On the Malliavin differentiability of BSDEs

Transcription

1 On he Malliavin differeniabiliy of BSDE Thibau Marolia Dylan Poamaï Anhony Réveillac Augu 25, 215 arxiv: v4 [mah.pr 24 Aug 215 Abrac In hi paper we provide new condiion for he Malliavin differeniabiliy of oluion of Lipchiz or quadraic BSDE. Our reul rely on he inerpreaion of he Malliavin derivaive a a Gâeaux derivaive in he direcion of he Cameron-Marin pace. Incidenally, we provide a new formulaion for he characerizaion of he Malliavin-Sobolev ype pace D 1,p. Key word: Malliavin calculu; abrac Wiener pace; BSDE. AMS 21 ubjec claificaion: Primary: 6H1; Secondary: 6H7. 1 Inroducion Backward Sochaic Differenial Equaion BSDE) have been udied exenively in he la wo decade a hey naurally arie in he conex of ochaic conrol problem for inance in Finance ee [9), and a hey provide a probabiliic repreenaion for oluion o emi-linear parabolic PDE, via a non-linear Feynman-Kac formula ee [2). Before going furher le u recall ha hi cla of equaion ha been inroduced in [4, 19, 2 and ha a BSDE can be formulaed a: Y = ξ + f,y,z )d Z dw, [,T, 1.1) where T i a fixed poiive number, W := W ) [,T i a one-dimenional Brownian moion defined on a probabiliy pace Ω,F T,P) wih naural filraion F ) [,T. The daa of he equaion are he F T -meaurable r.v. ξ, called he erminal condiion, and he mapping f : [,T Ω R 2 R which i a progreively meaurable proce and where according o he noaion ued in he lieraure we wrie f,y,z) for f,ω,y,z). A oluion o he BSDE 1.1) i hen a pair of predicable procee Y, Z), wih appropriae inegrabiliy properie, uch ha Relaion 1.1) hold P a.. Univerié Pari-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal De Lare De Taigny, Pari Cedex 16, FRANCE, marolia@ceremade.dauphine.fr Univerié Pari-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal De Lare De Taigny, Pari Cedex 16, FRANCE, poamai@ceremade.dauphine.fr IMT UMR CNRS 5219, Univerié de Touloue, INSA de Touloue, 135 avenue de Rangueil, 3177 Touloue Cedex 4, FRANCE, anhony.reveillac@ina-ouloue.fr 1

2 When dealing wih applicaion, one need o obain regulariy properie on he oluion Y, Z), uch a he Malliavin differeniabiliy of he random variable Y, Z a a given ime in [,T. Noe ha for he Z componen hi queion need o be clarified a lile bi becaue of he definiion of Z, cf. Theorem 5.1 for a precie aemen. More preciely, one need o anwer he following queion: Which condiion on he daa ξ and f in 1.1) enure ha Y, Z are Malliavin differeniable? Thi queion wa fir addreed in he paper [2 in a Markovian eing, ha i when ξ := gx T ) and f,ω,y,z) := h,x ω),y,z) where g : R R and h : [,T R 3 R are regular enough deerminiic funcion and X := X ) [,T i he unique oluion o a SDE of he form: X = X + σ,x )dw + b,x )d, [,T, wih regular enough coefficien σ,b : [,T R R. In ha framework, Pardoux and Peng proved in [2, Propoiion 2.2 ha, under eenially) he following condiion: PP1) g i coninuouly differeniable wih bounded derivaive. PP2) h i coninuouly differeniable in x,y,z) wih bounded derivaive uniformly in ime, Y i Malliavin differeniable a any ime, wih a imilar aemen for Z, and he Malliavin derivaive of Y and Z provide a oluion o an explici linear BSDE. To be more precie, in [2 he auhor make one aumpion for he whole paper which i ronger han PP1)- PP2) above. However a careful reading of he proof of [2, Propoiion 2.2 enable one o conclude ha Condiion PP1)-PP2) are ufficien o obain he Malliavin differeniabiliy of he oluion. Aumpion PP1)-PP2) look prey inuiive ince hey baically require he Malliavin differeniabiliy of he erminal condiion ξ and of he generaor f once he componen y,z) are frozen, i.e., of he proce,ω) f,ω,y,z) for given y,z). Hence, i i naural o expec ha he laer condiion can be eaily generalized o he non-markovian framework. Unforunaely, he fir reul in ha direcion, given by El Karoui, Peng and Quenez in [9, require more ringen condiion han he aforemenioned inuiive one. More explicily, he main reul in [9 concerning he Malliavin differeniabiliy of he oluion o he BSDE 1.1) eenially) involve he following condiion ee [9, Propoiion 5.3 for a precie aemen): EPQ1) ξ i Malliavin differeniable 1 and E[ ξ 4 < +. EPQ2) A any ime [,T, he r.v. ω f,ω,y,z ) i Malliavin differeniable 2 wih Malliavin derivaive denoed by D f,y,z ) uch ha here exi a predicable procek θ := K) θ [,T wih E[ Kθ 2 d) 2 dθ < +, and uch ha for any y 1,y 2,z 1,z 2 ) R 4 i hold for a.e. θ [,T ha: 1 i.e. ξ i in D 1,2 D θ f,ω,y 1,z 1 ) D θ f,ω,y 2,z 2 ) K θ ω) y 1 y 2 + z 1 z 2 ). 2 In fac a an adaped proce i belong o D 1,2, we refer o he pace L a 1,2 whoe precie definiion i recalled in [9, p. 58 2

3 Roughly peaking, hi mean ha ξ and ω f,ω,y,z) have o be Malliavin differeniable, bu in order o prove ha Y and Z are Malliavin differeniable, one need o enforce an exra regulariy condiion on each of he daa: ha i ξ ha a finie momen of order 4, and he Malliavin derivaive of he driver f i Lipchiz coninuou in y, z) wih a ufficienly inegrable ochaic Lipchiz conan K. Noe ha a careful reading of he proof allow one o conclude ha he momen condiion on ξ and Df can acually be relaxed o hold only in L 2+ε for ome ε >. Beide, a noed in [9, Remark a he boom of p. 59, if K i bounded hen he proof can be modified o ha he exra inegrabiliy condiion on ξ i.e. E[ ξ 4 < + ) can be dropped. However, even in hi cae, one can check ha in he Markovian framework, Condiion EPQ1)-EPQ2) are ricly ronger han Condiion PP1)-PP2). Since hee wo eminal paper, he mo noable exenion wa concerned wih he udy of he Malliavin differeniabiliy of Y,Z) in a quadraic eing, ha i o ay when he generaor f ha quadraic growh in he z variable, a problem addreed in [1, 11, 8, 13. Noice nonehele ha he proof in hee reference are rongly influenced by he one in he Lipchiz eing of [2, 9, a hey all ar by approximaing he quadraic generaor by Lipchiz one, o which hey apply he reul of [2, 9. The applicaion of he Malliavin differeniabiliy of BSDE alo received a lo of aenion in he lieraure. Hence, i wa ued in he conex of numerical cheme for BSDE in, among oher, [7, 1, or o udy he exience and regulariy of deniie for he marginal law of Y,Z) in [2, 3, 17. However, in all he above reference, he auhor alway refer o eiher [2, 9 in a Lipchiz conex or [1 in a quadraic conex, when aing differeniabiliy reul in he Malliavin ene ee for inance he enence before Theorem 2.2 in [3, or Sep 2 in he proof of Theorem 3.3 in [2, which refer o [3, or he proof of Par a) of Theorem 2.6 in [1, or Propoiion 3.2 in [17). The aim of hi paper i o provide an alernaive ufficien condiion o EPQ1)-EPQ2) for he Malliavin differeniabiliy of he oluion o a BSDE of he form 1.1) in he general non- Markovian eing. Our main reul in ha direcion i Theorem 5.1 below, uing a fundamenally differen approach from [9, 2, a well a differen ype of aumpion. Since hey involve ome noaion concerning he analyi on he Wiener pace, we refrain from deailing hem immediaely, and raher explain informally wha are he main difference beween our approach and he one of [9. A naural way o olve a BSDE of he form 1.1) when he driver f i Lipchiz in y,z) i o make ue of a Picard ieraion, ha i o ay a family Y n,z n ) of oluion o BSDE aifying Y n = ξ + f,y n 1,Z n 1 )d Z n dw, [,T, 1.2) where Y Z. Then, a fixed poin argumen allow one o conruc, in appropriae pace, a oluion Y, Z) o Equaion 1.1). If ξ and f, y, z) are Malliavin differeniable, hen o i Y n,z n ). Then, i ju remain o prove ha hi propery exend o he limi Y and Z of repecively Y n and Z n, in appropriae pace. More preciely hi i done by a uniform in n) conrol of he Sobolev norm of Y n,z n or equivalenly by proving ha he Malliavin derivaive DY n,dz n ) of Y n,z n ) converge o he oluion of a linear BSDE whoe oluion will be he Malliavin derivaive DY,DZ) of Y and Z. Thi la ep i exacly where he exra regulariy EPQ1)-EPQ2) i needed. I appear quie clearly ha for hi approach, he condiion of [9 canno be opimized in he general cae. Even hough hi idea eem prey 3

4 naural, i i baed on a choice omehow arbirary. Indeed, a neceary condiion for DY o be well-defined a a given ime, i ha here exi a equence of random variable F n ) n converging o Y in L 2 uch ha each variable F n i Malliavin differeniable wih derivaive DF n and uch ha DF n converge, wih repec o a uiable norm, o DY. A a conequence, in he approach decribed above, one believe ha hi equence F n ) n can be choen o be he Picard ieraion Y n ) n. Once again, hi idea look very naural, according o he ame ype of proof for SDE, bu hen one ee ha in he BSDE framework hi inuiive idea lead o prey heavy aumpion. We elaborae a lile bi more on hi poin in Secion 6.3. Regarding he dicuion above, one could hink of rying o find a equence of procee known o approximae he Malliavin derivaive of Y and Z) when Y i Malliavin differeniable. Thi approximaion i provided by he well-known inerpreaion of he Malliavin derivaive a a Gâeaux derivaive in he direcion of he Cameron-Marin pace. More preciely, a neceary condiion for Y o belong o D 1,2, i ha for any aboluely coninuou funcion h aring from a wih derivaive denoed ḣ, he difference quoien ε 1 Y ω +εh) Y ω)) converge, in a ene o be made precie, a ε goe o o DY,ḣ L 2 [,T). Thi fac wa iniially given by Malliavin and hen exended by Sroock, Shigekawa, Kuuoka and Sugia in a erie of paper [16, 23, 22, 15, 24. In addiion, Sugia proved in [24 ha a r.v. F i Malliavin differeniable if i i ray aboluely coninuou 1 and if i i ochaically Gâeaux differeniable. Uing he main idea of [24 we provide incidenally a new formulaion of he characerizaion of he Malliavin- Sobolev ype pace D 1,p in Theorem 4.1. Since we did no find explicily hi characerizaion in he lieraure, we believe ha hi reul i new and maybe inereing by ielf. The main poin i ha hi formulaion i epecially handy when dealing wih ochaic equaion like BSDE. Wih hi reul a hand, we obain new condiion ee Aumpion D), H 1 ) and H 2 ) a he beginning of Secion 5) for Y,Z o be Malliavin differeniable, ee Theorem 5.1. Our aumpion refine hoe of [2, 9 in he Markovian cae, and our approach i direcly applicable o quadraic growh BSDE ince we do no rely on any approximaion procedure. We refer he reader o Secion 6 for ome example and a dicuion on he difference beween our approach and he one of [2, 9. The re of he paper i organied a follow. We ar below wih ome preliminarie. Then we urn in Secion 3 o ome elemen of analyi on he Wiener pace. Our characerizaion of he e D 1,p i given in Secion 4, and he maerial on he Malliavin differeniabiliy of BSDE ielf i conained in Secion 5. We provide applicaion and a comparion of he reul in Secion 6. Finally, we exend our approach o quadraic growh BSDE in Secion 7. 2 Preliminarie 2.1 Noaion We fix hroughou he paper a ime horizon T > and d a poiive ineger. For any poiive ineger k, we denoe by he Euclidian norm in R k and by he inner produc, wihou menion of k which will be clear in he conex. For any poiive ineger n and m, we idenify R n m wih he pace of real marice wih n row and m column, endowed wih he Euclidean norm on R n m. Le M be in R n m, 1 j n and 1 l m, we denoe by M j,: R 1 m 1 we refer o Secion 4 where hi noion i recalled 4

5 rep. M :,l R n,1 ) i j-h row rep. i l-h column). We e M R m n o be he ranpoe of M. We alo idenify R k wih R 1,k. Le now Ω := C [,T,R d ) be he canonical Wiener pace of coninuou funcion ω := ω 1,...,ω d ) from [,T o R d uch ha ω) =,...,). Le W := W 1,...,Wd ) [,T be he canonical Wiener proce, ha i, for any ime in [,T, W denoe he evaluaion mapping: Wω) i := ω i for any elemen ω in Ω and i in {1,...,d}. We e F o he naural filraion of W. Under he Wiener meaure P, he proce W i a andard Brownian moion and we denoe by F := F ) [,T he uual augmenaion which i righ-coninuou and complee) of F o under P. Unle oherwie aed, all he expecaion conidered in hi paper will have o be underood a expecaion under P, and all noion of meaurabiliy for elemen of Ω will be wih repec o he filraion F or he σ-field F T. For any Hilber pace K, for any p 1 and for any [,T, we e L p [,T;K) o be following pace { } L p [,T;K) := f : [,T K, Borel-meaurable,.. f) p K d < +, where he norm K i he one canonically induced by he inner produc on K. We denoe, for impliciy, by H := L 2 [,T;R d ) and by, H i canonical inner produc, ha i o ay f,g H := f) g)d = d i=1 f i )g i )d, f,g) H 2. Le now H be he Cameron-Marin pace ha i he pace of funcion in Ω which are aboluely coninuou wih quare-inegrable derivaive and which ar from a : } H := {h : [,T R d, ḣ H, h) = ḣx)dx, [,T, For any h in H, we will alway denoe by ḣ a verion of i Radon-Nykodym deniy wih repec o he Lebegue meaure. Then, H i an Hilber pace equipped wih he inner produc h 1,h 2 H := h 1, h 2 H, for any h 1,h 2 ) H H, and wih aociaed norm h 2 H := ḣ,ḣ H. Define nex L p K) a he e of all F T -meaurable random variable F which are valued in an Hilber pace K, and uch ha F p L p K) < +, where F L p K) := E [ F p K) 1/p. Le now S be he e of cylindrical funcional, ha i he e of R-valued random variable F of he form F = fwh 1 ),...,Wh n )), h 1,...,h n ) H n, f C b Rn ), for ome n 1, 2.1) where Wh) := ḣ dw := d i=1 ḣi dw i for any h in H and where C b Rn ) denoe he pace of bounded mapping which are infiniely coninuouly differeniable wih bounded derivaive. For any F in S of he form 2.1), he Malliavin derivaive F of F i defined a he following H-valued random variable: F := n f xi Wh 1 ),...,Wh n ))h i, 2.2) i=1 5

6 where f xi := df dx i. I i hen cuomary o idenify F wih he ochaic proce F) [,T. More preciely, we define for any, ω) [, T Ω, Fω) := n f xi Wh 1 )ω),...,wh n )ω))h i ). i=1 Denoe hen by D 1,p he cloure of S wih repec o he Malliavin-Sobolev emi-norm 1,p, defined a: F 1,p := E[ F p +E [ F p H) 1/p. We e D 1, := p 2 D1,p. In order o link our noaion wih he one of he relaed paper [2, 9 we make ue of he noaion DF o repreen he derivaive of F a: F = D Fd, [,T. We denoe by δ : L p H) L p R) he adjoin operaor of by he following dualiy relaionhip: E[Fδu) = E[ F,u H, u domδ), where domδ) := { u L p H), c u >, E[ F,u H c u F L p R), F D 1,p}. δ i alo known under he name of Skorohod or divergence) operaor. Recall ha any elemen u of he form u := Gh wih G in S and h in H belong o domδ) and ha δgh) = GWh) G,h H, 2.3) ee for example [18, Relaion 1.46). Noe ha for any h in H, δh) = Wh). Noice ha in [24 he cylindrical pace, ha we will denoe by P in he following, i he pace of funcional F of he form 2.1) wih f a polynomial. More preciely le P be he e of polynomial cylindrical funcional, ha i he e of random variable F of he form F = fwh 1 ),...,Wh n )), h 1,...,h n ) H n, f R n [X, for ome n 1, 2.4) where R n [X denoe he e of polynomial of degree le or equal o n. However, he cloure of boh S and P wih repec o any 1,p coincide, a any polynomial ogeher wih i derivaive can be approximaed in L p R n ) ee Lemma 2.1 below). Lemma 2.1. Le G be in P. There exi a equence G N ) N 1 S uch ha lim N + G N = G in D 1,r for any r 1. Proof. Le G := fwh 1 ),,Wh n )) wih n 1, h i in H and f in R n [X. Wihou lo of generaliy, we aume ha he family h 1,...,h n ) i orhonormal in H. Le θ be a cuoff funcion, ha i a mapping θ : R n R + uch ha θx) = 1 if x < 1, θx) = for x 2, and uch ha θ Cb Rn ). For N 1, we e: G N := f N Wh 1 ),,Wh n )), f N x) := fx) θx/n), x R n. Noe ha each random variable G N belong o S. Fix r 1. We aim in proving ha lim N + G N G 1,r =. On he one hand, [ E[ G N G r = E G r Wh1 ) θ N,, Wh ) n) r 1 N 6

7 E[ G 2r E[ 1/2 θ Wh1 ) N,, Wh ) n) 1 N C e x 2 /2 dx, R n \B n,n) 2r 1/2 where C i a poiive conan. Hence, lim N + E[ GN G r =. We now urn o he proof of he convergence of he derivaive. We have G N = n i=1 f N x i Wh 1 ),,Wh n ))h i, wih fn x i x) = f x i θx/n)+n 1 fx) θ x i x/n). Hence: E [ G N G) 2r H = n i=1 f E[ N f x i x i 2r θ n f Wh1 ) C E[ Wh 1 ),,Wh n )) x i=1 i +N E[ 2r θ ) 2r f)wh 1 ),,Wh n )) x i. N + 2r Wh 1 ),,Wh n )) N,, Wh n) N ) 1 2r We conclude hi ecion by inroducing he following norm and pace which are of inere when udying BSDE. For any poiive ineger p,n, we e S p n he pace of R n -valued, coninuou and F-progreively meaurable procee Y.. [ Y p S p := E up Y p < +. n T We denoe by H p n,d he pace of Rn d -valued and F-predicable procee Z uch ha We e S p := S p 1 and Hp d := Hp 1,d. Z p H p := E n,d n Z j 2 d j=1 p 2 < +. 3 Some elemen of analyi on he Wiener pace One of he main ool ha we will ue hroughou hi paper i he hif operaor along direcion in he Cameron-Marin pace. More preciely, for any h H, we define he following hif operaor τ h : Ω Ω by τ h ω) := ω +h := ω 1 +h 1,...,ω d +h d ). 7

8 Noe ha he fac ha h belong o H enure ha τ h i a meaurable hif on he Wiener pace. In fac, one can be a bi more precie, ince according o [25, Lemma B.2.1 for any F T -meaurable r.v. F he mapping h F τ h i coninuou in probabiliy from H o L R d ), he pace of real-valued and F T -meaurable random variable, ee Lemma 3.2 below. Taking F = Id, one ge ha τ h i a coninuou mapping on Ω for any h in H. We li below ome oher properie of uch hif. Lemma 3.1 Appendix B.2, [25). Le X and Y be wo F T -meaurable random variable. If X = Y, P a.., hen for any h in H, X τ h = Y τ h, P a.. We recall, he quie urpriing reul ha any r.v. i coninuou in probabiliy in he direcion of he Cameron-Marin pace. More preciely: Lemma 3.2 Lemma B.2.1, [25). Le F be a F T -meaurable random variable. The mapping h F τ h i coninuou from H o L R d ) where he convergence i in probabiliy. One of he main echnique when working wih hif on he pah pace i he famou Cameron- Marin formula. Propoiion 3.1. Cameron-Marin Formula, ee e.g. [25, Appendix B.1) Le F be a F T - meaurable random variable and le h be in H. Then, when boh ide are well-defined [ E[F τ h = E F exp ḣ) dw 1 ) 2 ḣ) 2 d. For furher reference, we alo emphaize ha for any h H and for any p 1, he ochaic ) exponenial E ḣ) dw := exp ḣ)dw 1 ) 2 ḣ) 2 d verifie ) E ḣ) dw S p, p ) Lemma 3.3. Le in [,T and le F be a F -meaurable random variable. For any h in H, i hold ha F τ h = F τ h, P a., where h ) := ḣ 1 u)1 u du,..., In paricular, F τ h i F -meaurable. ḣ d u)1 u du). Proof. I i well-known ha by definiion of P, any F -meaurable random variable admi a F o -meaurable verion. Therefore, here exi ome meaurable map ϕ : Ω R, uch ha F = ϕw ), P a.. Hence, we deduce by Lemma 3.1 ha for P a.e. ω Ω F τ h ω) = ϕw ω)) τ h = ϕw τ h ω)) = ϕω )+h )) = F τ hω). 8

9 We conclude hi ecion wih he following lemma which migh be known. However ince we did no find i in he lieraure we provide a proof in order o make hi paper elf-conained. Lemma 3.4. Le Z H 2 d and h in H. I hold ha Z dw τ h = Z τ h dw + Z τ h ḣ)d, P a.. Proof. Le S be he cla of imple procee X of he form n j X := X 1,...,Xd ), X j = λ j i 1 j i,j i+1 ), j {1,...,d} i= where for any j {1,...,d}, n j N, j = < j 1 <... < j n = T and where for any i n j, λ j i ) i=1,...n j are F j-meaurable and in L 2 R). i We ar by proving he reul for Z in S and hen we prove he reul for any elemen Z in H 2 d uing a deniy argumen. Le Z S wih he decompoiion n j Z = Z 1,...,Zd ), Z j := λ j i 1 j i,j i+1 ), [,T, j {1,...,d}. i= Then, for any h H and for every ω Ω, n d j Z dw τ h )ω) = = = = j=1 i= n d j j=1 i= j=1 i= λ j i Wj W j j i+1 j i λ j i ω +h)wj j i+1 ) τ h ω) W j )ω +h) j i n d j ) λ j i τ hω) ω j j i+1 ) ωj i )+hj j i+1 ) hj j i ) Z τ h dw ω)+ Z τ h ω) dh, which give he deired reul ince h i aboluely coninuou. We exend hi reul o procee Z in H 2 d. Le Z H2 d, hen here exi a equence Zn ) n N in S which converge o Z in H 2 d. Hence, [ T E Z dw τ h [ T E Z dw τ h Z τ h dw Z τ h dh [ T Z n dw τ h +E [ T +E Z n τ h dh Z τ h dh [ T [ E Z Z n ) dw T τ h +E Z n Z ) τ h dw }{{}}{{} =:A n =:B n 9 Z n τ h dw Z τ h dw

10 [ T +E Z n Z ) τ h dh. } {{ } =:C n Le u eimae hee hree erm. Fir, uing Propoiion 3.1, Cauchy-Schwarz Inequaliy, hen Burkholder-Davi-Gundy Inequaliy, we have [ T A n = E Z Z n ) dw e T ḣ) dw 1 2 [ T T ḣ) 2 d E 1/2 Z Z n 2 d [ ) 2 1/2 E E ḣ) dw. T By 3.1), hi clearly goe o a n goe o infiniy. Similarly, uing Burkholder-Davi-Gundy Inequaliy, we have ) 1 B n E Z n Z ) τ h 2 2 T ) 1 d = E Z n Z 2 2 d τ h. Therefore, we can ue Propoiion 3.1 and Cauchy-Schwarz Inequaliy, o alo deduce ha B n. Finally, we have n + [ C n = E E ) T ḣ) dw Z n Z ) ḣ)d [ ) 2 1/2E[ ) 2 1/2 E E ḣ) dw Z n Z ḣ) d [ ) 2 1/2E[ 1/2 1/2 E E ḣ) dw Z n Z 2 d d) ḣ) 2, which alo goe o a n goe o infiniy. Therefore he proof i complee. Thi reul enail he following ueful conequence. Le in,t and h in H uch ha ḣ =,...,) for. Then for any Z in H 2 d, i hold ha: ince Z τ h ḣ)d =. Z dw τ h = Z τ h dw, P a.., 3.2) 4 A characerizaion of Malliavin differeniabiliy Before going furher, we would like o recall he main finding of [24. Any Malliavin-Sobolev ype pace D 1,p a defined in Secion 2 originally defined by Malliavin [16 and Shigekawa [22) agree wih he Sobolev pace due o Sroock [23 and Kuuoka [15) D 1,p coniing in he e of Ray Aboluely Coninuou RAC) and Sochaically Gâeaux Differeniable SGD) r.v. F in L p R), where hee noion are defined a follow: 1

11 RAC) For any h in H, here exi a r.v. Fh uch ha F h = F, P a.., and uch ha for any ω in Ω, R F h ω +h) i aboluely coninuou, where h := h 1,...,h d ). SGD) There exi DF in L p H) uch ha for any h in H, F τ εh F ε DF,h H, in probabiliy. 4.1) ε In addiion, for any F in D 1,p, F = DF, P a.. Noe ha according o he aemen of Sep 1 in he proof of [24, Theorem 3.1, if F i RAC) and SGD) hen for any h in H and any ε > i hold ha ε ε 1 F h τ εh F h ) = ε 1 F τ h,h H d, P a.. Furhermore, by Lemma 3.1, we have for any ε here exi a e A ε uch ha P [A ε = and F τ εh = F h τ εh and F = F h ouide A ε. Hence, for any ε in,1), he relaion above rewrie a: ε ε 1 F τ εh F) = ε 1 F τ h,h H d, P a.. 4.2) Remark 4.1. I ha acually been proved by Janon [14 ha 4.2) i equivalen o RAC) and SGD), for any p > 1, ee Lemma Noice ha [14 alo obained a imilar characerizaion for p = 1 ee Lemma 15.71). However, a aed in Remark 4 of [24, he idenificaion of he Kuuoka-Sroock and Shigekawa pace when p = 1 i ill an open reul, o ha we never conider he cae p = 1 in hi paper. The main reul of hi ecion i he following heorem whoe proof i poponed o he end of he ecion. Theorem 4.1. Le p > 1 and F L p R). The following properie are equivalen i) F belong o D 1,p. ii) There exi DF in L p H) uch ha for any h in H and any q [1,p) [ lim E F τ εh F ε ε DF,h H q =. iii) There exi DF in L p H) and here exi q [1,p) uch ha for any h in H [ lim E F τ εh F ε ε DF,h H q =. iv) There exi DF in L p H) uch ha for any h in H [ lim E F τ εh F ε ε DF,h H =. In ha cae, DF = F. Remark 4.2. The implicaion ii) i) when q = p = 2 already appear in [5 ee ). Thi i of coure conained in our reul. 11

12 We now give he following lemma which characerize he Malliavin derivaive uing he dualiy formula involving he Skorohod operaor alo called divergence operaor). Lemma 4.1. Le ε > and 1 < p < +. Suppoe ha F L 1+ε R) and aume ha here exi DF in L p H) uch ha: E[FδGh) = E[G DF,h H, for every G S and h H. Then, i hold ha F D 1,p, and DF = F, P a.. Proof. We know ee e.g. [24, Corollary 2.1) ha he reul i rue if S i replaced by P. Le G be in P. By Lemma 2.1 here exi G N ) in S uch ha G N approximae G in D 1,p. Le h in H. For any N 1, we have E[FδGh) = E[FGWh) G,h H ) Furhermore, by Lemma 2.1 = E[FG N Wh) G N,h H ) E[G N G)FWh) F G N G),h H = E[FδG N h) E[G N G)FWh) F G N G),h H = E[G N DF,h H E[G N G)FWh) F G N G),h H. E[G N G)FWh) E[ FWh) p 1/p E[ G N G p 1/ p N +, wih 1 < p < 1+ε and where p i he conjugae of p, and E[F G N G),h H E[ F p 1/p E[ G N G) p H 1/ p h H N +, by Lemma 2.1 again. Hence, and E[F δgh) = lim N + E[GN DF,h H = E[G DF,h H + lim N + E[GN G) DF,h H, lim N + E[GN G) DF,h H lim N + E[ GN G p 1/p E[ DF p H 1/ p h H. N + Thu we have proved ha for any G in P and for any h in H, E[FδGh) = E[G DF,h H, which give he reul by [24, Corollary 2.1. We now prove he following lemma for he Malliavin differeniabiliy of a given random variable. Lemma 4.2. Le p > 1. Le F be in D 1,p. Then, for any q in [1,p) and for any h in H, F τ εh F ε F,h H in L q R). ε 12

13 Proof. Fix q in [1,p), h in H and η > uch ha q +η < p. We know from [24, Theorem 3.1 ha ince F i in D 1,p, F i SGD), RAC), and Relaion 4.2) hold rue. We hu have uing Jenen Inequaliy ε E[ 1 F τ εh F) [ q+η ε = E ε q+η) q+η F τ h,h H d [ ε ε 1 E F τ h,h H q+η d ε = ε 1 E [ F,h H q+η τ h d = ε 1 ε [ E F,h H q+η E ḣ r dw r )d E[ F,h H p q+η p < +. up E[ E,1) ) p p q η p p q η ḣ r dw r Hence by de La Vallée Pouin Crierion, we deduce ha he family of random variable ε 1 F τ εh F) q) i uniformly inegrable which ogeher wih he convergence in probabiliy 4.1) give he ε,1) reul. Remark 4.3. Noe ha he concluion of he previou Lemma may fail for q = p 1 We can now proceed wih he proof of Theorem 4.1. Proof of Theorem 4.1. From Lemma 4.2 we have i) ii) and of coure ii) iii) iv). We urn o iv) i). Le F be uch ha here exi DF in L p H) uch ha [ lim E F τ εh F ε ε DF,h H =. The proof coni in applying Lemma 4.1 by proving he dualiy relaionhip By Lemma A.1 in he Appendix) wih ε =, E[FδGh) = E[G DF,h H, G S, h H. 4.3) E[FδGh) = d dε E[F τ εhg ε= = lim η E[F τ ηh F)G [ F τηh F = lim E DF,h H )G η η η 1 +E[ DF,h H G = E[ DF,h H G, 4.4) where he proof ha he fir erm on he righ-hand ide goe o i repored below. 1 Afer he fir verion of hi paper, a couner example ha been given in [12. 13

14 Noe ha E[ DF,h H G < + ince G i bounded and DF belong o L p H). The Equaliy 4.5) i juified by Hölder inequaliy ince [ F τηh F E DF,h H )G η [ F τ ηh F G E DF,h H. η ε Corollary 4.1. Le F be in D 1,p. For any ε > and any h in H, F τ εh belong o D 1,p and F τ εh ) = F) τ εh. Proof. Le F be in D 1,p. Uing Theorem 4.1, we know ha for any h in H and any q [1,p) [ lim E F τ εh F ε ε F,h H q =. By Lemma A.1 in he Appendix) i hold ha E[F τ εh δgh) = d dε E[F τ εhg 1 = lim η η E[ F τ ε+η)h F τ εh )G [ F τε+η)h F τ εh = lim E F) τ εh,h H )G η η +E[ F) τ εh,h H G = E[ F) τ εh,h H G, 4.5) where he proof ha he fir erm on he righ-hand ide goe o i repored below. Noe ha E[ F) τ εh,h H G < + ince F) τ εh,h H = F,h H τ εh, P a.., G belong o all he pace L r R) for r 1 and E[ F,h H p h p H E[ F p H < +. The Equaliy 4.5) i juified by Hölder inequaliy ince [ F τε+η)h F τ εh E F τ εh,h H )G η [ F τ ε+η)h F τ εh r1 r E F τ εh,h E[ G r H 1 r η [ F τ ηh F = E η [ F τ ηh F E η F,h H r F,h H q 1 q E [E τ εh 1 r E[ G r 1 r 1 )ᾱ rᾱ ε ḣ) dw E[ G r 1 r where 1 < r < q and α := q r and where r rep. ᾱ) i he Hölder conjugae of r rep. α). Conequenly, E[F τ εh δgh) = E[ F τ εh,h H G, and from Lemma 4.1 F τ εh ) = F) τ εh. 14

15 5 Malliavin differeniabiliy of BSDE In hi ecion we derive a ufficien condiion enuring ha he oluion o a BSDE i Malliavin differeniable. To implify he comparion of he reul wih he companion paper [9, 2 we adop he noaion ued in hee paper concerning he Malliavin calculu. More preciely, for any F in D 1,p for p > 1) we have defined he Malliavin derivaive F a an H-valued random variable. Recall ha denoing DF he derivaive of F ha i F = D rfdr, DF coincide wih he Malliavin derivaive inroduced in [9, 2, 18. In paricular F,h H = DF,ḣ H for any h in H. Le n be a poiive ineger, we conider now he following BSDE: Y = ξ + fr,y r,z r )dr Z r dw r, [,T, P a.., 5.1) whereξ := ξ 1,,ξ n ) i af T -meaurabler n -valued r.v. andf : [,T Ω R n R n d R n i a F-progreively meaurable proce where a uual he ω-dependence i omied. The aim of hi ecion i o how ha for any [,T, we can apply Theorem 4.1 under he following aumpion: L) The map y, z) f, y, z) i differeniable wih uniformly bounded and coninuou parial derivaive. We denoe by f y := f j y k )j {1,...,n}, k {1,...,n} he Jacobian marix of f wih repec o y, where j rep. k) indexe he column rep. he row) of f y and fy j denoe he gradien of f j. We denoe by fz, j for any j {1,...,n} he Jacobian marix of f j wih repec o z, ha i fz j = f j z k,l )k {1,...,n},. l {1,...,d} D) ξ belong o D 1,2 ) n, for any y,z) R n R n d, he map,ω) f,ω,y,z) i in L 2 [,T;D 1,2 ) n ), f,y,z) and Df,y,z) are F-progreively meaurable, and n E D f j,y,z ) 2 H d < +. j=1 H 1 ) There exi p 1,2) uch ha for any h H and for any j {1,...,n} [ lim E f j, +εh,y,z ) f j ),,Y,Z ) ε Df j,,y,z ε ),ḣ p H d =, H 2 ) Le ε k ) k N be a equence in,1 uch ha lim ε k =, and le Y k,z k ) k be a equence k + of random variable which converge in S p n H p n,d for any p [1,2) o ome Y,Z). Then for all h H and for all j {1,..., n}, he following convergence hold in probabiliy fy,ω j +ε k h,y k,z ) fy,ω,y,z ) j L 2 [,T;R n ) k + fz j,ω +ε kh,y k,z k ) fj z,ω,y,z ), 5.2) k + or fy j,ω +ε kh,y k,z k ) fj y,ω,y,z ) 15 L 2 [,T;R n d ) L 2 [,T;R n ) k +

16 fz j,ω +ε kh,y,z k ) fj z,ω,y,z ) L 2 [,T;R n d ). 5.3) k + Before urning o he main reul of hi ecion, we would like o commen on Aumpion H 2 ). In hi explanaion we e n = 1 for he ake of impliciy. On he one hand, by Lemma 3.2, a given,y,z), f y,ω +ε k h,y,z) converge in probabiliy o f y,ω,y,z) a n goe o infiniy. On he oher hand, f y,ω, ) i coninuou by aumpion. Thu, Condiion H 2 ) i ju requiring join coninuiy of f y in L 2 [,T). The ame commen hold for f z. Noe finally, ha ince f y i aumed o be bounded, a ufficien condiion for H 2 ) o hold rue i ha f y,y k,z ) converge in probabiliy o f y,y,z ) for d-almo every and he ame for f z ). Theorem 5.1. Le be in [,T. Under Aumpion L), D), H 1 ) and H 2 ), Y belong o D 1,2 ) n and Z j,: ) L 2 [,T;D 1,2 ) d ), j {1,...,n}. Proof. We only conider he cae where 5.2) hold in Aumpion H 2 ), ince he oher one can be reaed imilarly. We prove fir ha Y j belong o D 1,p for any j in {1,...,n} where p 1,2) i he exponen appearing in Aumpion H 1 ), and hen we exend he reul o D 1,2. To hi end we aim a applying Theorem 4.1. Fix j in {1,...,n}. Le h in H. Since Y j i F-progreively meaurable, by Lemma 3.3, we can aume wihou lo of generaliy ha ḣ = for >. Le ε >. By Lemma 3.1 and 3.4, i hold ha Y j τ εh = ξ j τ εh + f j r,y r,z r ) τ εh dr A a conequence, eing for he ake of impliciy Z j,: r ) τ εh dw r, [,T, P a.. Y ε := 1 ε Y τ εh Y ), Z ε := 1 ε Z τ εh Z ), ξ ε := 1 ε ξ τ εh ξ), [,T, we have for any j {1,...,n} wih T d Y ε )j = ξ ε ) j + Ã ε r +Ãy,ε r Yr ε + fz j ):,k r, +εh,y r τ εh, Z r k ) Zε r )dr ):,k k=1 Z ε r) j,: ) dw r, 5.4) Ã y,ε r := f j yr, +εh,ȳ ε,h r,z r ), Ã ε r := 1 ε fj r, +εh,y r,z r ) f j r,,y r,z r )), where Ȳ r ε,h i a convex combinaion of Y r and Y r τ εh and where for any k {1,...,d}, we have Z r k := Z:,1 r τ εh,...,z r :,k 1 τ εh, Z r :,k,z r :,k+1,...,z r :,d ) where Z r :,k i a convex combinaion of Z r :,k τ εh and Z r :,k. Under Aumpion D) and L), he following linear BSDE on [,T ha a unique oluion Ŷ h,ẑh ) in S 2 n H2 n,d wih Ŷ h := Ŷ h ) j ) j {1,...,n}, Ẑ h := Ẑh ) j,: ) j {1,...,n} and for any j {1,...,n} Ŷ h ) j = ξ j ),h H + f j )r,,y r,z r ),h H +fyr,,y j r,z r ) Ŷh r 16

17 + d fz j ):,k r, +εh,y r,z r ) Ẑh r ):,k) dr k=1 Ẑh r )j,: ) dw r. 5.5) Uing a priori eimae ee Propoiion 3.2 in [6) in L p, we have for ome conan C p, independen of ε E [ up Y ε )j Ŷ h )j +E p [,T [ C p E [ ξ ε ) j ξ j ),h H p +E ) p/2 Z ε )j,: ) Ẑh )j,: ) 2 d [ [ ) p +C p E Ãy,ε fy j,,y,z ) Ŷ h d [ d +C p E k=1 ) p ) d Ãε fj ),,Y,Z ),h H ) p fz) j :,k, +εh,y τ εh, Z ) f k z) j :,k, +εh,y,z ) Ẑh ) :,k d. Since ξ j i in D 1,2, lim ε E [ ξ ε ) j ξ j ),h H p = by Lemma 4.2. By Aumpion H 1 ), he econd erm in he righ-hand ide of 5.6) goe o a ε goe o. For he la wo erm, we will ue Aumpion H 2 ). Fir, he above eimae implie direcly ha Y j τ εh Y j,z j,: ) τ εh Z j,: ) ) ε goe o in S q H q d a ε goe o for any q 1,2). We can herefore conclude wih Aumpion H 2 ), ogeher wih he fac ha fy j i bounded, ha by he dominaed convergence heorem: [ ) p E Ãy,ε fy j,,y,z ) Ŷ h d CE [ 1 ) p [ 2 1 T ) p 2 Ãy,ε fy j,,y,z ) 2 d E Ŷ h 2 d. ε We can how imilarly ha he la erm on he righ-hand ide of 5.6) alo goe o, by uing he fac ha for any j {1,...,n}, Ẑh ) j,: ) H 2 d. I ju remain o prove ha for any j {1,...,n}, Ŷ h ) j i a random operaor on H or equivalenly ha here exi DY j an H-valued r.v. uch ha Ŷ h)j = DY j,h H for any h in H. To hi end, le h k ) k be an orhonormal yem in H, we e for any j {1,...,n}, l {1,...,d}: DY j := Ŷ h k ) j h k, DZ j,l := Ẑh k ) j,l )h k. k 1 k 1 5.6) Noe ha hee elemen are well-defined, ince one can prove ha DY j L 2 H) and ha DZ j,l L 2 [,T;H). Indeed uing once again a priori eimae for affine BSDE, here exi C > which may differ from line o line) uch ha: n n d E DY j 2 H + DZ j,l 2 H d j=1 j=1 l=1 17

18 = n Ŷ k 1E h k ) j 2 + j=1 n d j=1 l=1 C n ξ k 1E j ),h k H 2 + C j=1 j=1 n [ E ξ j ) 2 H + n j=1 Ẑh k ) j,l 2 d f j ),Y,Z ),h k H 2 d f j ),Y,Z ) 2 Hd < +, 5.7) by our aumpion on ξ and f. We now idenify Ŷ h)j repecively Ẑh )j,l ) wih he inner produc DY j,h H repecively DZ j,l,h H ). For any, i hold ha: DY j,h H = ξ j ),h k H h k,h H + h k,h H f j )r,y r,z r ),h k H dr k 1 k 1 + T d h k,h H fy j r,y r,z r ) Ŷh k r + fz j ):,l r, +εh,y r,z r ) Ẑh k r )dr ):,l k 1 + h k,h H Ẑh k r )j,: ) dw r k 1 T = ξ j ),h H + l=1 f j )r,y r,z r ),h H +f j yr,y r,z r ) + d h k,h H fz) j :,l r, +εh,y r,z r ) Ẑh k r ) )dr :,l k 1 + l=1 h k,h H Ẑh k r )j,: ) dw r, k 1 k 1 Ŷ h k r h k,h H where we juify he exchange beween he erie and he Riemann inegral by Fubini Theorem. Concerning he Wiener inegral we make ue of he ochaic Fubini Theorem ee e.g. [26) ince by a priori eimae: [ 1/2 E h k,h H Ẑh k ) j,: 2 d k 1 C [ 1/2 h k,h H E ξ j ),h k H 2 + f j )r,y r,z r ),h k H 2 dr k 1 CE [ ξ j ) 2H + f j )r,y r,z r ) 2 H dr < +, where C i a conan which may vary from line o line. By eing we obain DY h := DY j,h H ) j {1,...,n}, DZ h := DY j,h H = ξ j ),h H + DZ j,l,h H )1 j n, 1 l d, f j )r,y r,z r ),h H +f j y r,y r,z r ) DY h 18

19 + d fz j ):,l r, +εh,y r,z r ) DZr )dr h ):,l + l=1 DZ h r )j,: ) dw r, Thu, by uniquene of he oluion o affine BSDE wih quare inegrable daa, i hold ha Ŷ h)j = DY j,h H in L 2 R) and Ẑh ) j,: 1 [,T = DZ h ) j,: in H 2 d for any h in H. Thu, uing Eimae 5.6) we have proved ha for any h in H, [ lim E Y ε ε )j DY j,h H p =. Hence by Theorem 4.1, Y j belong o D 1,p and Y j = DY j. If we e D Z j,: ) dw := k 1 he ochaic Fubini Theorem implie ha: D Z j,: ) dw,h = H Ẑh k ) j,: ) dw h k, DZ h )j,: ) dw. Moreover, Burkholder-Davi-Gundy inequaliy implie ha here exi C p > uch ha E[ ε 1 Zr j,: ) dw r τ εh Zr j,: ) dw r ) D = E[ ε 1 Zr j,: ) dw r τ εh [ E up T [ ) p/2 C p E Zr ε )j,: ) DZr h )j,: ) 2 dr = C p E [ Z j,: r ) dw r ) ε 1 Z j,: r ) τ εh Z j,: r ) ) DZ h r )j,: ) ) dw r p ) p/2 Zr) ε j,: ) Ẑh r) j,: ) 2 dr, Zr j,: ) dw r,h H p DZr h )j,: ) dw r p where we have ued in he la inequaliy he fac ha Ẑh ) j,: 1 [,T = DZ h ) j,: in H 2 d in H. The righ-hand ide above end o a ε goe o, once again by 5.6). Therefore, for any h Z j,: ) dw D 1,p and Z j,: ) dw = D Z j,: ) dw. Furhermore, by he compuaion 5.7) we deduce ha Y j belong o D 1,2 and ha Zj,: ) dw belong o D 1,2 which, by [2, Lemma 2.3 implie ha Z j,: belong o L 2 [,T;D 1,2 ) d ). Finally, o mach wih he noaion of he paper [2, 9 le u denoe by DY j and DZ j,: he derivaive of repecively Y j and Z j,:. We alo define D Y R n d by D Y) j,l := D Y j ) l, 1 j n, 1 l d, 19

20 and imilarly for D ξ, D f and D Z :,k, 1 k d. Le u alo define for any 1 j n, D Z j,: R d n by D Z j,: ) k,l := D Z j,k ) l, 1 k d, 1 l n. We hu obain uing he chain rule formula D Y ) j,l = D ξ) j,l + + D f) j,l r,y r,z r )+f j yr,y r,z r ) D Y r ) j,: ) ) dr d fz j ):,k r,y r,z r ) D Zr k,: )j,: ) dr k=1 which can be inerpreed a an affine BSDE. D Z j,: r ):,l dw r, 5.8) Remark 5.1. We would like o poin ou ha ince each proce Z j i defined a a H-valued r.v., one may be careful no o udy Z direcly a a given ime, a Z i no well defined for a given. Hence, in he proof we raher udy a any ime he random variable Z dw and prove ha i belong o D 1,2. Then by [2, Lemma 2.3 he laer reul i equivalen o he fac ha Z belong o L 2 [,T;D 1,2 ). Remark 5.2. We emphaize ha our crierion can alo be ued o udy higher-order differeniabiliy properie for Y, Z). For inance, he pair DY, DZ) i ielf he oluion of a linear) BSDE. Therefore, a long a one i able o derive appropriae a priori eimae for hi BSDE, he mehodology above can hen be applied o obain condiion enuring econd-order Malliavin differeniabiliy of Y, Z). Noice nonehele ha when handling higher order derivaive, produc of lower order derivaive appear. One may hen need o add condiion on he coefficien enuring rong inegrabiliy properie of Y,Z) and heir Malliavin derivaive. 6 Applicaion and dicuion of he reul For impliciy, in hi ecion we will enforce ha n = d = Applicaion o FBSDE We conider in hi ecion a FBSDE of he form X = X + Y = gx T )+ b,x )d+ f,x,y,z )d σ,x )dw, [,T, P a.. where X R. We make he following Aumpion: Z dw, [,T, P a.., 6.1) A 1 ) b,σ : [,T R R are coninuou in ime and coninuouly differeniable in pace for any fixed ime and uch ha here exi k b,k σ > wih b x,x) k b, σ x,x) k σ, for all x R. Beide b,),σ,) are bounded funcion of. A 2 ) i) g i coninuouly differeniable wih polynomial growh. 2

21 ii) f : [,T R 3 R i coninuouly differeniable in x,y,z) wih bounded fir parial derivaive in y,z uniformly in, uch ha E[ f,,,) 2 d < + and aifying for ome C > q,κ) R + [,2), f x,x,y,z) C1+ y κ + z κ + x q ),,x,y,z) [,T R 3. Noice ha under A 1 ) and A 2 ), he FBSDE 6.1) admi a unique oluion X,Y,Z) ee [19). The well-known following lemma provide he exience of a Malliavin derivaive for X for all [,T under Aumpion A 1 ) ee e.g. [18, Theorem 2.2.1). Lemma 6.1. Under Aumpion A 1 ), for any p 1, X D 1,p for all [,T, and X S p. The following Propoiion how ha Aumpion A 1 ) and A 2 ), which are acually weaker han PP1) and PP2), imply our new aumpion H 1 ) and H 2 ). A a corollary, uing Theorem 5.1, we recover he original reul [2, Propoiion 2.2. Propoiion 6.1. Le X, Y, Z) be he unique oluion of FBSDE 6.1). Under Aumpion A 1 ), A 2 ), Aumpion L), D), H 1 ) and H 2 ) hold. Remark 6.1. We ini on he fac ha in he Markovian cae, he original aumpion PP1) and PP2) of [2 imply direcly our new aumpion H 1 ) and H 2 ) while Aumpion EPQ1) and EPQ2) of [9 are ricly ronger han PP1) and PP2). In oher word, in he Markovian cae our aumpion are enough o recover he original reul [2, Propoiion 2.2, wihou any addiional condiion. Proof. [Proof of Propoiion 6.1 From Lemma 6.1, Propery D) hold by he chain rule formula and L) follow from our aumpion. I remain o prove H 1 ) and H 2 ). We ar wih H 1 ). Le 1 < p < 2 and h in H. Below C denoe a poiive conan which can differ from line o line. Recall ha from our aumpion, E [ up Y r + [,T ) r/2 Z 2 d <, r ) Denoing by X a random poin beween X and X τ εh, where we uppreed he dependence on ε for noaional impliciy. We have for any in [, T, ha ε E[ 1 f,x τ εh,y,z ) f,x,y,z )) f x,x,y,z ) DX,ḣ H p [ X τ εh X = E f x, ε X,Y,Z ) f x,x,y,z ) DX,ḣ p H ε CE[ 1 X τ εh X ) DX,ḣ H p 1+ Y κp + Z κp + X pq + X τ εh pq ) fx +CE[, X,Y,Z ) f x,x,y,z ) p DX,ḣ H p ε CE[ 1 X τ εh X ) DX,ḣ H pr 1 r E [ 1+ Y κp + Z κp + X pq + X τ εh pq ) r1 r fx +CE[, X,Y,Z ) f x,x,y,z ) p DX,ḣ H p =: A 1,ε +A 2,ε, 21

22 where r > 1 and p are choen o ha pκ r < 2 and r denoe he Hölder conjugae of r. Uing he above eimae, we deduce [ ) p E ε 1 f,x τ εh,y,z ) f,x,y,z )) f x,x,y,z ) DX,ḣ Hd Then, we have A 1,ε +A 2,ε ) d. A 1,ε d C ε E[ 1 X τ εh X ) DX,ḣ H pr ) 2/rd 1/2 E [ ) 1/2 1+ Y κp + Z κp + X pq + X τ εh pq ) r 2/ r d. 6.3) In addiion by Lemma 3.4, we have ha M ε,h := X τ εh X i oluion o he linear SDE: dm ε,h = M ε,h b x,x )d+σ x,x )dw )+εσ,x τ εh )ḣd, where X denoe once again a random poin beween X and X τ εh. Hence uing Aumpion A 1 ) and andard eimae for SDE, we ge ha for any q 1, [ lim E ε up X τ εh X q =. [,T Following he ame line a above, and recalling ha N h := DX,ḣ H i oluion o he SDE: dn h = N h b x,x )d+σ x,x )dw )+σ,x )ḣd, we ge ha he proce P ε,h := ε 1 X τ εh X) DX,ḣ H i oluion o he affine SDE: dp ε,h = dh ε +P ε,h b x,x )d+σ x,x )dw ), wih ) dh ε := DX,ḣ Hb x,x ) b x,x ))+ḣσ,x τ εh ) σ,x )) d + DX,ḣ Hσ x,x ) σ x,x ))dw Uing he fac ha σ x,b x are bounded, σ ha linear growh and i coninuou, we ge by imilar compuaion han hoe done everal ime in hi paper ha: [ lim E ε up H ε q =, q 1, [,T from which we deduce uing he explici repreenaion of oluion o affine SDE ee e.g. [21, Theorem V.9.53) ha [ lim E ε up ε 1 X τ εh X) DX,ḣ H q =, q 1. [,T 22

23 A a conequence, combining hi eimae wih 6.3), we ge ha: A 1,ε d C E [ 2/r up ε 1 X τ εh X ) DX,ḣ H pr [,T which goe o a ε goe o, ince we recall ha we have choen p, r > 1 o ha κp r < 2, which implie by 6.2), Lemma 6.1 and he Cameron-Marin formula ha E [ 1+ Y κp + Z κp + X pq + X τ εh pq ) r 2/ r d <. Concerning he erm A2,ε d, chooing p > 1 o ha p p < 2, i hold by Hölder and by Jenen inequaliie ha ince A i hold ha A 2,ε fx d C E[, X,Y,Z ) f x,x,y,z ) p p 1/ p d), E [ lim E ε up DX,ḣ H q <, q > 1. [,T [ up X τ εh X q =, q 1 [,T limf x, X,Y,Z ) f x,x,y,z ) p p =, P d a.e. ε Furhermore, for any 2 > ρ > 1, up E[ f x, X,Y,Z ) f x,x,y,z ) ρp p d ε,1) C up ε,1) [ E 1+ X q + X τ εh q + Y κ + Z κ ) ρp p d <, by chooing p mall enough o ha κρp p 2. So by Lebegue dominaed convergence heorem, lim A 2,ε ε d =, which prove H 1 ). Concerning, H 2 ) we ju menion ha f y repecively f z ) i bounded, joinly coninuou in x,y,z) and we make ue of Lemma Affine BSDE The aim of hi ecion i o prove ha wih our condiion, we can provide weaker condiion compared o [9 for affine BSDE. We ake a driver of he form f,ω,y,z) := α ω)+β ω)y +γ ω)z 1/2, 23

24 wih bounded F-progreively meaurable procee uch ha α,β,γ L 2 [,T;D 1,2 ), and ξ in D 1,2. The condiion given in [9, Propoiion 5.3 for proving ha he aociaed oluion Y,Z) i Malliavin differeniable read a follow ogeher wih ome meaurabiliy condiion): η > uch ha E[ ξ 2+η < + and wih K θ ) := D θ β) + D θ γ). [ 2+η 1/2+η) E K θ ) d) 2 dθ < +, In our eing, one need o check aumpion L), D), H 1 ) and H 2 ). A menioned below by Lemma 3.2 Condiion H 2 ) come for free, and Aumpion D) and L) are alo rivially aified. The inereing poin i ha H 1 ) i rue a oon a 6.4) i replaced wih: ε η > uch ha lim E[ 1 µ τ εh µ ) Dµ,ḣ H 2+η d =, for µ {β,γ}. ε 6.5) Hence our condiion only involve a condiion on γ and β and no on ξ. For inance if β and γ are given a: β = ϕ 1 X ), γ := ϕ 2 X ), [,T, wih ϕ 1,ϕ 2 wo mooh funcion wih polynomial growh and X i he oluion o an SDE of he form of he one conidered in Secion 6.1, hen he requiremen of Condiion 6.4) and 6.5) are aified for β and γ, however in conradiincion o Condiion 6.4), Aumpion 6.5) doe no pu exra regulariy on he erminal condiion ξ. We make precie our reul. Propoiion 6.2. Le ξ in D 1,2, and α,β,γ bounded F-progreively procee in L 2 [,T;D 1,2 ) uch ha Dα,Dβ and Dγ are F-progreively meaurable. Aume ha Aumpion 6.5) i in force. Then for any in [,T, Y belong o D 1,2, Z L 2 [,T;D 1,2 ) where Y,Z) i he unique oluion in S 2 H 2 o he affine BSDE: Y = ξ + α +β Y +γ Z )d Z dw, [,T. Proof. Once again we check ha aumpion of Theorem 4.1 are in force. Properie D) and L) are immediaely aified. Le f,ω,y,z) := α ω) + β ω)y + γ ω)z. Since f y,ω,y,z) = β ω), and f z,ω,y,z) = γ ω) we immediaely ge by Lemma 3.2 and ince β,γ are bounded ha H 2 ) i aified. Concerning H 1 ), we have for any 1 < p < 2 and h in H, ha [ ) p E ε 1 f, +εh,y,z ) f,,y,z )) Df,,Y,Z ),ḣ H d CE +CE [ [ ) p ε 1 α τ εh α ) Dα,ḣ H d ) p Y ε 1 β τ εh β ) Dβ,ḣ H) d θ 6.4) 24

25 +CE [ ) p Z ε 1 γ τ εh γ ) Dγ,ḣ H) d =: A ε 1 +Aε 2 +Aε 3, 6.6) where C i a conan. By Lemma A.2 we have ha lim ε A ε 1 =. We conider he erm Aε 3. We have ha: [ A ε 3 E Z ε 1 γ τ εh γ ) Dγ,ḣ H) p d C C E [ Z 2 [ ε p/2 E 1 γ τ εh γ ) Dγ,ḣ H E [ Z 2 ) p/2 d 2p 2 p 2 p 2 d ε E[ 1 γ τ εh γ ) Dγ,ḣ H 2p 2 p ) 2 p 2 d. 2p Chooing p uch ha 2 p = 2 + η we ge ha Aε 3 converge o a ε end o by 6.5). Similarly, lim ε A ε 2 = for hi choice of p. Remark 6.2. Noe ha, ince he BSDE i affine, Y can be expreed explicily a: [ Y = E M,T ξ M, α d F, where M, := exp γ u dw u 1 2 γ u 2 du+ ) β u du, [,T. Hence, on he one hand, Y belong o D 1,2 if and only if he coefficien α,β,γ belong o L 2 [,T;D 1,2 ) and ξ i in D 1,2. The ame concluion follow for he Z componen. Hence, neiher our condiion 6.5) nor he one of [9, namely 6.4), are harp. However, boh are harp in he cae where β = γ =. On he oher hand, Condiion 6.4) or 6.5) give more informaion han he imple fac ha Y, Z are Malliavin differeniable, ince hey imply ha he BSDE olved by DY,DZ) i limi in S 2 H 2 of repecively DY n,dz n ) where Y n,z n ) i he oluion o he Picard ieraion equaion a order n approximaing Y,Z)) for 6.4), and of he difference quoien ε 1 Y τ εh Y),ε 1 Z τ εh Z)) in our cae 6.5). 6.3 Dicuion and comparion of he reul We would like before going o he quadraic BSDE cae o make a commen abou he difference beween our approach and he one of [2, 9 and our approach. In hee reference, he auhor conider he equence of BSDE: Y n = ξ + f,y n 1,Z n 1 )d Z n dw, [,T, which approximae in S 2 H 2 he oluion o he original BSDE: Y = ξ + f,y,z )d Z dw, [,T. 25

26 Now, under mild aumpion on f, he procee Y n,z n ) are Malliavin differeniable and i hold ha a verion of D r Y n,d rz n ) aifie for [,T, r : D r Y n = D r ξ + [D r f,θ n 1 D r Z n dw, )+ y f,θ n 1 )D r Y n 1 + z f,θ n 1 )D r Z n 1 d wih Θ n 1 := Y n 1,Z n 1 ). On he oher if Y, Z) where Malliavin differeniable we would have ha a verion of D r Y,D r Z ) would aify for [,T, r : D r Y = D r ξ+ [D r f,y,z )+ y f,y,z )D r Y + z f,y,z )D r Z d D r Z dw. In oher word, auming ha y f and z f o be coninuou, we would ge formally ha DY n,dz n ) converge o DY,DZ) in S 2 H 2 ) a n goe o infiniy provided ha a he limi one can replace D r f,y n 1,Z n 1 ) by D r f,y,z ) which i exacly where come he main aumpion in [9, 2 which impoe D r f o be ochaic) Lipchiz coninuou in y,z) wih inegrabiliy condiion on he Lipchiz conan o make he aforemenioned argumen rigorou. However, i i no a neceary condiion for Y,Z) o be Malliavin differeniable ha DY n,dz n ) o converge o DY,DZ). However, for Y o be in D 1,2, i i neceary and ufficien) ha ε 1 Y τ εh Y ) converge in L p for ome p < 2 o DY,ḣ H for any h in H according o Theorem 4.1). Hence, hi i an advanage of our condiion. 7 Exenion o quadraic growh BSDE The aim of hi ecion i o exend our previou reul o o-called quadraic growh BSDE. Some reul for hee equaion already exi in he lieraure, ee in paricular [1, 13 or he hei [8, however hey are generally limied o pecific form of he generaor or o a Markovian eing. We will how ha our approach o he Malliavin differeniabiliy i flexible enough o be able o rea hi problem wihou major modificaion o our proof. Since he wellpoedne heory for mulidimenional quadraic BSDE i ill an open problem, we enforce n = 1 hroughou hi ecion. We will now li our aumpion in hi quadraic eing D ) ξ i bounded, belong o D 1, and i Malliavin derivaive Dξ i bounded, for any y,z) R R d,,ω) f,ω,y,z) i in L 2 [,T;D 1, ), f,y,z) and Df,y,z) are F-progreively meaurable, Df,y,z) i uniformly bounded in y,z. Q) The map y, z) f, y, z) i coninuouly differeniable and here exi ome conan C > uch ha for any,ω,y,z,z ) [,T Ω R R d R d f,ω,y,z) f,ω,y,z ) C 1+ z + z ) z z, f,ω,,) C, f y,ω,y,z) C, f z,ω,y,z) C1+ z ), where f z = f z l denoe he gradien of f wih repec o he z variable. )l {1,...,d} 26

27 H 1, ) For any p > 1 and for any h H [ lim E f, +εh,y,z ) f,,y,z ) ε ε ) Df,,Y,Z ),ḣ p H d =. H 2, ) Le ε k ) k N be a equence in,1 uch ha lim ε k =, and le Y k,z k ) k be a equence k + of random variable which converge in S p H p d for any p > 1 o ome Y,Z). Then for all h H, he following convergence hold in probabiliy f y,ω +ε k h,y k,z ) f y,ω,y,z ) f z,ω +ε k h,y k,z k ) f z,ω,y,z ) or f y,ω +ε k h,y k,z k ) f y,ω,y,z ) f z,ω +ε k h,y,z k ) f z,ω,y,z ) L 2 [,T;R) L 2 [,T;R d ) L 2 [,T;R) L 2 [,T;R d ) k +, 7.1) n + k ) k + Le S be he e of F-progreively meaurable procee Y uch ha up [,T Y i bounded and H 2 BMO he e of Rd -valued predicable procee Z uch ha: eup τ T [ E τ Z 2 d F τ < +, P a.., where T denoe he e of F-opping ime wih value in [,T. We ar by recalling he following by now claical reul on quadraic growh BSDE and ochaic Lipchiz BSDE, which can be found among oher in [13. Propoiion 7.1. Under Aumpion D ) and Q), he BSDE 5.1) and 5.5) boh admi a unique oluion in S H 2 BMO. We have he following exenion of Theorem 5.1. Theorem 7.1. Le be in [,T. Under Aumpion D ), Q), H 1, ) and H 2, ), Y belong o D 1, and Z L [,T;D 1,2 ) d ). Proof. We follow he proof of Theorem 5.1), uing he ame noaion. Since he BSDE are now quadraic, we can ue he a priori eimae of Lemma A.1 in [13 o obain ha for any p > 1, here exi ome q > 1 uch ha E [ up [,T Y ε Ŷh 2p +E [ [ C p E [ ξ ε Dξ,ḣ H pq 1/q T +E ) p Z ε Ẑh 2 d [ ) pq 1/q +C p E Ãy,ε f y,,y,z ) Ŷ h d 27 ) pq 1/q à ε Df,,Y,Z ),ḣ Hd

28 +C p E where we e [ Ãz,ε f z,,y,z ) Ẑh d ) pq 1/q, 7.3) à y,ε := f y, +εh,ȳ ε,h,z ) à ε := 1 ε fr, +εh,y,z ) f,,y,z )), where Ȳ ε,h r i a convex combinaion of Y r and Y r τ εh and à z,ε := Ãz,ε )k ) k {1,...,d} wih Ãz,ε ) k := f z, +εh,y τ εh, Z k r), and Z k r i a in he proof of Theorem 5.1). Since ξ D 1,, he fir erm on he righ-hand ide above goe o hank o Theorem 4.1. Moreover, he econd erm alo goe o hank o Aumpion H ). Then, ince f y i bounded by Aumpion Q) and ince Ỹ h S by Propoiion 7.1, we can eaily conclude wih Aumpion H 2 ) and he dominaed convergence heorem ha he hird erm on he righ-hand ide alo goe o. Le u now concenrae on he fourh erm involving he conrol variable. By Cauchy-Schwarz inequaliy we have ha [ ) pq E Ãz,ε f z,,y,z ) Ẑh d [ pq 1/2 [ pq 1/2 E Ãz,ε f z,,y,z ) d) 2 E Z d) h ) Since Ŷ h,ẑh ) i he oluion o he ochaic linear BSDE 5.5) wih bounded coefficien Df and f y by D )) and f z,y,z ) i in H 2 BMO ince f z,y,z ) C1 + Z ) by Aumpion Q)), we deduce ha Ẑh H 2 BMO which implie ha Ẑh H m d for any m > 1 by he energy inequaliie. Furhermore, for any η > i hold ha [ ) pq+η E Ãz,ε f z,,y,z ) Ẑh d CE [ CE ) pq+η 1+ Z + Z τ εh ) Ẑh d ) pq+η 1+ Z + Z τ εh ) 2 2 T d Ẑh 2 d [ [ pq+η 1/2 CE 1+ Z + Z τ εh ) d) 2 E C 1+E [ ) pq+η 2 ) pq+η 1/2 Ẑh 2 d ) p 1/q [ pq+η 1/2 Z 2 d E Ẑh d) 2 < +, where p,q > 1 uing Hölder[ Inequaliy and Propoiion[ 3.1. Hence, aking limi a ε goe o ) in 7.4) we ge ha lim ε E up [,T Y ε Ŷh +E 2p T p Z ε Ẑh 2 d =. Following 28

29 he ame line a in he proof of Theorem 5.1, one can ue a priori eimae for quadraic growh BSDE o obain ha Ŷ h and Ẑh are linear operaor-valued r.v.. Thi prove ha Y and Z dw belong o D 1, by Theorem 4.1. In paricular, Z1 [,T belong o L 2 [,T;D 1,2 ) d ) ee [2). Moreover, ince D Y,D Z) i he oluion of he ochaic linear BSDE 5.8) for any [,T and Aumpion D ) and Q) hold, from he relaion D Y ) j = Z ) j for any j in {1,...,d}, and for all [,T we obain Z1 [,T L [,T;D 1,2 ) d ). Remark 7.1. We would like o poin ou ha our condiion cover he cae of Markovian quadraic BSDE preened in [11, Theorem 2.9. Indeed, aume ha we conider a forwardbackward yem of he form 6.1) where he oluion proce X o he forward SDE i m- dimenional wih m a poiive ineger o ha we mach wih he noaion and aumpion of [11, Theorem 2.9) under aumpion, D ), Q), A 1 ), A 2 )i) and where A 2 )ii) i replaced by he following aumpion: A 2 )ii ) f : [,T R 3 R m R R d i coninuouly differeniable in x,y,z) and aifying for ome C > q R +, f x,x,y,z) C1+ y + z 2 + x q ),,x,y,z) [,T R m R R d, where f x := f x l denoe he gradien of f wih repec o he variable x. Under hee )l {1,...,m} aumpion, we can check ha H 1, ) and H 2, ) are in force. To ee hi we ju make a commen abou how he proof of Propoiion 6.1 ha o be modified o obain H 1, ), wherea H 2, ) i me rivially. Uing he noaion of hi proof one can manage a erm of he form: [ ) p E ε 1 X τ εh X ) DX,ḣ H f x, X,Y,Z ) d a follow: [ ) p E ε 1 X τ εh X ) DX,ḣ H f x, X,Y,Z ) d CE CE [ [ up [,T [,T X τ εh X ε DX,ḣ H p 1/2 up ε 1 X τ εh X ) DX,ḣ H 2p [ 2p 1/2 E 1+ X q + X τ εh q + Y + Z )d) 2, ) p 1+ X q + X τ εh q + Y + Z 2 )d which goe o a ε goe o ince Z belong o H 2 BMO involving A 2,ε can be reaed imilarly. and ince Y i bounded. The erm A Appendix The following lemma wa remarked in [24, Remark 2 wih he e of polynomial cylindrical funcion P, we provide a proof of i wih he e of cylindrical funcion S. 29