On the Malliavin differentiability of BSDEs

Transcription

1 On he Malliavin differeniabiliy of BSDE Thibau Marolia Dylan Poamaï Anhony Réveillac Univerié Pari-Dauphine CEREMADE UMR CNRS 7534 Place du Maréchal De Lare De Taigny Pari cedex 16 France April 3, 214 hal , verion 1-3 Apr 214 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 5), and a hey provide a probabiliic repreenaion for oluion o emi-linear parabolic PDE, via a non-linear Feynman-Kac formula (ee 12). Before going furher le u recall ha hi cla of equaion ha been inroduced in 2, 11, 12 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 which i called he erminal condiion (a Y T = ξ) 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 marolia@ceremade.dauphine.fr poamai@ceremade.dauphine.fr anhony.reveillac@ceremade.dauphine.fr 1

2 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.. 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 12 in a Markovian eing, ha i when ξ := g(x 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: hal , verion 1-3 Apr 214 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 12, 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 12 he auhor make one aumpion for he whole paper which i ronger han (PP1)- (PP2) above, however a careful reading of he proof of 12, 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 excep ha he laer condiion can be eaily generalized o he non-markovian framework. Unforunaely, he fir reul in ha direcion which wa given by El Karoui, Peng and Quenez in 5 require more ringen condiion han he aforemenioned inuiive one. More explicily, he main reul in 5 concerning he Malliavin differeniabiliy of he oluion o he BSDE (1.1) (eenially) involve he following condiion (ee 5, 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 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 5, p. 58 2

3 Roughly peaking, hi mean ha ξ and ω f(, ω, y, z) have o be Malliavin differeniable (which i inuiively he minimal expeced requiremen), 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 ha he Malliavin derivaive of he driver f i Lipchiz coninuou in (y, z) wih a ochaic Lipchiz conan K which i ufficienly inegrable. 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 5, 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 ha cae, one can check ha in he Markovian framework, Condiion (EPQ1)-(EPQ2) are ricly ronger han Condiion (PP1)-(PP2). hal , verion 1-3 Apr 214 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, which ue a fundamenally differen approach from 5, 12, 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 5. 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 ZdW n,,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 (ha i in he domain of he Malliavin derivaive D 1,2 which i a Banach pace equipped wih a Sobolev ype norm), hen o i (Y n,z n ). Then, i ju remain o prove ha hi propery exend o Y,Z which are limi (in he appropriae pace) of repecively Y n and Z n. 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 5 canno be opimized in he general cae. Even hough hi idea eem prey naural, i i baed on a choice which i 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 which converge 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 a ene 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 lihe bi more on hi poin in Secion 6.3. Regarding he dicuion above, one could hink of rying o find a equence of procee which are known o approximae he Malliavin derivaive of Y (and Z) when Y i Malliavin differen- 3

4 hal , verion 1-3 Apr 214 iable. 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 9, 15, 14, 8, 16. In addiion, Sugia proved in 16 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 16 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 12, 5 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 12, 5. We proceed a follow, we ar below wih ome preliminarie. Then we urn o ome elemen of analyze on he Wiener pace in Secion 3. 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 >. Le Ω := C (,T,R) be he canonical Wiener pace of coninuou funcion ω from,t o R uch ha ω() =. Le W := (W ),T be he canonical Wiener proce, ha i, for any ime in,t, W denoe he evaluaion mapping: W (ω) := ω for any elemen ω in Ω. 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 1 we refer o Secion 4 where hi noion i defined 4

5 impliciy, by H := L 2 (,T;R) and by, H i canonical inner produc, ha i o ay f,g H := f()g()d, (f,g) L 2 (,T;R) L 2 (,T;R). 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, ḣ H, h() = ḣ(x)dx,,t, which 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. For any h in H, we will alway denoe by ḣ a verion of i Radon-Nykodym deniy wih repec o he Lebegue meaure. 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 hal , verion 1-3 Apr 214 F L p (K) := ( E F p K) 1/p. Le S be he e of polynomial cylindrical funcional, ha i he e of random variable F of he form F = f(w(h 1 ),...,W(h n )), (h 1,...,h n ) H n, f R n X, for ome n 1, (2.1) where W(h) := ḣdw for any h in H. 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 (W(h 1 ),...,W(h n ))h i, (2.2) i=1 where f xi := df dx i. I i hen cuomary o idenify F wih he ochaic proce ( F),T. 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 12, 5 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 he operaor by he following dualiy relaionhip: EFδ(u) = E F,u H, u dom(δ), where dom(δ) := { 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 dom(δ) and ha δ(gh) = GW(h) G,h H, (2.3) 5

6 ee for example 1, Relaion (1.46). We conclude hi ecion by inroducing he following norm and pace which are of inere when udying BSDE. For any p 1, we e S p he pace of R-valued, coninuou and F- progreively meaurable procee Y.. Y p S p := E up Y p < +. T We denoe by H p he pace of R-valued and F-predicable procee Z uch ha ( ) p 2 Z p H := E Z p 2 d < +. 3 Some elemen of analyi on he Wiener pace hal , verion 1-3 Apr 214 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. 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 17, 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), 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, 17). 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, 17). Le F be a F T -meaurable random variable. The mapping h F τ h i coninuou from H o L (R) 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. 17, 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 ( EF τ h = E F exp ḣ()dw 1 ) 2 ḣ() 2 d. 6

7 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 1. (3.1) 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 h1,, P a.. In paricular, F τ h i F -meaurable. Proof. I i well-known ha by definiion of P, any F -meaurable r.v. admi a F o - meaurable verion. Therefore, here exi ome meaurable map ϕ : Ω R, uch ha F = ϕ(w ), P a.. Hence, we deduce by Lemma 3.1 ha for P -a.e. ω Ω hal , verion 1-3 Apr 214 F τ h (ω) = ϕ(w (ω)) τ h = ϕ(w τ h (ω)) = ϕ(ω( )+h( )) = F τ h1, (ω). 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 and h in H. I hold ha Z dw τ h = Z τ h dw + Proof. Le S be he cla of imple procee X of he form n X = λ i 1 (i, i+1 (), i= Z τ h ḣ()d, P a.. where n N, = < 1 <... < n = T and where for any i n, (λ i ) i=1,...n are F i -meaurable and in L 2 (R). We ar by proving he reul for Z in S and hen we prove he reul for any elemen Z in H 2 uing a deniy argumen. Le Z S wih he decompoiion n Z = λ i 1 (i, i+1 (),,T. Then, for any h H and for every ω Ω, ( ( T n ) Z dw τ h )(ω) = λ i (W i+1 W i ) τ h (ω) = = = i= i= n λ i (ω +h)(w i+1 W i )(ω +h) i= n λ i τ h (ω)(ω( i+1 ) ω( i )+h( i+1 ) h( i )) i= Z τ h dw (ω)+ Z τ h (ω)dh, 7

8 hal , verion 1-3 Apr 214 which give he deired reul ince h i aboluely coninuou. We exend hi reul o procee Z in H 2. Le Z H 2, hen here exi a equence (Z n ) n N in S which converge o Z in H 2. 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 Z n τ hdw Z τ h dw T +E (Z n Z ) τ h dh. } {{ } =:C n Le u eimae hee hree erm. Fir, uing Propoiion 3.1, Burkholder-Davi-Gundy Inequaliy, hen Cauchy-Schwarz Inequaliy, we have T A n = E (Z Z n )dw e T ḣ()dw 1 2 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 τh. Therefore, we can ue Propoiion 3.1 and Cauchy-Schwarz Inequaliy, o alo deduce ha B n. Finally, we have n + ( ) T C n = E E 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, i hold ha: ince Z τ h ḣ()d =. Z dw τ h = Z τ h dw, P a.., (3.2) 8

9 4 A characerizaion of Malliavin differeniabiliy Before going furher we would like o recall he main finding of 16 which i ha any Malliavin- Sobolev ype paced 1,p a defined in Secion 2 (originally defined by Malliavin 9 and Shigekawa 14) agree wih he Sobolev pace (due o Sroock 15 and Kuuoka 8) D 1,p which coni in he e of r.v. F in L p (R) which are Ray Aboluely Coninuou (RAC) and Sochaically Gâeaux Differeniable (SGD) where hee noion are defined a follow: (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, F h (ω +h) i aboluely coninuou. (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 hal , verion 1-3 Apr 214 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 16, 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) 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. F belong o D 1,p if and only if here exi DF in L p (H) and q (1,p) uch ha for any h in H lim E F τ εh F ε ε DF,h H q =. We recall he following lemma which can be found e.g. in 16, Corollary 2.1 and 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: EFδ(Gh) = EG DF,h H, for every G S and h H. Then, i hold ha F D 1,p, DF = F, P a.. We now prove he following lemma for he Malliavin differeniabiliy of a given random variable. 9

10 hal , verion 1-3 Apr 214 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 Proof. Fix q in (1,p), h in H and η > uch ha q +η < p. We know from 16, Theorem 3.1 ha ince F i in D 1,p, F i (SGD), (RAC), and Relaion (4.2) hold rue. We hu have by 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 4.1. Noe ha he concluion of he previou Lemma may fail for q = p. We can now proceed wih he proof of Theorem 4.1. Proof of Theorem 4.1. The neceary par follow from Lemma 4.2. We urn o he convere implicaion. Le F be uch ha here exi DF in L p (Ω;H) and q (1,p) uch ha lim E F τ εh F q DF,h H =. ε ε The proof coni in applying Lemma 4.1 by proving he dualiy relaionhip EFδ(Gh) = EG DF,h H, G S, h H (4.3) which, by aking ε =, i ielf a conequence of he following relaion By Lemma A.1 (in he Appendix) i hold ha EF τ εh δ(gh) = E (DF) τ εh,h H G. (4.4) EF τ εh δ(gh) = d dε EF τ εhg 1 = lim η η E (F τ (ε+η)h F τ εh )G ( F τ(ε+η)h F τ εh = lim E (DF) τ εh,h H )G η η +E (DF) τ εh,h H G = E (DF) τ εh,h H G, (4.5) 1

11 where he proof ha he fir erm on he righ-hand ide goe o i repored below. Noe ha E (DF) τ εh,h H G < + ince (DF) τ εh,h H = DF,h H τ εh, P a.., G belong o all he pace L r (R) for r 1 and E DF,h H p h p H E DF p H < +. hal , verion 1-3 Apr 214 The Equaliy (4.5) i juified by Hölder inequaliy ince ( F τ(ε+η)h F τ εh E (DF) τ εh,h H )G η F τ (ε+η)h F τ εh r1 r E (DF) τ εh,h E G r H 1 r η F τ ηh F = E η F τ ηh F E η DF,h H r DF,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), which eablihe (4.4). Remark 4.2. Noe ha according o Lemma 4.2 he neceary par of Theorem 4.1 can be renghen ince he convergence of he difference quoien hold in any pace L q (R) wih q < p. Remark 4.3. From (4.4) we prove ha for any ε > and any h in H, F τ εh belong o D 1,p and ha (F τ εh ) = ( F) τ εh. 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 5, 12 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 Fd, DF coincide wih he Malliavin derivaive inroduced in 5, 12, 1. In paricular F,h H = DF,ḣ H for any h in H. We conider now he following BSDE: Y = ξ + f(,y,z )d Z dw,,t, P a.., (5.1) where ξ i a F T -meaurable r.v. and f :,T Ω R 2 R 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. 11

12 (D) ξ belong o D 1,2, for any (y,z) R 2, (,ω) f(,ω,y,z) i in L 2 (,T;D 1,2 ), f(,y,z) and Df(, y, z) are F-progreively meaurable, and E D f(,y,z ) 2 H d < +. (H 1 ) There exi p (1,2) uch ha 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 n ) n N be a equence in(,1 uch ha lim n + ε n =, and le(y n,z n ) n be a equence of random variable which converge in S p H p for any p 1,2) o ome (Y,Z). Then for all h H, he following convergence hold in probabiliy hal , verion 1-3 Apr 214 or f y (,ω +ε n h,y n,z ) f y (,ω,y,z ) H n + f z (,ω +ε n h,y n,z n ) f z (,ω,y,z ) H, (5.2) n + f y (,ω +ε n h,y n,z n ) f y (,ω,y,z ) H n + f z (,ω +ε n h,y,z n ) f z(,ω,y,z ) H. (5.3) n + Before urning o he main reul of hi ecion, we would like o commen on Aumpion (H 2 ). On he one hand, by Lemma 3.2, a given (,y,z), f y (,ω +ε n 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,R). 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 n,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 and Z L 2 (,T;D 1,2 ). Proof. We proceed in wo ep and only conider he cae where (5.2) hold in Aumpion (H 2 ), ince he oher one can be reaed imilarly. Sep 1: We prove ha Y belong o D 1,p where p (1,2) i he exponen appearing in Aumpion (H 1 ). To hi end we aim a applying Theorem 4.1. Le h in H. Since Y 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 1 Y τ εh = ξ τ εh + f(r,y r,z r ) τ εh dr Z r τ εh dw r,,t, P a.. 1 noe ha by Cameron-Marin formula he proce (Y τ εh ) i coninuou which enable u o ge ha he Ω-excepion e doe no depend on. 12

13 A a conequence, eing for impliciy 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 ha (Y ε,z ε ) olve he BSDE: wih Y ε = ξ ε + Ã ε r +Ãy,ε r Yr ε +Ãz,ε r Zrdr ε ZrdW ε r, (5.4) Ã ε r := 1 ε (f(r, +εh,y r,z r ) f(r,,y r,z r )), hal , verion 1-3 Apr 214 where Ȳ r ε,h Z r τ εh ). Ã y,ε r := f y (r, +εh,ȳ ε,h r,z r ), Ã z,ε ε,h r := f z (r, +εh,y r τ εh, Z r ), (rep. Zε,h r ) i ome inermediary poin beween Y r and Y r τ εh (rep. Z r and Under Aumpion (D) and (L), he following linear BSDE on,t i alo well-poed Ỹ h = Dξ,ḣ H+ Df(r,,Y r,z r ),ḣ H+Ỹ r h f y (r,,y r,z r )+ Z rf h z (r,,y r,z r )dr Z h rdw r. (5.5) Beide, we have (Ỹ h, Z h ) S 2 H 2. Uing a priori eimae (ee Propoiion 3.2 in 3) in L p, we have for ome conan C p, independen of ε ( E up Y ε Ỹh +E p,t C p (E ξ ε Dξ,ḣ H p +E ) p/2 Z ε Z h 2 d ( ) p +C p E Ãy,ε f y (,,Y,Z ) Ỹ h d ( ) p ) Ãε Df(,,Y,Z ),ḣ Hd ( ) p +C p E Ãz,ε f z (,,Y,Z ) Z h d. (5.6) Since ξ i in D 1,2, lim ε E ξ ε Dξ,ḣ H p = by Lemma 4.2. By Aumpion (H 1 ), he econd erm in he righ-hand ide of (5.6) goe oaεgoe o. For he la wo erm, we will ue Aumpion (H 2 ). Fir, he above eimae implie direcly ha (Y τ εh Y,Z τ εh Z) ε goe o in S q H q for any q (1,2). We can herefore conclude wih Aumpion (H 2 ) ogeher wih he fac ha f y i bounded ha by he dominaed convergence heorem: ( ) p E Ãy,ε f y (,,Y,Z ) Ỹ h d ( ) p/2 ( CE Ãy,ε f y (,,Y,Z ) 2 T ) p/2 d E Ỹ h 2 d ε. 13

14 We can how imilarly ha he la erm on he righ-hand ide of (5.6) alo goe o, by uing he fac ha Z h H 2. Hence by Theorem 4.1, Y belong o D 1,p. hal , verion 1-3 Apr 214 Moreover, Burkholder-Davi-Gundy inequaliy implie ha here exi C p > uch ha ( p E ε 1 Z r dw r τ εh Z r dw r ) Z rdw h r E up ε 1 (Z r τ εh Z r ) Z p rdw h r T ( ) p/2 C p E Zr ε Z r h 2 dr, which end o a ε goe o. Thu Z dw i in D 1,p. Sep 2: Le in (,T. We now prove ha (Y,Z1,T ) D 1,2 L 2 (,T;D 1,2 ). Le (h n ) n be a complee orhonormal yem of H. We have ha E Y 2H + Z dw 2 H = Y,h n H n 1E 2 T 2 + Z dw,h n H = n 1E = n 1E C n 1E 2 + Ỹ hn Ỹ hn 2 + ξ,h n H 2 + Z hn dw 2 Z hn 2 d f(,y,z ),h n H 2 d = CE ξ 2 H+ E f(,y,z ) 2 Hd = CE Dξ 2 H + E Df(,Y,Z ) 2 H d < +, where we have ued he fac ha Y,h n H = DY,ḣn H = Ỹ hn (and he ame for he Z componen) from Sep 1, and a priori eimae (ee e.g. 5, Propoiion 2.1) in S 2 H 2 (which involve a generic conan C > ) for BSDE of he form (5.5) ince he daa are quare inegrable. Hence, Y, and Z dw belong o D 1,2, and by 12, Lemma 2.3, Z belong o L 2 (,T;D 1,2 ). Finally noe ha a verion of (D Y,D Z ), T i given a he oluion (a priori in S p H p ) o he linear BSDE: D Y = D ξ+ D f(r,,y r,z r )+f y (r,,y r,z r )D Y r +f z (r,,y r,z r )D Z r dr which admi in fac a unique oluion in S 2 H 2 by our aumpion. D Z r dw r, Remark 5.1. We would like o poin ou ha ince he proce Z 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 12, Lemma 2.3 he laer reul i equivalen o he fac ha Z belong o L 2 (,T;D 1,2 ). (5.7) 14

15 6 Applicaion and dicuion of he reul 6.1 Applicaion o FBSDE We conider in hi ecion a FBSDE of he form X = X + Y = g(x T )+ b(,x )d+ We make he following Aumpion: f(,x,y,z )d σ(,x )dw,,t, P a.. 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. hal , verion 1-3 Apr 214 Beide b(,),σ(,) are bounded funcion of. (A 2 ) (i) g i coninuouly differeniable wih polynomial growh. (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) C(1+ y κ + z κ + x q ), (,x,y,z),t R 3. The well-known following lemma provide he exience of a Malliavin derivaive for X for all,t under Aumpion (A 1 ) (ee e.g. 1, 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 heorem below how ha in he Markovian cae, Theorem 5.1 hold direcly under Aumpion (A 1 ) and (A 2 ). Theorem 6.1. Le,T and aume ha (A 1 ) and (A 2 ) hold. Then, Y D 1,2 and Z L 2 (,T;D 1,2 ). Proof. We aim a applying Theorem 4.1. 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 2. (6.2) 15

16 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 ) +CE f x (, X,Y,Z ) f x (,X,Y,Z ) p DX,ḣ H p ε CE 1 (X τ εh X ) DX,ḣ H pr1 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, hal , verion 1-3 Apr 214 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 ) 1/2 2/r d ( E (1+ Y κp + Z κp + X pq + X τ εh pq ) r ) 1/2 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 +Pε,h (b x (,X )d+σ x (,X ))dw, 16

17 wih ( ) dh ε := DX,ḣ H(b x (,X ) b x (,X ))+ḣ(σ(,x τ εh ) σ(,x )) d + DX,ḣ H(σ 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. 13, Theorem V.9.53) ha lim E ε up ε 1 (X τ εh X) DX,ḣ H q =, q 1.,T hal , verion 1-3 Apr 214 A a conequence, combining hi eimae wih (6.3), we ge ha: T 2/r A 1,ε d C E 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 d C E f x (, 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 lim fx (, X,Y,Z ) f x (,X,Y,Z ) p p =, P d a.e. ε Furhermore, for any 2 > ρ > 1, 1/2 up ε (,1) C up ε (,1) E fx (, X,Y,Z ) f x (,X,Y,Z ) ρp p d E (1+ X q + X τ εh q + Y κ + Z κ ) ρp p d <, 17

18 by chooing p mall enough o ha κρp p 2. So by Lebegue dominaed convergence heorem, lim A 2,ε d =, ε 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 5 for affine BSDE. We ake a driver of he form f(,ω,y,z) := α (ω)+β (ω)y +γ (ω)z hal , verion 1-3 Apr 214 wih bounded F-progreively meaurable procee, and ξ in D 1,2. The condiion given in 5, 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, θ (6.4) 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 6.1. 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. 18

19 hal , verion 1-3 Apr 214 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 in force. 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 +CE ( ( ( ) p ε 1 (α τ εh α ) Dα,ḣ H d ) p Y (ε 1 (β τ εh β ) Dβ,ḣ H) d ) p Z (ε 1 (γ τ εh γ ) Dγ,ḣ H) d =: A ε 1 +A ε 2 +A ε 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 ε Z 2 p/2 E 1 (γ τ εh γ ) Dγ,ḣ H E ) p/2 ( Z 2 T 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 6.1. Noe ha, ince he BSDE i affine, Y can be expreed explicily a: where ( M, := exp γ u dw u 1 2 Y = E M,T ξ M, α d F, 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 5 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 ha 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). 19

20 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 12, 5 and our approach. In hee reference, he auhor conider he equence of BSDE: Y n = ξ + f(,y n 1,Z n 1 )d ZdW n,,t, which approximae in S 2 H 2 he oluion o he original BSDE: Y = ξ + f(,y,z )d Z dw,,t. 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 r Z n ) aifie for,t, r : hal , verion 1-3 Apr 214 D r Y n = ξ+ D r f(,θ n 1 )+ y f(,θ n 1 )D r Y n 1 + z f(,θ n 1 )D r Z n 1 d Z n dw, wih Θ n 1 := (Y n 1,Z n 1 )., y,z 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 = ξ + D r f(,y,z )+ y f(,y,z )D r Y + z f(,y,z )D r Z d 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 5, 12 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), hi i why hi aumpion i omehow arbirary. However, for Y o be in D 1,2, i i neceary (and ufficien) o have ha ε 1 (Y τ εh Y ) o converge in L p for ome p < 2 o DY,ḣ H for any h in H (according o Theorem 4.1). Hence, we feel ha our condiion are more precie. 7 Exenion o quadraic growh BSDE The aim of hi ecion i o exend our previou reul o o-called quadraic growh BSDE. Such reul already exi in he lieraure, ee in paricular 1, 7 or he hei 4, 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. 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 2, (,ω) f(,ω,y,z) i inl 2 (,T;D 1, ), f(,y,z) anddf(,y,z) aref-progreively meaurable, Df(, y, z) i uniformly bounded in y, z. 2

21 (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 3 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) C(1+ z ). (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 n ) n N be a equence in(,1 uch ha lim n + ε n =, and le(y n,z n ) n be a equence of random variable which converge in S p H p for any p > 1 o ome (Y,Z). Then for all h H, he following convergence hold in probabiliy hal , verion 1-3 Apr 214 or f y (,ω +ε n h,y n,z ) f y (,ω,y,z ) H n + f z (,ω +ε n h,y n,z n ) f z (,ω,y,z ) H, (7.1) n + f y (,ω +ε n h,y n,z n ) f y (,ω,y,z ) H n + f z (,ω +ε n h,y,z n ) f z(,ω,y,z ) H. (7.2) n + 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 predicable procee Z uch ha: eup τ T E τ Z 2 df 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 7. 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 ). 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 7 o obain ha for any 21

22 p > 1, here exi ome q > 1 uch ha ( ) p E up Y ε Ỹh +E 2p Z ε Z h 2 d,t ( C p E ξ ε Dξ,ḣ H pq 1/q T ) pq 1/q +E Ãε Df(,,Y,Z ),ḣ Hd ( ) pq 1/q +C p E Ãy,ε f y (,,Y,Z ) Ỹ h d ( pq 1/q +C p E Ãz,ε f z (,,Y,Z ) Z h d). (7.3) hal , verion 1-3 Apr 214 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 ) Z h d ( pq 1/2 ( E Ãz,ε f z (,,Y,Z ) d) 2 T pq 1/2 E Z h d) 2. (7.4) Since (Ỹ h, Z h ) 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 ) C(1+ Z ) (by Aumpion (Q)), we deduce ha Z h H 2 BMO which implie ha Z h H m for any m > 1 by he energy Inequaliie. Furhermore, for any η > i hold ha E ( CE ) pq+η Ãz,ε f z (,,Y,Z ) Z d h ( ( CE ) pq+η (1+ Z + Z τ εh ) Z d h ) pq+η ( (1+ Z + Z τ εh ) 2 2 T d Z h 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 Z h 2 d ) p 1/q ( pq+η 1/2 Z 2 d E Z d) h 2 <, 22

23 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 Ỹh 2p T + E Z ε Z ) p h 2 d =. 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 ) (ee 12). Moreover, ince (D Y,D Z) i he oluion of he ochaic linear BSDE (5.7) for any,t and Aumpion (D ) and (Q) hold, from he relaion D Y = Z for all,t we obain Z1,T L (,T;D 1,2 ). Remark 7.1. We would like o poin ou ha our condiion cover he cae of Markovian quadraic BSDE preened in 6, Theorem 2.9. Indeed, aume ha we conider a forwardbackward yem of he form (6.1) 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 i coninuouly differeniable in (x,y,z) and aifying for ome C > q R +, f x (,x,y,z) C(1+ y + z 2 + x q ), (,x,y,z),t R 3. hal , verion 1-3 Apr 214 Under hee 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 Theorem 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 ) p (1+ X q + X τ εh q + Y + Z 2 )d ( 2p 1/2 E (1+ X q + X τ εh q + Y + Z )d) 2, 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 16, Remark 2, we provide a proof for he comfor of he reader. Lemma A.1. Le p > 1 and F be in L p (R), G S and h H. The mapping EF τ h G i differeniable in and d d EF τ h G = EF τ h δ(gh). (A.1) 23

24 Proof. Le >, by he Cameron-Marin formula, we have ha 1( EF τ (+)h G EF τ h G ) ( G τ h exp = E F ḣ(u)dw ) u 2 T 2 ḣ(u) 2 du G τ h. Hence lim 1( EF τ (+)h G EF τ h G ) ( G τ hexp = E F ḣ(u)dw ) u 2 T 2 ḣ(u) 2 du G τ h lim ( = E F τ h lim G τ h G exp ḣ(u)dw u 2 T 2 du) ḣ(u) 2 1 +G τ h, hal , verion 1-3 Apr 214 where he exchange beween he limi and he expecaion i juified by he fac ha ( ) up E q G τ hexp ḣ(u)dw u 2 q (,1 2 ḣ(u) 2 du G < (A.2) for any q > 1 and by he fac F τ h i in L p by he Cameron-Marin formula. Indeed for any r in (1,p) we have by Hölder Inequaliy: ( G τ h exp E F τ ḣ(u)dw ) u 2 T 2 ḣ(u) 2 du G r h ( E F τ h p 1 r/p G τ h exp 1E ḣ(u)dw ) u 2 T rp 2 ḣ(u) 2 du G 2 1/p 2 }{{}, =:E 1 }{{} =:E 2 where r < p 1 < p and p 2 i he Hölder conjugae of p 1 /r. Uing Cameron-Marin Formula for E 1, Relaion (3.1) and Hölder Inequaliy wih r 1 = p p 1 and r 2 uch ha 1 r r 2 = 1, we deduce ha: We now urn o E 2, for any q > 1 up (,1 up G τ h E E ( ) r 2 1/r2 E 1 E F p E r/p E ḣ dw < +. ( ) T ḣ(u)dw u G q ( G τ h G T E E (,1 }{{} =:A 1 ḣ(u)dw u ) q ( E ) T E u 1 q G, } {{ } =:A 2 + up (,1 24

25 hence, on he one hand here exi α 1,α 2 > 1 uch ha: ) 1 A 1 up q E G τ h G qα 1 1 qα 2 α 2 α 1 E ( E ḣ(u)dw u < +, (,1 uing he fac ha G i polynomial, o G i locally Lipchiz and we conclude by Relaion (3.1). On he oher hand, uing he mean value Theorem and Relaion 3.1, we obain alo A 2 < +. We deduce ha Relaion (A.2) hold. Moreover, given ha G S i polynomial, we deduce ha G τ h G G,h H a... Hence, lim 1( EF τ (+)h G EF τ h G ) ( = E F τ h lim G τ h G exp ḣ(u)dw ) u 2 T 2 ḣ(u) 2 du 1 +G τ h hal , verion 1-3 Apr 214 = EF τ h ( G,h H +Gδ(h)) = EF τ h δ(gh), by (2.3), o (A.1) hold. Lemma A.2. Le α in L 2 (,T;D 1,2 ). Then for any p in (1,2), lim E ε α τ εh α ε α,h H pd =. Proof. Noe fir ha he pace L 2 (,T;D 1,2 ) can be idenified wih he pace D 1,2 (H) which i he compleion of he e of H-valued r.v. of he form: n F i u i, F i S, u i L 2 (,T), n 1, i=1 wih repec o he norm 1,2,2 defined a: u 2 1,2,2 := E u 2 L 2 (,T) +E u 2 H L 2 (,T). Alernaively, an elemen u in D 1,2 (L 2 (,T)) i idenified wih a ochaic proce uch ha for almo avery in,t, u belong o D 1,2 and uch ha E u 2 H L 2 (,T) = E D u 2 dd < +. Hence we can aume ha α belong o D 1,2 (L 2 (,T)). Thu by 16, Theorem 3.1, α aifie (RAC) and (SGD), which enail in hi eing ha for any h in H, here exi a L 2 (,T)- valued r.v. α h uch ha α h = α in L 2 (,T), P -a.., and for any ε > α h τ εh α h ε ε = ε 1 α τ h,h H d, in L 2 (,T), P a... 25

26 Uing Lemma 3.1 we hu ge ha for any r (p,2), i hold ha: E ε 1 (α τ εh α ) r d = E E C CE ε 1 (( α h )() τ εh ( α h )()) r d ε 1 ε α τ uh,h H r dud E α,h H p r/p d r/p α,h H p d C h r H E α p H H r/p < +, hal , verion 1-3 Apr 214 where we have ued Cameron-Marin formula and imilar compuaion o hoe of he proof of Lemma ( 4.2, and C denoe a poiive conan which can differ from line o line. Hence, he family ε 1 (α τ εh α ) α,h H p d i uniformly inegrable. In addiion, by )ε (,1) Propery (SGD), ε 1 (α τ εh α) converge in probabiliy o α,h H (wih repec o he norm L 2 (,T)) which implie ha ε 1 (α τ εh α ) α,h H p d converge in probabiliy o a ε goe o, which provide he reul. Acknowledgmen Thibau Marolia i graeful o Région Ile-De-France for financial uppor. Reference 1 S. Ankirchner, P. Imkeller, and G. Do Rei. Claical and variaional differeniabiliy of bde wih quadraic growh. Elecron. J. Probab, 12(53): , J.-M. Bimu. Conrôle de yème linéaire quadraique: applicaion de l inégrale ochaique. In Séminaire de Probabilié, XII (Univ. Srabourg, Srabourg, 1976/1977), volume 649 of Lecure Noe in Mah., page Springer, Berlin, P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Soica. L p oluion of backward ochaic differenial equaion. Sochaic Proce. Appl., 18(1):19 129, G. do Rei. On ome properie of oluion of quadraic growh BSDE and applicaion in finance and inurance. PhD hei, Humbold Univeriy, N. El Karoui, S. Peng, and M.C. Quenez. Backward ochaic differenial equaion in finance. Mahemaical finance, 7(1):1 71, P. Imkeller and G. Do Rei. Pah regulariy and explici convergence rae for BSDE wih runcaed quadraic growh. Sochaic Proce. Appl., 12(3): , P. Imkeller, A. Réveillac, and A. Richer. Differeniabiliy of quadraic bde generaed by coninuou maringale. Ann. Appl. Probab., 22(1): ,

27 8 S. Kuuoka. Dirichle form and diffuion procee on Banach pace. J. Fac. Sci. Univ. Tokyo Sec. IA Mah., 29(1):79 95, P. Malliavin. Sochaic calculu of variaion and hypoellipic operaor. In Proceeding of he Inernaional Sympoium on Sochaic Differenial Equaion (Re. In. Mah. Sci., Kyoo Univ., Kyoo, 1976), page Wiley, New York-Chicheer-Bribane, D. Nualar. The Malliavin calculu and relaed opic. Probabiliy and i Applicaion (New York). Springer-Verlag, Berlin, econd ediion, E. Pardoux and S. Peng. Adaped oluion of a backward ochaic differenial equaion. Syem Conrol Le., 14(1):55 61, E. Pardoux and S. Peng. Backward ochaic differenial equaion and quailinear parabolic parial differenial equaion. In Sochaic parial differenial equaion and heir applicaion, page Springer, hal , verion 1-3 Apr P. E. Proer. Sochaic inegraion and differenial equaion, volume 21 of Sochaic Modelling and Applied Probabiliy. Springer-Verlag, Berlin, 25. Second ediion. Verion 2.1, Correced hird prining. 14 I. Shigekawa. Derivaive of Wiener funcional and abolue coninuiy of induced meaure. J. Mah. Kyoo Univ., 2(2): , D.W. Sroock. The Malliavin calculu and i applicaion o econd order parabolic differenial equaion. I. Mah. Syem Theory, 14(1):25 65, H. Sugia e al. On a characerizaion of he obolev pace over an abrac wiener pace. Journal of Mahemaic of Kyoo Univeriy, 25(4): , A.S. Üünel and M. Zakai. Tranformaion of meaure on Wiener pace. Springer, 2. 27