Y π } E Z s Z π 2 ds

16 18 A. Aman Ierang he las nequaly, we ge E δ π Y C +1 E δ π Z s 2 ds β 1 + C π T/ π {E δ π Y T 2 + π + n =1 1 E Z s Z π 1 2 ds Moreover, follows from Lemma 4.2, he Lpschz condon on g and Proposon 4.1 ha E δ π Y C +1 E δ π Z s 2 ds β C π T/π {E δ π Y T 2 + π +C π } C π for small π. On he oher hand, summng up nequaly 4.11 wh =, we ge 1 Cβ 1 + Cβ π E δ π Z s 2 ds 1 + Cβ π C β π +1 + Cβ π 1 + C π E δπ Y T Cβ π Cβ π 1 E δ π Y Cβ π 1 + C π + Cβ n 1 π 1 E δ π Y Cβ π C β n 1 +1 = E Z s Z π 2 ds. Therefore, by nequaly 4.12 and agan Lemma 4.2, one derves ha and hen E δ π Z s 2 ds C π sup δ π Y 2 C π. T To end hs secon, le us gve he followng bound on Y π s, whch wll be used n he approxmang of he dscree condonal expecaon E π, for all n 1. Lemma 4.3. Assume b,σ,f and g sasfy he assumpons H1, H2 and H3, and le h W 1, R d such ha h s bounded by K. For all n 1, defne he sequences of =1 }.

17 A numercal scheme for backward doubly SDEs 19 random varables by backward nducon: Then, for all n, α π n = 2C, β n = C, α π = 1 C π C 2 π 1/2 {1 + 2C π 1 + C π B +1 β π +1 + C π B +1 + C π }, β π = 1 C π C 2 π 1/2 {1 + C π B +1 α π C2 π 1 + 2C π B C π }. E π 1 Y π E π 1 Y π Y π α π + β π Xπ 2, g,x π,y π π B + g,x π,y π π B 2 1/ C π {1 + C π B +1 β π +1 + C π B +1 } X π C π B +1 α+1 π + 6C2 π 1 + 2C π B +1, E π 1 Y π + g,x π,y π π B π W π 1 + 2C π {1 + C π B +1 β π +1 + C π B +1 } X π π 1 + C π B +1 α π C2 π 1 + 2C π B +1. Moreover, lm sup π max n απ + β π < a.s. Proof. Frs, snce h s K-Lpschz, h s bounded by K, Nex, we assume ha Y n = hx π T C Xπ T +1 2C + C Xπ T 2 = α n + β n X π T 2. Y +1 α π +1 + βπ +1 Xπ +1 2, 4.16 for some fxed n 1. Then by he defnon of Y π n 4.4, here exss a F -measurable random varable ζ such ha 1 C π Y π E π Y π +1 + g +1,X π +1,Y π +1 π B ζ π W +1 + C π 2 + X π 4.17 E π Y π +1 + g +1,X π +1,Y π +1 π B /2 E π +ζ π W /2 + C π 3 + X π 2.

18 11 A. Aman I show n Bouchard and Touz 5 ha E π 1 + ζ π W C 2 π. Ths provdes, from 4.17, 1 C π Y π 1 + C 2 π 1/2 E π Y π +1 + g +1,X π +1,Y π +1 π B /2 + C π 3 + X π Bu follows from he Lpschz propery of g ha E π Fnally 4.18 becomes Y π +1 + g +1,X π +1,Y π +1 π B /2 1 + C π B +1 E π Y π /2 + C π B +1 E π Xπ /2 1 + C π B +1 {α π +1 + βπ C π Xπ 2 + 6K 2 π } + C π B C π X π 2 + 6K 2 π = 1 + 2C π {1 + C π B +1 β π +1 + C π B +1 } X π C π B +1 α π C2 π 1 + 2C π B +1. Y π 1 C π C 2 π 1/2 {1 + 2C π 1 + C π B +1 β π +1 + C π B +1 + C π } X π C π C 2 π 1/2 {1 + C π B +1 α π C2 π 1 + 2C π B C π } = α π + β π Xπ Rae of convergence of he regresson approxmaon In hs secon, we ry o gve some deas for a mehod of smulang a numercal scheme derved n he above secon. I well known ha he process X π defned by 4.1 s smulaed by he classcal Mone Carlo mehod. We are reduced o smulae he process Y π,z π defned n 4.3 and 4.4. In pracce, he man ool used o defne an approxmaon of Y π, and hen of Z π, s o replace he condonal expecaon E π by s esmaor Ê π n he backward scheme 4.3 and 4.4. We frs esablsh he followng bound on he Y π s whch help us o derve hs smulaon.

19 A numercal scheme for backward doubly SDEs 111 For he regresson approxmaon, we consder {P π } n, {R π } n and {J π } n defned by P π = α π + β π Xπ 2, R π = 1 + 2C π {1 + C π B +1 β π +1 + C π B +1 } X π C π B +1 α π C2 π 1 + 2C π B +1, J π = π 1 + 2C π {1 + C π B +1 β π +1 + C π B +1 } X π 2 + π 1 + C π B +1 α π C2 π 1 + 2C π B +1. Therefore, hanks o Lemma 4.3,wehave P π Xπ, π B +1 Y π P π Xπ, π B +1, 5.1 R π Xπ, π B +1 E π Ỹ π +1 R π Xπ, π B +1, 5.2 J π X π, π B +1 EỸ π +1 π W +1 J π X π, π B Nex, for an R-valued random varable ξ, we defne T Pπ ξ = P π Xπ, π B +1 ξ P π Xπ, π B +1, T Rπ ξ = R π Xπ, π B +1 ξ R π Xπ, π B +1, T J π ξ = J π X π, π B +1 ξ J π X π, π B +1. Gven an approxmaon Ê π of E π, we are ready o ge he process Ŷ π, Ẑ π defned by followng backward nducon scheme, Ŷ π n = gx π T, Ẑ π 1 = 1 π Ê π Ŷ π 1 + g,x π, Ŷ π π B π W for all 1 n. Y π 1 = Ê π 1 Ŷ π + g,x π, Ŷ π π B +f 1,X π 1,Y π 1, Ẑ π 1 π 5.6 Ŷ π 1 = T Pπ 1 Y π 1, 5.7 Remark 5.1. Usng he above noaon and replacng X π by U π = X π,b π, one can sae analogously o Examples 4.1 and 4.2, appearng n Bouchard and Touz 5. To end hs secon, we derve he followng L p esmae of he error Ŷ π Y π n erms of he regresson errors Ê π E π.

20 112 A. Aman Theorem 5.1. Le p 1 be gven and P π be a sequence defned above. Then here s a consan C dependng only on T,K and p such ha Ŷ π C π Y π L p max { Êj E j Ŷ π 1 j n 1 j+1 + g j+1,x π j+1, Ŷ π j+1 π B j+1 L p + Ê j E j Ŷ π j+1 + g j+1,x π j+1, Ŷ π j+1 π B j+1 π W j+1 L p}. Proof. For n 1 o be fxed, wh calculus smlar o ha used n he proof of Theorem 4.1 n Bouchard and Touz 5, we have 1 Cπ Y π Ŷ π ε +1 + C π B +1 E Y π Ŷ π p 1/p 5.8 E π 1 + ζ π W +1 2k 1/2k, where k s an arbrary neger greaer han he conjugae of p and ε = Ê E Ŷ π +1 + g +1,X π +1, Ŷ π Snce and + π +1 { f,x π,y π, π +1 1 E π f,x π,y π Ê π, π π B +1 Ŷ π +1 + g +1,X π +1, Ŷ π +1 π B +1 π W +1 Ŷ π +1 + g +1,X π +1, Ŷ π +1 π B +1 π W +1 }. ε L p η = C Ê E Ŷ π +1 + g +1,X π +1, Ŷ π +1 π B +1 L p we ge from Cπ Y π + Ê E Ŷ π +1 + g +1,X π +1, Ŷ π +1 π B +1 π W +1 L p, E π 1 + ζ π W +1 2k 1/2k 1 + C π, Ŷ π L p η C π 1/2k Y π + Ŷ π +1 L p, 5.9 and hus he resul follows as n Bouchard and Touz 5. Remark 5.2. Wh he foregong resuls, s possble, wh some no very dffcul adjusmens, o oban smlar resuls o hose obaned by Bouchard and Touz see Secons 5 and 6 of Bouchard and Touz 5.

21 A numercal scheme for backward doubly SDEs 113 Acknowledgemens The auhor would lke o hank I. Boufouss and Y. Ouknne for her valuable commens and suggesons and o express hs deep graude o UCAM Mahemacs Deparmen for her frendly hospaly durng hs say n Cad Ayyad Unversy. We also hank a anonymous referee whose suggesons and commens have been very useful n mprovng he orgnal manuscrp. Ths work s enrely suppored by AUF pos docoral Gran 7-8, Réf:PC-42/246 and parally performed when he auhor vs Cad Ayyad Unversy of Marrakech Maroc. References 1 Aman, A. 21. Refleced generalzed backward doubly SDEs drven by Lévy processes and applcaons. J. Theore. Probab. DOI:1.17/s Aman, A., N z, M. and Owo, J.M. 21. A noe on homeomorphsm for backward doubly SDEs and applcaons. Soch. Dyn Bahlal, S. and Gherbal, B. 21. Opmaly condons of conrolled backward doubly sochasc dfferenal equaons. Random Oper. Soch. Equ MR Bally, V Approxmaon scheme for soluons of BSDE. In Backward Sochasc Dfferenal Equaons Pars, Pman Res. Noes Mah. Ser Harlow: Longman. MR Bouchard, B. and Touz, N. 24. Dscree-me approxmaon and Mone-Carlo smulaon of backward sochasc dfferenal equaons. Sochasc Process. Appl MR Brand, P., Delyon, B. and Mémn, J. 21. Donsker-ype heorem for BSDEs. Elecron. Commun. Probab elecronc. MR Buckdahn, R. and Ma, J. 21. Sochasc vscosy soluons for nonlnear sochasc paral dfferenal equaons. I. Sochasc Process. Appl MR Buckdahn, R. and Ma, J. 21. Sochasc vscosy soluons for nonlnear sochasc paral dfferenal equaons. II. Sochasc Process. Appl MR Chevance, D Numercal mehods for backward sochasc dfferenal equaons. In Numercal Mehods n Fnance L.C.G. Rogers and D. Talay, eds.. Publ. Newon Ins Cambrdge: Cambrdge Unv. Press. MR Douglas, J. Jr., Ma, J. and Proer, P Numercal mehods for forward backward sochasc dfferenal equaons. Ann. Appl. Probab MR Karazas, I. and Shreve, S.E Brownan Moon and Sochasc Calculus. Graduae Texs n Mahemacs 113. New York: Sprnger. 12 Kloeden, P.E. and Plaen, E Numercal Soluon of Sochasc Dfferenal Equaons. Applcaons of Mahemacs New York 23. Berln: Sprnger. MR Lemor, J.P., Gobe, E. and Warn, X. 26. Rae of convergence of an emprcal regresson mehod for solvng generalzed backward sochasc dfferenal equaons. Bernoull MR Ma, J., Proer, P., San Marín, J. and Torres, S. 22. Numercal mehod for backward sochasc dfferenal equaons. Ann. Appl. Probab MR Ma, J., Proer, P. and Yong, J.M Solvng forward backward sochasc dfferenal equaons explcly A four sep scheme. Probab. Theory Relaed Felds MR Ma, J. and Zhang, J. 22. Pah regulary for soluons of backward SDE s. Probab. Theory Relaed Felds

22 114 A. Aman 17 Mlsen, G.N. and Treyakov, M.V. 26. Numercal algorhms for forward backward sochasc dfferenal equaons. SIAM J. Sc. Compu elecronc. MR Pardoux, É. and Peng, S.G Backward doubly sochasc dfferenal equaons and sysems of quaslnear SPDEs. Probab. Theory Relaed Felds MR Zhang, J. 21. Some fne properes of backward sochasc dfferenal equaons. Ph.D. hess, Purdue Unv. MR Zhang, J. 24. A numercal scheme for BSDEs. Ann. Appl. Probab MR22327 Receved June 211