Discrete time approximation of decoupled Forward-Backward SDE with jumps

Transcription

1 Discree ime approximaion of decoupled Forward-Backward SD wih jumps Bruno BOUCHARD LPMA CNRS, UMR 7599 Universié Paris 6 bouchard@ccr.jussieu.fr and CRST Romuald LI CRMAD CNRS, UMR 7534 Universié Paris 9 elie@ensae.fr and CRST-NSA This version : March 27 Absrac We sudy a discree-ime approximaion for soluions of sysems of decoupled forward-backward sochasic differenial equaions wih jumps. Assuming ha he coefficiens are Lipschiz-coninuous, we prove he convergence of he scheme when he number of ime seps n goes o infiniy. The rae of convergence is a leas n 1/2+ε, for any ε >. When he jump coefficien of he firs variaion process of he forward componen saisfies a non-degeneracy condiion which ensures is inversibiliy, we achieve he opimal convergence rae n 1/2. The proof is based on a generalizaion of a remarkable resul on he pah-regulariy of he soluion of he backward equaion derived by Zhang 25 in he no-jump case. Key words : Discree-ime approximaion, forward-backward SD s wih jumps, Malliavin calculus. MSC Classificaion (2: 65C99, 6H7, 6J75. 1 Inroducion In his paper, we sudy a discree ime approximaion scheme for he soluion of a sysem of decoupled Forward-Backward Sochasic Differenial quaions (FBSD in shor wih jumps of he form { X = X + b(x rdr + σ(x rdw r + β(x r,e µ(de,dr, Y = g(x T + T h(θ r dr T Z r dw r T U (1.1 r(e µ(de,dr where Θ := (X,Y,Z,Γ wih Γ := ρ(eu(eλ(de. Here, W is a d-dimensional Brownian moion and µ an independen compensaed Poisson measure µ(de, dr = µ(de, dr λ(dedr. Such equaions naurally appear in hedging problems, see e.g. yraud-loisel 13, or in sochasic conrol, see e.g. Tang and Li 23 and he 1

2 recen paper Becherer 3 for an applicaion o exponenial uiliy maximizaion in finance. Under sandard Lipschiz assumpions on he coefficiens b, σ, β, g and h, exisence and uniqueness of he soluion have been proved by Tang and Li 23, hus generalizing he seminal paper of Pardoux and Peng 2. The main moivaion for sudying discree ime approximaions of sysems of he above form is ha hey provide an alernaive o classical numerical schemes for a large class of (deerminisic PD s of he form Lu(,x + h(,x,u(,x,σ(,x x u(,x, Iu(,x =, u(t,x = g(x, (1.2 where Lu(,x := Iu(,x := u (,x + xu(,xb(x + 1 d (σσ (x ij 2 u 2 x i x j (,x i,j=1 + {u(,x + β(x,e u(,x x u(,xβ(x,e}λ(de, {u(,x + β(x,e u(,x}ρ(eλ(de. Indeed, is well known ha, under mild assumpions on he coefficiens, he componen Y of he soluion can be relaed o he (viscosiy soluion u of (1.2 in he sense ha Y = u(,x, see e.g. 1 or 9. Thus solving (1.1 or (1.2 is essenially he same. In he so-called four-seps scheme, his relaion allows o approximae he soluion of (1.1 by firs esimaing numerically u, see e.g. 1. Here, we follow he converse approach. Since classical numerical schemes for PD s generally do no perform well in high dimension, we wan o esimae direcly he soluion of (1.1 so as o provide an approximaion of u. In he no-jump case, i.e. β =, he numerical approximaion of (1.1 has already been sudied in he lieraure, see e.g. Zhang 25, Bally and Pages 2, Bouchard and Touzi 7 or Gobe e al. 16. In 7, he auhors sugges he following implici scheme. Given a regular grid π = { = it/n, i =,...,n}, hey approximae X by is uler scheme X π and (Y,Z by he discree-ime process (Ȳ π i, Z π i i n defined backward by Z π i = n T Ȳ π i+1 W i+1 F i Ȳ π = Ȳ π i+1 F i + T n h( X π,ȳ π, Z π where Ȳ π n := g(x π n and W i+1 := W i+1 W i. In he no-jump case, i urns ou ha he discreizaion error rr n (Y,Z := { max i<n sup,+1 is inimaely relaed o he quaniy n 1 i+1 i= Y Ȳ π 2 + n 1 i+1 i= Z Z i 2 d where Zi := n 2 Z Z π 2 d i+1 Z d F i }1 2.

3 Under Lipschiz coninuiy condiions on he coefficiens, Zhang 24 was able o prove ha he laer is of order of n 1. This remarkable resul allows o derive he bound rr n (Y,Z Cn 1/2. Observe ha his rae of convergence can no be improved in general. Consider for example he case where X is equal o he Brownian moion W, g is he ideniy and h =. Then, Y = W and Ȳ π i = W i. In his paper, we exend he approach of Bouchard and Touzi 7 and approximae he soluion of (1.1 by he backward scheme Z π i = n T Ȳ π i+1 W i+1 F i Ȳ π = Ȳ π i+1 F i + T n h( X π,ȳ π, Z π, Γ π, Γ π = n T Ȳ π i+1 ρ(e µ(de,(,+1 F i where Ȳ π n := g(x π n. By adaping he argumens of Gobe e al. 16, we firs prove ha our discreizaion error rr n (Y,Z,U defined as { max i<n sup,+1 Y Ȳ π 2 + n 1 i+1 i= Z Z π 2 + Γ Γ π 2 d converges o as he discreizaion sep T/n ends o. We hen provide upper bounds on max i<n sup,+1 Y Y i 2 n 1 + i= i+1 Z Z i 2 + Γ Γ i 2 d, where Γ i+1 i := (n/t Γ d F i. When he coefficiens are Lipschiz coninuous, we obain and max i<n sup,+1 n 1 i+1 i= Y Y i 2 n 1 + i= i+1 } 1 2 Γ Γ i 2 d C n 1 Z Z i 2 d C ε n 1+ε, for any ε >. Under some addiional condiions on he inversibiliy of β + I d, see H, we hen prove ha he previous inequaliy holds rue for ε =. This exends o our framework he remarkable resul derived by Zhang 25 in he no-jump case. I allows us o show ha our discree-ime scheme achieves, under he sandard Lipschiz condiions, a rae of convergence of a leas n 1/2+ε, for any ε >, and he opimal rae n 1/2 under he addiional assumpion H. Observe ha, in opposiion o algorihms based on he approximaion of he Brownian moion by discree processes aking a finie number of possible values (see e.g. 17 and he references herein, our scheme does no provide a fully implemenable numerical procedure since involves he compuaion of a large number of condiional expecaions. However, he implemenaion of he above menioned schemes 3

4 in high dimension is quesionable and, in our seing, his issue could be solved by approximaing he condiional expecaion operaors numerically in an efficien way, see 2, 7, 16 and 11 for an adapaion o our seing of he echnics suggesed in 16. The res of he paper is organized as follows. In Secion 2, we describe he approximaion scheme and sae our main convergence resul. Secion 3 conains some resuls on he Malliavin derivaives of Forward and Backward SD s. Applying hese resuls in Secion 4, we derive some regulariy properies for he soluion of he backward equaion under addiional smoohness assumpions on he coefficiens. We finally use an approximaion argumen o conclude he proof of our main heorem. Noaions : Any elemen x R d will be idenified o a column vecor wih i-h componen x i and uclidian norm x. For x i R d i, i n and d i N, we define (x 1,...,x n as he column vecor associaed o (x 1 1,...,xd 1 1,...,x1 n,...,x dn n. The scalar produc on R d is denoed by x y. For a (m d-dimensional marix M, we noe M := sup{ Mx ; x R d, x = 1}, M is ranspose and we wrie M M d if m = d. Given p N and a measured space (A, A,µ A, we denoe by L p (A, A,µ A ; R d, or simply L p (A, A or L p (A if no confusion is possible, he se of p- inegrable R d -valued measurable maps on (A, A,µ A. For p =, L (A, A,µ A ; R d is he se of essenially bounded R d -valued measurable maps. The se of k-imes differeniable maps wih bounded derivaives up o order k is denoed by Cb k and Cb := k 1 Cb k. For a map b : Rd R k, we denoe by b is Jacobian marix whenever i exiss. In he following, we shall use hese noaions wihou specifying he dimension when is clearly given by he conex. 2 Discree ime approximaion of decoupled FBSD wih jumps 2.1 Decoupled forward backward SD s As in 5, we shall work on a suiable produc space Ω := Ω W Ω µ where Ω W is he se of coninuous funcions w from,t ino R d, and Ω µ is he se of ineger-valued measures on,t wih := R m for some m 1. For ω = (w,η Ω, we se W(w,η = w and µ(w,η = η and define F W = (F W T (resp. F µ = (F µ T as he smalles righ-coninuous filraion on Ω W (resp. Ω µ such ha W (resp. µ is opional. We le P W be he Wiener measure on (Ω W, FT W and P µ be he measure on (Ω µ, F µ T under which µ is a Poisson measure wih inensiy ν(d,de = λ(ded, for some finie measure λ on, endowed wih is Borel ribe. We hen define he probabiliy measure P := P W P µ on (Ω, FT W Fµ T. Wih his consrucion, W and µ are independen under P. Wihou loss of generaliy, we can assume ha he naural filraion F = (F T induced by (W,µ is complee. We denoe by 4

5 µ := µ ν he compensaed measure associaed o µ. Given K >, wo K-Lipschiz coninuous funcions b : R d R d and σ : R d M d, and a measurable map β : R d R d such ha sup e β(, e K and sup β(x,e β(x,e K x x x, x R d, (2.1 e we define X as he soluion on,t of X = X + b(x r dr + σ(x r dw r + β(x r,e µ(de,dr, (2.2 for some iniial condiion X R d. The exisence and uniqueness of such a soluion is well known under he above assumpions, see he Appendix for sandard esimaes for soluions of such SD. Before inroducing he backward SD, we need o define some addiional noaions. Given s and some real number p 2, we denoe by S p he se of real valued s, adaped càdlàg processes Y such ha Y S p s, 1 := sup Y r p p s r <, H p s, is he se of progressively measurable Rd -valued processes Z such ha Z H p s, := ( s p 1 p Z r 2 2 dr <, L p λ,s, is he se of P measurable maps U : Ω,T R such ha U L p λ,s, := s 1 U s (e p p λ(deds < wih P defined as he σ-algebra of F-predicable subses of Ω,T. The space B p s, := S p s, Hp s, Lp λ,s, is endowed wih he norm (Y,Z,U B p s, := ( 1 Y p S p + Z p H p + U p p L p. s, s, λ,s, In he sequel, we shall omi he subscrip s, in hese noaions when (s, = (,T. For ease of noaions, we shall someimes wrie ha an R n -valued process is in S p s, or L p λ,s, meaning ha each componens in he corresponding space. Similarly an elemen of M m is said o belong o H p s, if each column belongs o Hp s,. The norms are hen naurally exended o such processes. 5

6 The aim of his paper is o sudy a discree ime approximaion of he riple (Y,Z,U soluion on,t of he backward sochasic differenial equaion T T T Y = g(x T + h(θ r dr Z r dw r U r (e µ(de,dr, (2.3 where Θ := (X, Y, Z, Γ and Γ is defined by Γ := ρ(eu(eλ(de, for some measurable map ρ : R m saisfying sup ρ(e K. (2.4 e By a soluion, we mean a riple (Y,Z,U B 2 saisfying (2.3. In order o ensure he exisence and uniqueness of a soluion o (2.3, we assume ha he map g : R d R and h : R d R R d R m R are K-Lipschiz coninuous (see Lemma 5.2 in he Appendix. For ease of noaions, we shall denoe by C p a generic consan depending only on p and he consans K, λ(, b(, σ(, h(, g( and T. We wrie C p if i also depends on X. In his paper, p will always denoe a real number greaer han 2. Remark 2.1 For he convenience of he reader, we have colleced in he Appendix sandard esimaes for he soluions of Forward and Backward SD s. In paricular, hey imply (X,Y,Z,U p S p B p C p (1 + X p, p 2. (2.5 The esimae on X is sandard, see (5.2 of Lemma 5.1 in he Appendix. Plugging his in (5.6 of Lemma 5.2 leads o he bound on (Y,Z,U B p. Using (5.3 of Lemma 5.1, we also deduce ha sup X u X s p C p (1 + X p s, (2.6 s u while he previous esimaes on X combined wih (5.7 of Lemma 5.2 implies { } sup Y u Y s p C p (1 + X p s p + Z p H p + U p L p. (2.7 s u s, λ,s, 2.2 Discree ime approximaion We firs fix a regular grid π := { := it/n, i =,...,n} on,t and approximae X by is uler scheme X π defined by { X π := X X π i+1 := X π i + T n b(xπ + σ(x π i W i+1 + (2.8 β(xπ,e µ(de,(,+1 6

7 where W i+1 := W i+1 W i. Is well known ha max sup X X π i<n i 2 C2 n 1. (2.9,+1 We hen approximae (Y,Z,Γ by (Ȳ π, Z π, Γ π defined by he backward implici scheme Z π := n Ȳ T π i+1 W i+1 F i, Γ π := n T Ȳ π i+1 ρ(e µ(de,(,+1 F i Ȳ π π := Ȳi+1 F i + T n h( X π i,ȳ π i, Z π i, Γ π (2.1 on each inerval,+1, where Ȳ π n := g(x π n. Observe ha he resoluion of he las equaion in (2.1 may involve he use of a fixed poin procedure. However, h being Lipschiz and muliplied by 1/n, he approximaion error can be negleced for large values of n. Remark 2.2 The above backward scheme is a naural exension of he one considered in 7 in he case β =. By he represenaion heorem, see e.g. Lemma 2.3 in 23, here exis wo processes Z π H 2 and U π L 2 λ saisfying Ȳ π π i+1 Ȳi+1 F i = i+1 Z π s dw s + i+1 U π s (e µ(ds,de. Observe ha Z π and Γ π defined in (2.1 saisfy Z π i = n i+1 T Zs π ds F and Γπ i = n i+1 T Γ π s ds F (2.11 and herefore coincide wih he bes H 2,+1 -approximaions of (Zπ <+1 and (Γ π i <+1 := ( ρ(euπ (eλ(de i <+1 by F i -measurable random variables (viewed as consan processes on,+1. Finally, observe ha we can define Y π on,+1 by seing Y π := Ȳ π i ( h(x π i,ȳ π i, Z π i, Γ π + Zs π dw s + Us π (e µ(ds,de. 2.3 Convergence of he approximaion scheme In his subsecion, we show ha he approximaion error { rr n (Y,Z,U := sup }1 Y Ȳ π 2 + Z Z π 2 H + Γ Γ 2 π 2 2 H 2 T converges o. Le us firsnroduce he processes ( Z, Γ defined on each inerval,+1 by Z := n i+1 T Z s ds F i and Γ := n i+1 T Γ s ds F i. 7

8 Remark 2.3 Observe ha Z i and Γ i are he counerpars of Zπ i and Γ π for he original backward SD. They can also be inerpreed as he bes H 2,+1 - approximaions of (Z i <+1 and (Γ i <+1 by F i -measurable random variables (viewed as consan processes on,+1. Proposiion 2.1 We have n 1 i= i+1 Y Y i 2 d C 2 n 1 and Z Z H 2 + Γ Γ H 2 ǫ(n(2.12 where ǫ(n as n. Moreover, rr n (Y,Z,U C 2 so ha rr n (Y,Z,U n. Proof. Since Y solves (2.3, Y Y i 2 C 2 ( n 1/2 + Z Z H 2 + Γ Γ H 2, (2.13 h(x r,y r,z r,γ r 2 + Z r 2 + U r (e 2 λ(de dr. Combining he Lipschiz propery of h wih (2.5, i follows ha n 1 i+1 i= Y Y i 2 d C 2 n. This is exacly he firs par of (2.12. The proof of (2.13 hen follows exacly he same argumens as in 7, see p 99 in 11 for deails. I remains o prove he second par of (2.12. Since Z is F-adaped, here is a sequence of adaped processes (Z n n such ha Z n = Z n i on each,+1 and Z n converges o Z in H 2. By Remark 2.3, we observe ha Z Z 2 H Z Z n 2 2 H, and applying he same reasoning o Γ 2 concludes he proof. 2.4 Pah-regulariy and convergence rae In view of Proposiion 2.1, he discreizaion error converges o zero. In order o conrol is speed of convergence, i remains o sudy Z Z 2 H 2 + Γ Γ 2 H 2. Before saing our main resul, le us inroduce he following assumpion: H : For each e, he map x R d β(x,e admis a Jacobian marix β(x,e such ha he funcion (x,ξ R d R d a(x,ξ;e := ξ ( β(x,e + I d ξ saisfies one of he following condiion uniformly in (x,ξ R d R d a(x,ξ;e ξ 2 K 1 or a(x,ξ;e ξ 2 K 1. 8

9 Remark 2.4 Observe for laer use ha he condiion H implies ha, for each (x,e R d, he marix β(x,e + I d is inverible wih inverse bounded by K. This ensures he inversibiliy of he firs variaion process X of X, see Remark 3.6. Moreover, if q is a smooh densiy on R d wih compac suppor, hen he approximaing funcions β k, k N, defined by β k (x,e := R d k d β( x,eq(kx xd x are smooh and also saisfy H. In Secion 5 of 9, he auhors imposes a similar condiion: de( β(x,e + I d 1 δ, x R d λ(de a.e. for some δ >. Under mild addiional assumpions, his allows o prove he exisence of a bounded soluion, in a suiable weighed Sobolev space, o a PD of he form (1.2 which can hen be relaed o Y. Our main heorem is saed for a suiable version of (Z,U,Γ. Observe ha does no change he quaniy rr n (Y,Z,U. Theorem 2.1 The following holds. (i For all i < n, sup Y Y i 2 C2 n 1 and sup Γ Γ i 2 C2 n 1 (2.14,+1,+1 so ha Γ Γ 2 S 2 C 2 n 1 and Γ Γ 2 H 2 C 2 n 1. Moreover, for any ε >, (ii Assume ha H holds. Then Z Z 2 H 2 C ε n 1+ε. (2.15 Z Z 2 H 2 C 2 n 1. (2.16 This regulariy propery will be proved in he subsequen secions. Combined wih Proposiion 2.1, i provides an upper bound for he convergence rae of our backward implici scheme. Corollary 2.1 For any ε >, rr n (Y,Z,U C ε n 1/2+ε. If H holds, hen rr n (Y,Z,U C 2 n 1/2. Remark 2.5 One could also use an explici scheme as in e.g. 2 or 16. In his case, (2.1 has o be replaced by ρ(e µ(de,(,+1 F i Z π := n T Ỹ π i+1 W i+1 F i Ỹ π, Γ π := n T Ỹ π +1 π := Ỹi+1 F i + T ( h n X π i,ỹ π i+1, Z π i, Γ π F i (2.17 wih he erminal condiion Ỹ π n = g(x π n. The advanage of his scheme is ha does no require a fixed poin procedure. However, from a numerical poin of view, adding a erm in he condiional expecaion defining Ỹ π i makes i more difficul o esimae. We herefore hink ha he implici scheme may be more racable in pracice. The above convergence resuls can be easily exended o his scheme, see 11 for deails. 9

10 Remark 2.6 In he unpublished paper 9, he auhors discuss he regulariy of (X,Y,Z,U wih respec o he iniial condiion X in a case where he coefficiens b,σ,h and g are C 3, wih linear growh and bounded derivaives for he wo firs ones and derivaives having polynomial growh for he wo las ones. Under hese regulariy assumpions, hey show ha he map (,x u(,x = Y,x belongs o C,2 (,T R d wih 1 2-Hölder coninuiy in ime and derivaives having polynomial growh in space, see heir Proposiion 3.5 and heir Corollary 3.6. Similar resuls are obained for (,x (Z,x,U,x which can be idenified o ( u(,xσ(x,u(,x + β(x, u(,x. This readily implies he properies saed in Theorem 2.1 which can be seen as weak versions of he regulariy resuls of 9. The imporan poin here is ha: 1. Our resuls do no require all he regulariy assumpions of 9; 2. This is all we need o provide he convergence raes of Corollary 2.1. Remark 2.7 I will be clear from he proofs ha all he resuls of his paper hold if we le he maps b, σ, β, and h depend on whenever hese funcions are 1/2-Hölder in and he oher assumpions are saisfied uniformly in. The uler approximaion X π of X could also be replaced by any oher adaped approximaion saisfying (2.9. Remark 2.8 We refer o 11 for exensions o he approximaion of sysems of semilinear PDs hrough heir relaion wih BSDs wih jumps, see 21, and for similar convergence resuls wihou H bu under addiionnal regulariy assumpions. 3 Malliavin calculus for FBSD In his secion, we prove ha he soluion (Y,Z,U of (2.3 is smooh in he Malliavin sense under he addiional assumpions C X : b, σ and β(,e are C 1 b uniformly in e, CY : g and h are C 1 b. This will allow us o provide represenaion and regulariy resuls for Y, Z and U in Secion 4. Under C X -C Y, hese resuls will immediaely imply he firs asserion of (i of Theorem 2.1, while he second one (resp. (ii will be obained by adaping he argumens of 6 (resp. 25 under he addiional assumpion H. 3.1 Generaliies The consrucion of he Malliavin derivaives on he Wiener space is sandard, see e.g. 18, and can be easily exended o our seing by observing ha here is an isomery beween L 2 (Ω W Ω µ and L 2 (Ω W,L 2 (Ω µ, wih obvious noaions. Le S denoe he se of random variables of he form ( T T F = φ f 1 ( dw,..., f κ ( dw,µ, where κ 1, f i :,T R d is a bounded measurable map for each i κ, φ is a real-valued measurable map on R κ Ω µ and φ(,η C b, P µ(dη-a.e. 1

11 We denoe by D he Malliavin derivaive operaor wih respec o he Brownian moion. For F S as above and s T, is defined as D s F := ( T T i φ f 1 ( dw,..., f κ ( dw,µ f i (s, i κ where i φ is he derivaive of φ wih respec o is i-h argumen. We hen denoe by ID 1,2 he closure of S wih respec o he norm F ID 1,2 := { F 2 T } 1 + D s F 2 2 ds and define H 2 (ID 1,2 as he se of elemens ξ H 2 such ha ξ ID 1,2 for almos all T and such ha, afer possibly passing o a measurable version, T ξ 2 := ξ 2 H 2 (ID 1,2 H + D 2 s ξ 2 H ds <. 2, Observe ha for ψ in L 2 λ (Fµ, he se of elemens of L 2 λ which are independen of W, we have Dψ =. We finally define L 2 λ (ID1,2 as he closure of he se } L 2 λ (ID 1,2 := Vec {ψ = ξϑ : ξ H 2 P (ID1,2, F W, ϑ L 2 λ (Fµ, ψ L 2λ (ID 1,2 < for he norm T ψ 2 := ψ 2 L 2 λ (ID1,2 L + D 2 s ψ 2 λ L. 2 λds Here, H 2 P (ID1,2, F W denoes he se of F W -predicable elemens of H 2 (ID 1,2 and D s (ξϑ = (D s ξϑ for ξ H 2 P (ID1,2, F W, ϑ L 2 λ (Fµ. Here again, we exend he definiion of H 2 (ID 1,2 and L 2 λ (ID1,2 o processes wih values in Md and R d in a naural way. From now on, given a marix A, we shall denoe by A i is i-h column. For k d, we denoe by D k he Malliavin derivaive wih respec o W k, meaning ha D k F = (DF k for F ID 1,2. Remark 3.1 Wih his consrucion, he operaor D enjoys he usual properies of he Malliavin derivaive operaor on Wiener spaces. In paricular, if ξ H 2 (ID 1,2 and f C 1 b (Rd, hen D s ( T T T f(ξ d + ξ dw = s f(ξ D s ξ d + ξ s + d j=1 T s D s ξ j dw j for all s T. Here denoes ransposiion. I follows from he same argumen as in 18, which we refer o for more deails. 11

12 Remark 3.2 Fix ξ H 2 P (ID1,2, F W. By Lemma in 18, here exiss a family of deerminisic measurable kernels f m ( 1,..., m, in L 2 (,T m+1, m, such ha ξ = m I m(f m (, and D s ξ = m 1 mi m 1(f m (,s, where I m denoes he m-ieraed Wiener inegral, see Proposiion in 18. Therefore, if τ is a random ime bounded by T and independen of W, we have ξ τ = m I m(f m (,τ and, by he same argumen as in he proof of Proposiion in 18, ξ τ ID 1,2 whenever τ has a bounded densiy and D s (ξ τ = m 1 mi m 1(f m (,s,τ = (D s ξ τ. Remark 3.3 For ξ H 2 (ID 1,2 he inegrals T D sξ d and T D sξ dw have T o be undersood as inegrals wrien on he process D s ξ, i.e. (D sξ d and T (D sξ dw. Similarly, for ψ L 2 λ (ID1,2, T D sψ (e µ(de,d has o be undersood as T (D sψ (e µ(de,d. However, i follows from Remark 3.2 ha coincides wih T D s(ψ (e µ(de,d, so ha here is no ambiguiy. The wo following Lemmas are generalizaions of Lemma 3.3 and Lemma 3.4 in 21 which correspond o he case where is finie, see also Lemma 2.3 in 2 for he case of Iô inegrals. Lemma 3.1 Assume ha ψ L 2 λ (ID1,2. Then, H := T ψ (e µ(de,d ID 1,2 and D s H = T D sψ (e µ(de,d for all s T. Proof. Firs noice ha suffices o prove he required resul when ψ L 2 λ (ID 1,2. Indeed, we can hen rerieve he general case by considering a sequence (ψ n n in L 2 λ (ID 1,2 which converges o ψ in L 2 λ (ID1,2, so ha H n := T ψn (e µ(de,d is a Cauchy sequence in ID 1,2 which converges o H and (D s H n s T converges o ( T D sψ (e µ(de,d s T in H 2. We herefore assume ha ψ = ξϑ where ξ H 2 P (ID1,2, F W, ϑ L 2 λ (Fµ and ψ L 2 λ (ID 1,2 <. Then, T ψ (e µ(de,d = T T ξ ϑ (eµ(de,d ξ ϑ (eλ(ded, where, by Remark 3.1 and he fac ha ϑ (eλ(de is independen of W, T ( D s ξ I remains o prove ha T D s ϑ (eλ(de d = ξ ϑ (eµ(de,d = = T T T D s ξ ϑ (eλ(ded (D s ξ ϑ (eλ(ded. (D s ξ ϑ (eµ(de,d. 12

13 To see his, we define N by N := µ(,ds for T, (τ i i 1 as he sequence of jump imes of N and ( i i 1 by i := N τi N τi. Wih hese noaions, we have o show ha D s ξ τi ϑ τi ( i 1 τi T = s ξ τi ϑ τi ( i 1 τi T, (3.1 i 1(D i 1 see Remark 3.3. Using Remark 3.2, we now oberve ha D n s i=1 ξ τ i ϑ τi ( i 1 τi T = n i=1 (D sξ τi ϑ τi ( i 1 τi T, for each n 1. Passing o he limi leads o (3.1 and concludes he proof. Indeed, he previous ideniy implies ha he sequence (F n n 1 defined by F n := n i=1 ξ τ i ϑ τi ( i 1 τi T belongs o H 2 (ID 1,2 and saisfies F n i 1 ξ τ i ϑ τi ( i 1 τi T as well as D s F n i 1 (D sξ τi ϑ τi ( i 1 τi T which implies ha sup n 1 F n 2 H 2 (ID 1,2 is bounded by ( T 2 T 2 2 ξ ϑ (e µ(de,d + ξ ϑ (e λ(ded T +2 T +2 T T C 2 ψ 2 L 2 λ (ID1,2 <, 2 D s ξ ϑ (e µ(de,d ds 2 D s ξ ϑ (e λ(ded ds where he second and he fourh erms on he righ hand-side are bounded by using Jensen s inequaliy and he assumpion λ( <. Moreover, by dominaed convergence, F n F 2 + T D sf n D s F 2 ds where F := i 1 ξ τ i ϑ τi ( i 1 τi T and D s F := i 1 (D sξ τi ϑ τi ( i 1 τi T saisfy, by he same argumens as above, F 2 H 2 (ID 1,2 C 2 ψ 2 L 2 λ (ID1,2 <. Remark 3.4 Similar argumens as in he above proof shows ha for ψ L 2 λ (ID1,2 and f L (, we have, for almos every s T, ψ s(ef(eλ(de ID 1,2 and D ( ψ s (ef(eλ(de := D ψ s (ef(eλ(de. Lemma 3.2 Le S(W denoe he se of random variables of he form H W ( T T = φ f 1 ( dw,..., f κ ( dw where κ 1, φ Cb and f i :,T R d is a bounded measurable map for each i κ. Then, Vec{S(W L (Ω µ, F µ T } is dense in ID1,2 for he norm ID 1,2. 13

14 Proof. I suffices o prove ha Vec{S(W L (Ω µ, F µ T } is dense in S. Fix H S of he form ( T T H = φ f 1 ( dw,..., f κ ( dw,µ. Observe ha Ω µ can be idenified o he space of finie (possibly empy sequences (,e i 1 i n of,t, n, such ha ( i 1 is increasing. Le G n denoe he se of such sequences of lengh n and G := n G n. Given η Ω u, we denoe by ( η i,eη i i 1 he associaed sequence, and we idenify φ wih a measurable map on R κ G. We denoe by φ n is resricion o R κ G n, n. Le ψ n denoe he gradien ( of φ n wih respec o is firs κ componens and se f := (f 1,...,f κ, T G := f1 ( dw,..., T fκ ( dw. Since (H,D s H = n (φ n (G,( µ i,eµ i 1 i n,ψ n (G,( µ i,eµ i 1 i n f(s1 µ(,,t=n, i suffices o prove ha each H n := φ n (G,( µ i,eµ i 1 i n can be approximaed by linear combinaions of elemens of S(W L (Ω µ, F µ T. Moreover, we can always assume ha φ n is Cb on R κ G n. Indeed, φ is already Cb in is firs κ componens, a.e., and we can replace φ n by is convoluion wih a sequence of smooh kernels acing only is las n componens. Since boh funcions are coninuous, we can hen approximae (φ n,ψ n poinwise by linear combinaions of funcions of he form (φ n,ψ n (,(,e i 1 i n 1 A where A is a Borel se of G n and (,e i 1 i n G n. The required resul hen follows from he fac ha D s φ n (G,(,e i 1 i n 1 A (( µ i,eµ i 1 i n = (ψ n (G,(,e i 1 i n f(s1 A (( µ i,eµ i 1 i n. Lemma 3.3 Fix (ξ,ψ H 2 L 2 λ and assume ha T T H := ξ dw + ψ (e µ(de,d ID 1,2. Then, (ξ,ψ H 2 (ID 1,2 L 2 λ (ID1,2 and T D s H := ξs + d i=1 where ξ denoes he ranspose of ξ. T D s ξ dw + D s ψ (e µ(de,d, Proof. One easily deduce from Lemma 3.2 ha H := Vec{H W H µ : H W S(W, H µ L (Ω µ, F µ T, H W H µ = } is dense in ID 1,2 {H L 2 (Ω, F, P : H = } for ID 1,2. Thus, i suffices o prove he resul for H of he form H W H µ where H W S(W, H µ L (Ω µ, F µ T and H W H µ =. By he represenaion heorem, see e.g. 8, here exiss ψ L 2 λ such ha H µ = H µ + T ψ (e µ(de,d and by Ocone s formula, see e.g. Proposiion in 18, H W = H W + T D H W F W dw. Thus i follows from Iô s Lemma ha H = T H µ D H W F W dw + T HW ψ (e µ(de,d where H µ = H µ F and H W = H W F. Furhermore, easy compuaions show ha he wo inegrands belong respecively o H 2 (ID 1,2 and L 2 λ (ID1,2. Thus, Remark 3.1 and Lemma 3.1 conclude he proof. 14

15 3.2 Malliavin calculus on he Forward SD In his secion, we recall well-known properies concerning he differeniabiliy in he Malliavin sense of he soluion of a Forward SD. In he case where β = he following resuls saed in e.g. 18. The exension o he case β is easily obained by condiioning by µ, see e.g. 14 for explanaions in he case where is finie, or by combining Remark 3.1, Lemma 3.1 wih a fixed poin procedure as in he proof of Theorem in 18, see also Proposiion 3.2 below. Proposiion 3.1 Assume ha C X holds, hen X ID 1,2 for all T. For all s T and k d, D k sx admis a version χ s,k which solves on s,t χ s,k = σ k (X s + + s s b(x r χ s,k r dr + s β(x r,eχ s,k r µ(dr,de. d j=1 σ j (X r χ s,k r dw j r Remark 3.5 Fix p 2 and r s u T. Under C X, i follows from Lemma 5.1 applied o X and χ s ha χ s p S p C p (1 + X p, χ s u χ s p C p u (1 + X p and χ s χ r p S p C p s r (1 + X p. Remark 3.6 Under C X, we can define he firs variaion process X of X which solves on,t X = I d + + b(x r X r dr + d σ j (X r X r dwr j j=1 β(x r,e X r µ(dr,de. (3.2 Moreover, under H, see Remark 2.4, ( X 1 is well defined and solves on,t d ( X 1 = I d ( X 1 r b(x r σ j (X r σ j (X r dr + j=1 ( X 1 r β(x r,eλ(dedr d ( X 1 r σj (X r dwr j j=1 ( X 1 r ( β(x r,e + I d 1 β(x r,eµ(de,dr. (3.3 Remark 3.7 Fix p 2. Under H-C X, i follows from Remark 2.4 and Lemma 5.1 applied o X and ( X 1 ha X S p + ( X 1 S p C p. Remark 3.8 Assume ha H-C X holds and observe ha χ s = (χ s,k k d and X solve he same equaion up o he condiion a ime s. By uniqueness of he soluion on,t, i follows ha χ s r = X r ( X s 1 σ(x s 1 s r for all s,r T. 15

16 3.3 Malliavin calculus on he Backward SD In his secion, we generalize he resul of Proposiion 3.1 in 21. Le us denoe by B 2 (ID 1,2 he se of riples (Y,Z,U B 2 such ha Y ID 1,2 for all T and (Z,U H 2 (ID 1,2 L 2 λ (ID1,2. Proposiion 3.2 Assume ha C X -C Y holds. Then, he riples (Y,Z,U belongs o B 2 (ID 1,2. For each s T and k d, he equaion Υ s,k T T = g(x T χ s,k T + h(θ r Φ s,k r dr T ζr s,k dw r Vr s,k (e µ(de, dr (3.4 wih Φ s,k := (χ s,k,υ s,k,ζ s,k,γ s,k and Γ s,k := ρ(ev s,k (eλ(de, admis a unique soluion. Moreover, (Υ s,k,ζ s,k,v s,k s, T is a version of (DsY k,dsz k,dsu k s, T. Proof. Wih he help of Lemma 3.3 and he esimaes of Lemma 5.2, we can reproduce exacly he proof of Proposiion 5.3 in 12 up o minor modificaions. See 11 p 113 for deails. Proposiion 3.3 Assume ha C X -C Y holds. For each k d, he equaion T T T Y k = g(x T XT+ k h(θ r Φ k rdr Zr k dw r Ur k (e µ(de,dr (3.5 wih Φ k = ( X k, Y k, Z k, Γ k and Γ k := ρ(e Uk (eλ(de, admis a unique soluion ( Y k, Z k, U k. Moreover, here is a version of (ζ s,k,υ s,k,v s,k given by {( Y, Z, U ( X s 1 σ k (X s 1 s } s, T where Y s he marix whose k-column is given by Y k and Z, U are defined similarly. Proof. In view of Proposiion 3.2 and Remark 3.8, his follows immediaely from he uniqueness of he soluion of (3.4. Remark 3.9 Combined wih C X -C Y and Remark 3.5, Lemma 5.2 below implies ha sup (Υ s,ζ s,v s p B C p p (1 + X p for all p 2. (3.6 s T I follows from Lemma 5.2 and Remark 3.7 ha ( Y, Z, U B p C p for all p 2. (3.7 s, T 4 Represenaion resuls and pah regulariy for he BSD In his secion, we use he above resuls o obain some regulariy for he soluion of he BSD (2.3 under C X -C Y, C X -C Y -H. Similar resuls wihou C X -C Y will hen be obained by using an approximaion argumen. 16

17 Fix (u,s,,x,t 3 R d and k,l d. In he sequel, we shall denoe by X(,x he soluion of (2.2 on,t wih iniial condiion X(,x = x, and by (Y (, x, Z(, x, U(, x he soluion of (2.3 wih X(, x in place of X. We define similarly (Υ s,k (,x, ζ s,k (,x, V s,k (,x and ( Y (,x, Z(,x, U(,x. Observe ha, wih hese noaions, we have (X(,X,Y (,X, Z(,X, U(,X = (X,Y,Z, U. 4.1 Represenaion We sar his secion by proving useful bounds for he (deerminisic maps defined on,t R d by u(,x := Y (,x, u(,x := Y (,x, v s,k (,x := Υ s,k (,x, where s,t and k d. Proposiion 4.1 Assume ha C X and C Y hold, hen, u(,x + v s,k (,x C 2 (1 + x and u(,x C 2 (4.1 for all s, T, k d and x R d. Proof. When (,x = (,X, he resul follows from (2.5 in Remark 2.1, (3.6 and (3.7. The general case is obained similarly by changing he iniial condiion on X. Proposiion 4.2 Assume ha C X and C Y hold. Then, here is a version of Z given by (Υ T which saisfies Z p S p C p (1 + X p. Proof. Here again we only consider he case d = 1 and omi he indexes k,l. By Proposiion 3.2, (Y,Z,U belongs o B 2 (ID 1,2 and i follows from Lemma 3.3 ha D s Y = Z s s h(θ r D s Θ r dr + s D s Z r dw r + s D s U r (e µ(de,dr, (4.2 for < s T. Taking s = leads o he represenaion of Z. Thus, afer possibly passing o a suiable version, we have Z = D Y = Υ. By uniqueness of he soluion of (2.2-(2.3-(3.4 for any iniial condiion in L 2 (Ω, F a, we have Υ = v (,X. The bound on Z hen follows from Proposiion 4.1 combined wih (2.5 of Remark 2.1. Proposiion 4.3 (i Define Ũ by Ũ(e := u(,x + β(x,e lim r u(r,x r = Y Y. Then Ũ is a version of U and i saisfies sup Ũ(e p S C p p (1 + X p. (4.3 e (ii Assume ha C X and C Y hold. Define Ũ by Ũ(e := u(,x + β(x,e lim r u(r,x r. Then Ũ is a version of U and i saisfies sup Ũ(e p S C p p. (4.4 e 17

18 Remark 4.1 We will see in Proposiion 4.4 below ha u is coninuous under C X and C Y so ha Ũ(e = u(,x + β(x,e u(,x. A similar represenaion is derived in 21, in a case where is finie, and in 9, in he case where is no finie, bu under more regulariy assumpions on he coefficiens. Proof of Proposiion 4.3. We only provide he proof of (i, he oher asserion is proved similarly. 1. By uniqueness of he soluion of (2.2-(2.3 for any iniial condiion in L 2 (Ω, F a ime, one has Y = u(,x a.s. for each T. We shall prove in sep 2. below ha u is joinly coninuous in x and righ-coninuous in. This implies ha (u(,x T is righ-coninuous so ha Y = u(,x and Y = lim r u(r,x r for each T a.s., see Theorem I.2 in 22 and recall ha X and Y are càdlàg. Thus U (eµ(de, {} = Y Y = u(,x lim r u(r,x r = Ũ(eµ(de, {}, for each T a.s. and T Ũ(e U (e 2 µ(de,d =, which implies, by aking expecaion, Ũ(e U (e =. L 2 λ 2. We now prove ha u is coninuous in x and righ-coninuous on. Fix 1 2 T and (x 1,x 2 R 2d. For A denoing X,Y,Z or U, we se A i := A(,x i for i = 1,2 and δa := A 1 A 2. By (5.4 of Lemma 5.1, we derive δx 2 S 2 2,T C 2 { x1 x (1 + x }. (4.5 Plugging his esimae in (5.8 of Lemma 5.2 leads o (δy,δz,δu 2 B 2 2,T C 2 { x1 x (1 + x }. (4.6 Now, observe ha Y u( 1,x 1 u( 2,x 2 2 = Y 1 1 Y C 2 2 Y Y 1 2 Y (4.7. Since Y 1 is righ-coninuous and bounded in S 2, he firs erm on he righ-hand side goes o as 2 1, while he second is conrolled by ( Pah regulariy Proposiion 4.4 Assume ha C X and C Y hold. Then, for 1 2 T and (x 1,x 2 R 2d, u( 1,x 1 u( 2,x 2 2 C 2 { (1 + x x 1 x 2 2}. Proof. I suffices o plug he esimae of Proposiion 4.2 and (4.3 in (2.7, which is possible since he norms in (2.7 do no change afer passing o suiable versions, and appeal o (4.6 and (4.7. Remark 4.2 A similar resuls obained in 21 when λ has a finie suppor. The coninuiy of u is proved in 1 in a case where h is bounded, see also 9. 18

19 Corollary 4.1 Assume ha C X and C Y hold. Then, here is a version of U such ha for all s T sup Y r Y s 2 + sup sup U r (e U s (e 2 C 2 (1 + X 2 s. r s, e r s, Proof. Recall from he proof of Proposiion 4.3 ha Y = u(,x on,t. Thus, plugging (2.5 and (2.6 in Proposiion 4.4 gives he upper-bound on he quaniy sup r s, Y r Y s 2. The upper-bound on sup e sup r s, U r (e U s (e 2 is obained similarly by passing o he version of U given in Remark 4.1. Proposiion 4.5 Assume ha H-C X -C Y holds. Then here is a version of Z such ha, for all n 1, n 1 i+1 i= Z Z i 2 C2 n 1. Proof. 1. We denoe by x h (resp. y h, z h, γ h he gradien of h wih respec o is x variable ( (resp. y, z, γ. We firsnroduce he processes Λ and M defined by Λ := exp yh(θ r dr, M := 1+ M r z h(θ r dw r. Since h has bounded derivaives, i follows from Iô s Lemma and Proposiion 4.2 ha T Λ M Z = M T (Λ T g(x T χ T + ( x h(θ r χ r + γh(θ r Γ r Λr dr F By Remark 3.8 and Proposiion 3.3, we deduce ha ( T Λ M Z = M T Λ T g(x T X T + F r Λ r dr F ( X 1 σ(x where he process F is defined by F r := x h(θ r X r + γ h(θ r Γ r for r T. I follows ha { } Λ M Z = G F F r Λ r dr ( X 1 σ(x (4.8 ( where G := M T Λ T g(x T X T + T F r Λ r dr. By Remark 3.7 and (4.4, we deduce ha. G p Cp for all p 2. (4.9 Se m s := G F s and le ( ζ,ṽ H2 L 2 λ (wih values in Md R d be defined such ha m s = G T ζ s r dw r T s Ṽr(e µ(de,dr. Applying (4.9 and Lemma 5.2 o (m, ζ,ṽ implies ha (m, ζ,ṽ B p C p for all p 2. (4.1 Using C X, Remark 3.7, (4.4, (4.1, applying Lemma 5.1 o M 1 and using Iô s Lemma, we deduce from he las asserion ha Z := (ΛM (m 1 F r Λ r dr ( X 1 19

20 can be wrien as Z = Z + µ rdr + σ rdw r + β r (e µ(de,dr, where and µ, σ and β are adaped processes saisfying Z p S p C p for all p 2, (4.11 A p,t C p for all p 2 (4.12 where A p s, := µ p H p + σ p H p + β p L p, s T. s, s, λ,s, 2. Observe ha Z = Z σ(x P a.s. since he probabiliy of having a jump a ime s equal o zero. I follows ha, for all i n and,+1, and I 2 i, := σ(x σ(x i 2 Z 2. Ob- where I 1 i, := Z Z i 2 σ(x i 2 serving ha I 1, Z Z i 2 ( C 2 I 1 i, + I2, = Z Z i 2 F i σ(x i 2 ( i+1 C 2 µ r 2 + σ r 2 + (4.13 β r (e 2 λ(de dr σ(x i 2 we deduce from Hölder inequaliy, (2.5 and he linear growh assumpion on σ ha n 1 i+1 i= C 2 n 1 I 1,d ( T µ r 2 + σ r 2 + β r (e 2 λ(de dr sup σ(x 2 T C 2(A 4,T 1 2 n 1. (4.14 Using he Lipschiz coninuiy of σ, we obain I 2 i, C 2 X X i 2 Z 2. (4.15 Now observe ha for each k,l d (X k Xk i 2 ( Z ( l 2 C 2 Arguing as above, we obain n 1 i+1 i= ( Z l Z l 2 (X k 2 + (X k Z l Xk Zl i 2.(4.16 ( Z l Z l i 2 (X k i 2 ( C2 1 + (A 4,T 1 2 n 1. (4.17 Moreover, we deduce from he linear growh condiion on b, σ, β and (2.5, (4.11 and (4.12 ha X k Zl can be wrien as X k Z l = X k Z l + ˆµ kl r dr + ˆσ r kl dw r + 2 ˆβ kl r (e µ(de, dr,

21 where ˆµ kl, ˆσ kl and ˆβ kl are adaped processes saisfying ˆµ kl H 2+ ˆσ kl H 2+ ˆβ kl L 2 λ C2. I follows ha n 1 i+1 i= (X k Z l Xk Zl i 2 ( C 2 n 1 ˆµ kl 2 H + ˆσ kl 2 2 H + ˆβ kl 2 2 L 2 λ which combined wih (4.15, (4.16 and (4.17 leads o n 1 i+1 i= I 2,d C 2(1 + (A 4,T 1 2 n 1. (4.18 The proof is concluded by plugging (4.14-(4.18 in (4.13 and recalling (4.12. Proposiion 4.6 Assume ha C X -C Y holds. Then here is a version of Z such ha, for all ε > and n 1, n 1 i+1 i= Z Z i 2 Cε n 1+ε. Proof. We adap he argumens of 6. Le Λ and M be defined as in he proof of Proposiion 4.5 and recall ha, afer possibly passing o a suiable version, Z = I where, for s, T, T Is := MT (Λ T g(x T χ T Λ M + ( x h(θ r χ r + γh(θ r Γ r Λr dr F s For,+1, i n 1, we herefore have Z Z i 2 ( C 2 I I 2 + I I 2 where, by Remark 3.5, (5.8 below applied o (3.4, recall ha ρ is bounded, and sandard esimaions on ΛM, sup i n 1, i,+1 I I 2 C2 n 1. Thus i suffices o prove ha n 1 i+1 i= I I 2 C ε n 1+ε, where ε > is now fixed. To his purpose, we firs observe ha I is a maringale on,+1, which implies ha I I 2 I +1 2 I 2. (4.19 Remark ha n 1 i= I +1 2 I 2 = Z T 2 Z 2 + n i=1 I 1 2 I 2 which, combined wih Proposiion 4.2 and (4.19, leads o n 1 i+1 i= I I 2 = C 2n 1 (1 + n i=1 I 1 2 I 2.., To conclude he proof, i remains o show ha I 1 2 I 2 I 1 I I 1 + I C ε n 1+ε. 21

22 which follows from Hölder inequaliy, Remark 3.5 and Lemma 5.2 as above. We now complee he proof of Theorem 2.1. Proof of Theorem We sar wih (ii. We firs show ha (2.16 holds under H and C Y. We consider a Cb densiy q on R d wih compac suppor and se (b k,σ k,β k (,e(x = k d R d (b,σ,β(,e( xq (kx x d x. For large k N, hese funcions are bounded by 2K a. Moreover, hey are K-Lipschiz and C 1 b. Using he coninuiy of σ, one also easily checks ha σk is sill inverible. By H and Remark 2.4, for each e and x R d, I d + β k (x,e is inverible wih uniformly bounded inverse. We denoe by (X k,y k,z k,u k he soluion of (2.2-(2.3 wih (b,σ,β replaced by (b k,σ k,β k. Since (b k,σ k,β k converges poinwise o (b,σ,β, one easily deduces from Lemma 5.1 and Lemma 5.2 ha (X k,y k,z k,u k converges o (X,Y,Z,U in S 2 B 2. Since he resul of Proposiion 4.5 holds for (X k,y k,z k,u k uniformly in k, his shows ha (ii holds under H and C Y, recall Remark 2.3. We now prove ha (2.16 holds under H. Le (X,Y k,z k,u k be he soluion of (2.2-(2.3 wih h k insead of h, where h k is consruced by considering a sequence of molifiers as above. For large k, h k ( is bounded by 2K. By Lemma 5.2, (Y k,z k,u k converges o (Y,Z,U in S 2 B 2 which implies (ii by arguing as above. 2. Since ρ is bounded and λ( <, Corollary 4.1 and Proposiion 4.6 imply (2.14 and (2.15 under C X -C Y, recall Remark 2.3. Now observe ha sup Γ Γ i 2 2 sup Γ Γ i Γ i Γ i 2,+1,+1 where, by Jensen s inequaliy and he fac ha Γ i is F i -measurable, Γ i Γ i 2 n T i+1 2 (Γ i Γ s ds n T i+1 Γ i Γ s 2 ds. Thus, (2.14 implies Γ Γ 2 S C 2 2 n 1 and Γ Γ 2 H C 2 2 n 1. We conclude by he same approximaion argumen as above. 5 Appendix: A priori esimaes For sake of compleeness, we provide in his secion some a priori esimaes on soluions of forward and backward SD s wih jumps. In he following, we consider some measurable maps b i : Ω,T R d R d, σ i : Ω,T R d M d, β i : Ω,T R d R d and f i : Ω,T R 22

23 R d L 2 (,,λ; R, i = 1,2. Here L 2 (,,λ; R is endowed wih he naural norm ( a(e 2 λ(de 1 2. Omiing he dependence of hese maps wih respec o ω Ω, we assume ha for each T b i (,, σ i (,, β i (,,e and f i (, are a.s. K-Lipschiz coninuous uniformly in e for β i. We also assume ha ( f i (,, b i (, is F-progressively measurable, and ( σ i (,, β i (, is F-predicable, i = 1,2. Given some real number p 2, we assume ha b i (,, σ i (, and f i (, are in H p, and ha β i (,, is in L p λ. For 1 2 T, Xi L 2 (Ω, F i, P; R d for i = 1,2, we now denoe by X i he soluion on,t of X = X i + bi (s,xs i ds + σ i (s,xs i dw s + Lemma 5.1 β i (s,e,xs i µ(de,ds. (5.1 X 1 p S p 1,T C p X 1 p + b 1 (, p H p + σ 1 (, p H p + β 1 (,, p L p. 1,T 1,T λ, 1,T (5.2 Moreover, for all 1 s T, sup Xu 1 Xs 1 p s u C p A 1 p s, (5.3 where A 1 p is defined as X 1 p + sup b 1 (s, p + sup σ 1 (s, p + sup 1 s T 1 s T 1 s T { β 1 (s,,e p λ(de}, and, for 2 T, δx p S p C p ( X 1 X 2 p + A 1 p 2 1 2,T ( T p + C p ( δ b d + δ σ p H p + δ β p L p 2 2,T λ, 2,T (5.4 where δx := X 1 X 2, δ b = ( b 1 b 2 (,X 1 and δ σ, δ β are defined similarly. Lemma 5.2 (i Le f be equal o f 1 or f 2. Given Ỹ Lp (Ω, F T, P; R, he backward SD T T T Y = Ỹ + f(s,y s,z s,u s ds + Z s dw s + U s (e µ(de,ds (5.5 has a unique soluion (Y,Z,U in B 2. I saisfies ( T (Y,Z,U p B C p p Ỹ p + p f(, d. (5.6 23

24 Moreover, if A p := Ỹ p + sup T f(, p <, hen { } sup Y u Y s p C p A p s p + Z p H p + U p L p s u s, λ,s,. (5.7 (ii Fix Ỹ 1 and Ỹ 2 in L p (Ω, F T, P; R and le (Y i,z i,u i be he soluion of (5.6 wih (Ỹ i, f i in place of (Ỹ, f, i = 1,2. Then, for all T, ( T p (δy,δz,δu p B p C p δỹ p + δ f r dr (5.8,T where δỹ := Ỹ 1 Ỹ 2, δy := Y 1 Y 2, δz := Z 1 Z 2, δu := U 1 U 2 and δ f := ( f 1 f 2 (,Y 1,Z 1,U 1. The proof of he previous resuls is sandard, see e.g. 15 and 12. We refer o pages in 11 for more deails. The jump pars handled by using he following proposiion which plays he same role as he usual Burkholder-Davis- Gundy s Lemma. I follows from an inducion argumen already used in 4. The proof can be found in 11, see page 125. Proposiion 5.1 Given ψ L 2 λ, le M be defined by M = ψ s(e µ(ds,de on,t. Then, for all p 2, k p ψ p L p M p S p K p ψ p L p where k p, K p λ,,t,t λ,,t, are posiive numbers ha depend only on p, λ( and T. References 1 Barles G., R. Buckdahn and. Pardoux (1997. Backward sochasic differenial equaions and inegral-parial differenial equaions. Sochasics Sochasics Repors, 6, Bally V., and G. Pages (22. A quanizaion algorihm for solving discree ime mulidimensional opimal sopping problems. Bernoulli, 9 (6, Becherer D. (25. Bounded soluions o Backward SD s wih Jumps for Uiliy Opimizaion and Indifference Hedging. Preprin, Imperial College London. 4 Bicheler K. and J. Jacod (1983. Calcul de Malliavin pour des diffusions avec sau: exisence d une densié dans le cas unidimensionel. Séminaire de Probabilié, 17, Bicheler K., Gravereaux J.-B. and J. Jacod (1987. Malliavin calculus for processes wih jumps. Gordon and Breach Science Publishers, New York. 6 Bouchard B. and J.-F. Chassagneux (26. Discree ime approximaion for coninuously and discreely refleced BSD s. Preprin LPMA, Universiy Paris 6. 7 Bouchard B. and N. Touzi (24. Discree-Time Approximaion and Mone-Carlo Simulaion of Backward Sochasic Differenial quaions. Sochasic Processes and heir Applicaions, 111 (2, Brémaud P. (1981. Poin Processes and Queues - Maringale Dynamics, Springer- Verlag, New-York. 24

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