Weak error analysis via functional Itô calculus

Transcription

1 Weak error analysis via funcional Iô calculus Mihály Kovács and Felix Lindner arxiv: v2 [mah.pr] 14 Jun 216 Absrac We consider auonomous sochasic ordinary differenial equaions SDEs and weak approximaions of heir soluions for a general class of sufficienly smooh pah-dependen funcionals f. Based on ools from funcional Iô calculus, such as he funcional Iô formula and funcional Kolmogorov equaion, we derive a general represenaion formula for he weak error EfX f X, where X and X are he pahs of he soluion process and is approximaion up o ime. he funcional f : C[, ], R d R is assumed o be wice coninuously Fréche differeniable wih derivaives of polynomial growh. he usefulness of he formula is demonsraed in he one dimensional seing by showing ha if he soluion o he SDE is approximaed via he linearly ime-inerpolaed explici Euler mehod, hen he rae of weak convergence for sufficienly regular f is 1. Keywords: Funcional Iô calculus, sochasic differenial equaion, Euler scheme, weak error, pah-dependen funcional MSC 21: 6H1, 6H35, 65C3 1 Inroducion Le W be an m-dimensional Wiener process and X be he srong soluion o a sochasic differenial equaion SDE, for shor of he form dx = bx d + σx dw 1.1 wih iniial condiion X = ξ R d. he funcions b: R d R d and σ : R d R d m are assumed o be smooh i.e., C -funcions such ha all derivaives of order 1 are bounded and σ saisfies a non-degeneracy condiion, see Secion 2 for deails. Fix, and le Y [, ] be a process wih coninuous sample pahs arising from a numerical discreizaion of 1.1 which approximaes X on [, ]. Le X and Y denoe he C[, ], R d -valued random variables ω X, ω [, ] and ω Y, ω = Y, ω [, ], where X, ω and Y, ω are he rajecories X, ω and Y, ω. In his aricle, we are ineresed in analyzing he weak approximaion error E fy fx, 1.2 for sufficienly smooh pah-dependen funcionals f : C[, ], R d R. o his end, suppose ha we are given a furher process X [, ] solving an SDE of he form d X = b, X d + σ, X dw 1.3 1

2 wih iniial condiion X = X = ξ R d, where b, : C[, ], R d R d and σ, : C[, ], R d R d m, [, ], are pah-dependen coefficiens and X denoes he C[, ], R d -valued random variable ω X, ω [,]. If he coefficiens b and σ are chosen in such a way ha he error EfY f X has a simple srucure and can be handled relaively easily, hen he problem of analyzing 1.2 essenially reduces o analyzing he weak error E f X fx. 1.4 Our main resul, heorem 7.2, provides a represenaion formula for he error 1.4 which is suiable o derive explici convergence raes for numerical discreizaion schemes. I is valid under he assumpion ha f : C[, ], R d R is wice coninuously Fréche differeniable and f and is derivaives have a mos polynomial growh, C[, ], R d being endowed wih he uniform norm. he proof is based on ools from funcional Iô calculus, such as he funcional Iô formula and funcional backward Kolmogorov equaion, cf. [2, 5, 6, 7, 8, 1, 11]. As a concree applicaion, we consider for d = m = 1 he explici Euler-Maruyama scheme wih maximal sep-size δ >. In order o consruc a process Y wih compuable sample pahs, we linearly inerpolae he oupu of he scheme beween he nodes. his process, however, does no saisfy an SDE such as 1.3. herefore, we also consider a sochasic inerpolaion X of he scheme via Brownian bridges which is no feasible for numerical compuaions bu saisfies 1.3 wih suiably chosen coefficiens b and σ. Using a Lévy-Ciesielsky ype expansion of Brownian moion, we show in Proposiion 8.3 ha he error EfY f X is Oδ whenever f : C[, ], R R is wice coninuously Fréche differeniable and is derivaives have a mos polynomial growh. For he analysis of he error Ef X fx, we use he error represenaion formula from heorem 7.2 and show, in Proposiion 8.4, ha i is also Oδ if f : C[, ], R R is four imes coninuously Fréche differeniable wih derivaives of polynomial growh. As a direc consequence, our main resul concerning he linearly inerpolaed explici Euler- Mayurama scheme, heorem 8.1, is ha if f : C[, ], R R is four imes coninuously Fréche differeniable and is derivaives have a mos polynomial growh, hen he weak error EfX fy is of order Oδ. he resul can be used, for insance, o show ha he bias CovY 1, Y 2 CovX 1, X 2 for he approximaion of covariances of he soluion process is Oδ, see Example 8.2. here exiss an exensive lieraure on srong and weak convergence raes of numerical approximaions schemes for SDEs, see, e.g., [13, 17, 26] and he references herein. he inerplay of srong and weak approximaion errors is paricularly imporan for he analysis of mulilevel Mone Carlo mehods. I is well-known ha for various discreizaion schemes and sufficienly smooh es funcions he order of weak convergence exceeds he order of srong convergence and is, in many cases, wice he srong order. However, he weak error analysis of SDEs is ofen resriced o funcionals which only depend on he value of he soluions process a a fixed ime, say. Such funcionals are of he form fx = ϕx for a funcion ϕ: R d R. here are no so many publicaions reaing convergence raes of weak approximaion errors for pah-dependen funcionals of he soluion process as in 1.2 and 1.4. In [12] Malliavin calculus mehods are used o derive esimaes for he convergence of he densiy of he soluion o he Euler-Maruyama scheme, leading o Oδ weak convergence for a specific class of inegral ype funcionals. Composiions of smooh funcions and non-smooh inegral ype funcionals are reaed in [18] for an exac simulaion of he soluion process a he ime discreizaion poins in a one-dimensional seing. Weak convergence raes for Euler-Maruyama approxima- 2

3 ions of non-smooh pah-dependen funcionals of soluions o SDEs wih irregular drif and consan diffusion coefficien are derived in [27] via a suiable change of measure, he obained order of convergence being a mos Oδ 1/4. Weak convergence resuls for approximaions of pah-dependen funcionals of SDEs wihou explici raes of convergence can be found in several aricles, e.g., in [3, 29]. In [8] he auhors use mehods from funcional Iô calculus o analyze Euler approximaions of pah-dependen funcionals of he form fx and o derive convergence raes for he corresponding srong error E fx f X 2p, p 1. his lis of references is only indicaive and we also refer o he references in he menioned aricles. In his paper, we presen a new and general mehod for he weak error analysis of numerical approximaions of a large class of sufficienly smooh, pah-dependen funcionals of soluions o SDEs of he ype 1.1. Our approach is based on he funcional Iô calculus as presened in [2, 7, 11] and is in a sense a naural, albei highly nonrivial, generalizaion of he classical approach o he analysis of weak approximaion errors based on Iô s formula and backward Kolmogorov equaions, cf., e.g., [31] or [17, Secion 14.1]. Le us remark ha weak error esimaes are also available for SPDEs, see, e.g., [9, 19, 2, 21, 22, 24]. In paricular, pah-dependen funcionals of soluions o semilinear SPDEs wih addiive noise are considered in [1, 4]. he analysis in [1] is based on Malliavin calculus and applies o cerain composiions of smooh funcions and inegral ype funcionals. A quie general class of pah-dependen C 2 -funcionals is reaed in [4], based on a second order aylor expansion of he composiion of he es funcion and an underlying Iô map. A difference o our resuls apar from he infinie dimensionaliy of he sae space is ha he analysis in [4] is resriced o spaial discreizaions, addiive noise, and he es funcions are assumed o be bounded. o presen he main idea behind our approach, le and for a deerminisic càdlàg pah x D[, ], R d le he process X,x = X,x s s be defined by X,x xs if s [, s := X,x s if s [,, where X,x s s [, is he srong soluion o Eq. 1.1 sared a ime from x R d. For ε > define a family of funcionals F ε = F ε [, ] by F ε x := Ef ε X,x, x D[, ], Rd, 1.5 where X,x denoes he pah of X,x up o ime and f ε is a suiably regularized version of he pah-dependen funcional f such ha E f X fx = lim ε E f ε X f ε X. hen, as we assume ha X = X = ξ R d, i follows ha E f ε X f ε X = E F ε X F ε X. Afer proving ha F ε is regular enough in a suiable sense we apply he funcional Iô formula from heorem 3.6 o F ε X F ε X and use a backward funcional Kolmogorov equaion from heorem 3.7 o eliminae a erm which canno be conrolled as ε. Finally we arrive a our explici represenaion formula for he weak error 1.4 in erms of 3

4 Ef ε X f ε X, saed in heorem 7.2, for f : C[, ], R d R wice coninuously Fréche differeniable wih a mos polynomially growing derivaives: E f ε X f ε X d = E j=1 d i,j,k=1 E [ Df ε X,x { [ E D 2 f ε X,x 1 [, ]D e j X,x ] x= bj, X b j X d X 1 [, ] D e i X,x, 1 [, ] D e j X,x + Df ε X,x 1 [, ] D e i+e j X,x ] σik σ jk, X σ ik σ jk X } d. x= X Here, e i i {1,...,d} is he canonical orhonormal basis of R d and, for a muli-index α N d, D α X,x = Dξ α X,ξ ξ=x denoes he corresponding parial derivaive of he soluion process X,ξ sared a ime w.r.. he iniial condiion ξ R d, evaluaed a ξ = x, see Secion 4 for deails. he paper is organized as follows. In Secion 2 we inroduce some general noaion used hroughou he aricle, sae he main assumpions on he coefficiens in 1.1 and 1.3 and also inroduce he regularized versions f ε, ε >, of a funcional f : C[, ], R d R via a mollificaion operaor. Secion 3 conains a shor inroducion o he noions and noaions of he funcional Iô calculus and a he end of he secion we also recall he funcional Iô formula as well he funcional backward Kolmogorov equaion. In Secion 4 we prove resuls, crucial for wha follows afer, concerning he regulariy of he soluion of 1.1 wih respec o he iniial daa mainly in he uniform opology. Secion 5 is devoed o he sudy of he regulariy of he funcional F ε and he explici compuaion of is verical and horizonal derivaives, so ha he funcional Iô formula and he funcional backward Kolmogorov equaion can be applied; he main findings are summarized in heorem 5.7. In Secion 6, using he regulariy resuls from Secion 5 and he maringale propery of F εx [, ] from Proposiion 6.1, we show in Corollary 6.2 ha F ε saisfies a funcional backward Kolmogorov equaion. heorem 7.2 in Secion 7 conains our main resul concerning he represenaion of he weak error Ef ε X f ε X. As an imporan applicaion of heorem 7.2, in Secion 8 we analyse he order of he weak error for he linearly inerpolaed explici Euler-Maruyama scheme and he main resul here is presened in heorem 8.1. Finally, in he Appendix, we presen a general convergence lemma, Lemma A.1, which is used exensively hroughou he paper and also a resul from he lieraure, Lemma A.2, concerning he opological suppor of he disribuion P X of X in C[, ], R d. 2 Preliminaries In his secion we describe some general noaion used hroughou he aricle, formulae he precise assumpions on he SDEs 1.1 and 1.3 for X and X, and inroduce a mollificaion operaor M ε ha allows us o define suiable smooh approximaions f ε of a given pah-dependen funcional f : C[, ], R d R. General noaion. he naural numbers excluding and including zero are denoed by N = {1, 2,...} and N = {, 1,...}, respecively. Norms in finie dimensional real 4

5 vecor spaces are denoed by. We usually consider he Euklidean norm, e.g., ξ = ξ ξ 2 d for a vecor ξ = ξ 1,..., ξ d R d or A = i,j a 2 ij for a marix A = a ij, bu he specific choice of he norm will no be imporan. he only excepion are muliindices α = α 1,..., α d N d, for which we se α := α α d. he canonical orhonormal basis in R d is denoed by e i i {1,...,d}. By C[a, b], R d and D[a, b], R d we denoe he spaces of coninuous funcions and càdlàg righ coninuous wih lef limis funcions defined on an inerval [a, b] wih values in R d, respecively. Boh spaces are endowed wih he uniform norm, e.g., x C[a,b];R d = sup [a,b] x. For a càdlàg pah x D[, ], R and [, ], we denoe by x := x [,] D[, ], R d he resricion of x o [, ], whereas x R d denoes he value of x a. Consisen wih his noaion, we will occasionally also wrie x insead of x for a given pah x D[, ], R d in order o indicae he domain of definiion. More generally, if x is a càdlàg pah defined on an arbirary inerval I [, and if I, hen x := x I [,] denoes he resricion of x o I [, ]. Accordingly, if Z = Zs s [a,b] or Z = Zs s a is an R d -valued sochasic process wih càdlàg pahs and if [a, b] or a, we wrie Z for he D[a, ]; R d -valued random variable ω Z, ω [a,], where Z, ω is a rajecory of Z. For insance, if X,ξ is he srong soluion o 1.1 sared a ime [, ] from ξ R d, hen X,ξ denoes he D[, ]; Rd -valued random variable ω X,ξ, ω [, ]. Le U, U and V, V be wo normed real vecor spaces. We denoe by L U, V he space of bounded linear operaors : U V, endowed wih he operaor norm L U,V := sup u U 1 u V. For n N, we wrie L n U, V for he space of bounded n-fold mulilinear operaors : U n V, endowed wih he norm L n U,V := sup u1 U 1,..., u n U 1 u 1,..., u n V. If g : U V is n-imes Fréche differeniable, we wrie D n gu for he n-h Fréche derivaive of g a u U and consider i as an elemen of L n U, V ; by D n guu 1,..., u n V we denoe he evaluaion of D n gu a u 1,..., u n U n. Specifically, if g : R d V is n-imes Fréche differeniable and α N d is a muli-index wih α n, we wrie D α gξ := D α ξ gξ := α ξ α 1 1,..., ξα d d gξ V for he corresponding parial derivaive a a poin ξ R d. We wrie C n U, V for he space space of n-imes coninuously Fréche-differeniable funcions from U o V, and C n p U, V is he subspace of n-imes coninuously Fréche-differeniable funcions g : U V such ha g and is derivaives up o order n have a mos polynomial growh a infiniy, i.e., C n p U, V = {g Cn U, V : C, q 1 such ha D n gu L n U,V C1 + u q U for all u U}. hroughou he aricle, C, denoes a finie consan which may change is value wih every new appearance. Main assumpions. hroughou he aricle, we suppose ha he following assumpions hold. All random variables and sochaic processes are assumed o be defined on a common filered probabiliy space Ω, F, F, P saisfying he usual condiions. he process W is a R m -valued Wiener process w.r.. he filraion F. Concerning he coefficiens appearing in he SDEs 1.1 and 1.3 for X and X we assume he following. Assumpion 2.1. he funcions b: R d R d and σ : R d R d m in Eq. 1.1 are C - funcions such ha all derivaives of order 1 are bounded. here exiss a consan c > such ha σxy c y for all x R d and y R m. 5

6 Assumpion 2.1 implies ha for every iniial condiion ξ R d and s here exiss a unique song soluion X s,ξ [s, o Eq. 1.1 saring a ime s from ξ. By X = X,ξ we denoe he soluion o 1.1 saring a ime zero in a fixed given saring poin ξ R d. Assumpion 2.2. he funcions b, : C[, ], R d R d and σ, : C[, ], R d R d m, [, ], in Eq. 1.3 are such ha he mapping, x b, x, σ, x defined on [, ] C[, ], R d is Borelmeasurable recall ha x = x [,] ; here exiss a unique srong soluion X [, ] o Eq. 1.3 saring from ξ ; he linear growh condiion b, x + σ, x C1 + sup s xs, [, ], x C[, ], R d, is fulfilled wih C, independen of x and s. We noe ha he boundedness of he derivaives Db and Dσ and he linear growh assumpion on b and σ imply ha E sup X p + E sup X p < 2.1 [, ] [, ] for all p 1. his is a consequence of he Burkholder inequaliy and Gronwall s lemma. A mollificaion operaor. In order o be able o apply he funcional Iô calculus presened in Secion 3 o our problem in a convenien way, we associae o every pah-dependen funcional f : C[, ], R d R a family of regularized versions f ε : D[, ], R d R, ε >, by seing Here, f ε := f M ε. 2.2 M ε : D[, ], R d C [, ], R d 2.3 is he mollificaion operaor defined as follows: Le η C c R be a sandard mollifier nonnegaive, ηd = 1, supp η [ 1, 1] and se η ε := ε/2 1 ηε/2 1 as well as η ε = η ε ε/2. Le x denoe he exension of a pah x D[, ], R d o R given by hen we se x := x1, + x1 [, ] + x1,. M ε x := η ε x [, ], x D[, ], R d, 2.4 where denoes convoluion, i.e., η ε x = R η ε sxs ds. Noe ha, in fac, M ε x = ε η ε sxs ds = ε η ε sxs ds, [, ], and ha we have he convergence M ε x εց x in C[, ], R d for all x C[, ], R d. 6

7 3 Funcional Iô calculus In his secion we presen some of he main noions and resuls from funcional Iô calculus, see [7] and compare also [2, 5, 6, 8, 1, 11]. On D[, ], R d we consider he canonical σ-algebra B generaed by he cylinder ses of he form A = {x D[, ], R d : xs 1 B 1,..., xs n B n }, where s 1... s n, B i BR d, i = 1,..., n, and n N. Noe ha B coincides wih he Borel-σ-algebra induced by he uniform norm D[,];R d. Definiion 3.1. A non-anicipaive funcional on D[, ], R d is a family F = F [, ] of mappings F : D[, ], R d R, x F x such ha every F is B /BR-measurable. We also consider non-anicipaive funcionals wih index se [, as well as R n - valued non-anicipaive funcionals. hese are defined analogously wih he obvious modificaions. Recall ha for a pah x D[, ], R and [, ], we denoe by x = x [,] D[, ], R d he resricion of x o [, ] and ha, consisen wih his noaion, we may also wrie x insead of x for a given pah x D[, ], R d in order o indicae he domain of definiion. For h, he horizonal exension x,h D[, + h], R d of a pah x D[, ], R d o [, + h] is defined by x s if s [, x,h s := x if s [, + h]. For h R d, he verical perurbaion x h D[, ], R d of a pah x D[, ], R d is defined by x h s := x s if s [, x + h if s =. Definiion 3.2. Le F = F [, ] be a non-anicipaive funcional on D[, ], R d. i For [, and x D[, ], R d, he horizonal derivaive of F a x is defined as D Fx := lim hց F +h x,h F x h 3.1 provided ha he limi exiss. If 3.1 is defined for all [, and x D[, ], R d, hen F is called horizonally differeniable. In his case, he mappings D F : D[, ], R d R, x D Fx, [,, define a non-anicipaive funcional DF = D F [,, he horizonal derivaive of F. ii For [, ] and x D[, ], R d, he verical derivaive of F a x is defined as F x he 1 F x x F x := lim,..., F x he d F x h h h 3.2 7

8 provided ha he limi exiss, where e j j=1,...,d is he canonical orhonormal basis in R d. If 3.2 is defined for all [, ] and x D[, ], R d, hen F is called verically differeniable. In his case, he mappings x F : D[, ], R d R d, x x F x, [, ], define a non-anicipaive funcional x F = x F [, ], he verical derivaive of F. In order o inroduce a proper noion of lef-coninuiy for non-anicipaive funcionals, one considers he following disance beween wo pahs which are possibly defined on differen ime inervals. For, [, ], x D[, ], R d and x D[, ], R d we define d x, x := + sup x, s x., s We remark ha d is a meric on he se Λ := [, ] s [, ] D[, ], R d. Definiion 3.3. Le F = F [, ] be a non-anicipaive funcional on D[, ], R d. i F is coninuous a fixed imes if, for all [, ], he mapping F : D[, ], R d R is coninuous w.r.. he uniform norm D[,],R d. ii F is coninuous if he mapping Λ x F x R is coninuous w.r.. he meric d on Λ, i.e., if [, ] x D[, ], R d ε > δ > [, ] x D[, ], R d d x, x < δ F x F x < ε. he class of coninuous non-anicipaive funcionals is denoed by C, [, ]. iii F is lef-coninuous if [, ] x D[, ], R d ε > δ > [, ] x D[, ], R d d x, x < δ F x F x < ε. he class of lef-coninuous non-anicipaive funcionals is denoed by C, l [, ]. iv F is boundedness-preserving if R > C > [, ] x D[, ], R d sup xs R F x C. s [,] he class of boundedness-preserving non-anicipaive funcionals is denoed by B[, ]. We will also use he above noions for R n -valued non-anicipaive funcionals; he corresponding definiions are analogous wih he obvious modificaions. 8

9 Definiion 3.4. For k N, we denoe by C 1,k b [, ] be he class of all lef-coninuous, boundedness-preserving, non-anicipaive funcionals F = F [, ] C, l [, ] B[, ] such ha F is horizonally differeniable, he horizonal derivaive DF = D F [, is coninuous a fixed imes, and he exension D F [, ] of D F [, by zero belongs o he class B[, ], F is k imes verically differeniable wih j x F = j x F [, ] C, l [, ] B[, ] for all j = 1,..., k. Remark 3.5. We remark ha in [7], our main reference for funcional Iô calculus, he slighly differen class of boundedness-preserving funcionals B[, wih index se [, is considered insead of he class B[, ] inroduced in Definiion 3.3 above. In conras o he laer he boundedness assumpion for funcionals in he former class in no uniform in ime. Our definiion corresponds o he one in [8]. Similarly, he class [, of regular and boundedness-preserving funcionals considered in [7] differs from he class C 1,k b [, ] inroduced in Definiion 3.4 above. As a consequence, he choice = is admissible in he funcional Iô formula below. C 1,k b Nex, we sae a funcional version of Iô s formula, compare [7, heorem 4.1] or [2, 6, 5, 8, 1, 11]. heorem 3.6. Le Y = Y [, ] be an R d -valued coninuous semimaringale defined on Ω, F, P and F = F [, ] a non-anicipaive funcional belonging o he class [, ]. hen, for all [, ], C 1,2 b F Y = F Y + D s FY s ds + x F s Y s dy s r 2 x F sy s d[y ]s. he following resul concerning funcional Kolmogorov equaions is aken from [11, heorem 3.7], compare also [2, Chaper 8]. heorem 3.7. Le X = X be he soluion o Eq. 1.1 and F C 1,2 b [, ]. he process F X [, ] is a maringale w.r.. F [, ] if, and only if, F saisfies he funcional parial differenial equaion D Fx = bx x F x 1 2 r 2 x F x σx σ x for all, and all x C[, ], R d belonging o he opological suppor of P X C[, ], R d,. in 4 Smoohness wih respec o he iniial condiion Here we collec and derive several auxiliary resuls concerning he regulariy of he soluion o Eq. 1.1 wih respec o he iniial condiion. hey are crucial for he regulariy properies of he funcional F ε and he explici represenaion of is derivaives as proved in Secion 5. Recall ha X s,ξ = X s,ξ [s, denoes he soluion o 1.1 sared a ime s from ξ R d. Given p 1 and a random variable Y L p Ω, F s, P; R d, we use he 9

10 analogue noaion X s,y = X s,y [s, for he soluion o 1.1 sared a ime ime s wih iniial condiion X s,y s = Y. For s [, ] and Y, Z L p Ω, F s, P; R d, he Burkholder inequaliy and Gronwall s lemma hen yield he sandard esimaes E sup X s,y p C 1 + E Y p, 4.1 [s, ] E sup X s,y X s,z p C E Y Z p, 4.2 [s, ] where C = C p,,σ,b, does no depend on Y, Z or s. Moreover, under our assumpions on σ and b i is well known ha, for fixed s, he random field X s,ξ [s,, ξ R d has a modificaion such ha, for P-almos all ω Ω, he mapping [s, R d, ξ X s,ξ, ω R d is coninuous and for all [s, he mapping R d ξ X s,ξ, ω R d is infiniely ofen differeniable, see, e.g. [15, Secion V.2]. In paricular, every coninuous modificaion of X s,ξ [s,, ξ R d saisfies his propery of smoohness w.r.. he iniial condiion. he reasoning in he proof of Proposiion V.2.2 in [15] and he imehomogeneiy of Eq. 1.1 also yield ha for every muli-index α N d, p 1 and all bounded ses O R d he parial derivaives D α X s,ξ, ω = Dξ α X s,ξ, ω saisfy he esimae E sup D α X s,ξ p <. 4.3 sup s [, ] ξ O, [s, ] For he proof of our error expansion we need o check ha he parial derivaives Dξ α X s,ξ, ω, α N d, can be aken uniformly wih respec o [s, ] and ha he L p Ω; C[s, ], R d -norms of hese derivaives are bounded in ξ R d. As already menioned in Secion 2, we use he noaion D α also for he parial derivaives of general Banach space-valued funcions. ha is, if B is a Banach space, g : R d B a sufficienly ofen Fréche-differeniable funcion, ξ = ξ 1,..., ξ d R d and α = α 1,..., α d N d, hen D α gξ := Dξ α gξ := ξ α 1 1,..., ξ α gξ B d d denoes he corresponding parial derivaive of order α = α α d of f a ξ. In he sequel, we use his noaion boh in he case B = R d and gξ = X s,ξ, ω wih fixed s and in he case B = C[s, ], R d and gξ = X s,ξ, ω. α heorem 4.1. For s [, ] fix a coninuous modificaion of X s,ξ [s, ], ξ R d. i For P-almos all ω Ω, he mapping R d ξ X s,ξ, ω C[s, ], R d is infiniely ofen Fréche-differeniable. In paricular, he parial derivaives D α X s,ξ, ω = Dξ α Xs,ξ, ω, α N d, ξ Rd, exis as C[s, ], R d -limis of he corresponding C[s, ], R d -valued difference quoiens. 1

11 ii For all α N d \ {} and p [1, we have sup E D α X s,ξ p C[s, ],R <. 4.4 d s [, ], ξ R d In he proof of heorem 4.1 and in Corollary 4.6 we will encouner cerain higher order chain rules of Faà di Bruno ype, for which he following noaion will be convenien. Noaion 4.2. For a given muli-index α N d \ {} we denoe by Π{1,..., α } PP{1,..., α } he se of all pariions of he se {1,..., α }. By π we denoe he size of a pariion π Π{1,..., α }, i.e., he number of subses of {1,..., α } conained in π. he disjoin subses of {1,..., α } conained in a pariion π Π{1,..., α } are denoed by π 1,..., π π, i.e., π = {π 1,..., π π }. Finally, we associae o every subse S {1,..., α } a muli-index α S N d by seing α S := {k S : 1 k α 1 } e 1 + dj=2 {k S : j 1 i=1 α i < k j i=1 α i } e j. Proof of heorem 4.1. i he proof of Proposiion V.2.2 in [15] implies ha, for almos all ω Ω, he mappings R d ξ X s,ξ, ω R d, s, are infiniely ofen differeniable, he mappings [s, R d, ξ D α X s,ξ, ω R d, α N d, are coninuous and, due o 4.3, sup ξ O, [s, ] D α X s,ξ, ω < 4.5 for all bounded domains O R d and α N d. Fix such an ω Ω and le e i i=1,...,d be he canonical orhonormal basis of R d. By aylor s formula, for h >, D sup α X s,ξ+he i, ω D α X s,ξ, ω D α+e i X s,ξ, ω [s, ] h h 4.6 sup D α+2e i X s,ξ, ω. 2 [s, ] ξ [ξ,ξ+he i ] Combining 4.5 wih α = 2e i and 4.6 wih α = and using he coninuiy of he mappings [s, ] R d, ξ D e i X s,ξ, ω R d, i {1,..., d}, one obains he Fréche-differeniabiliy of R d ξ X s,ξ, ω C[s, ], R d and he ideniy D α X s,ξ, ω = D α X s,ξ, ω 4.7 for all ξ R d, [s, ] and α N d wih α = 1. In 4.7, we have a derivaive of he funcion R d ξ X s,ξ, ω R d on he lef hand side and a derivaive of he funcion R d ξ X s,ξ, ω C[s, ], R d on he righ hand side. By repeaing his argumen for he higher derivaives we finish he proof of i via inducion over α. ii For a beer readabiliy, we fix s = for a momen and omi he explici noaion of he iniial condiion by wriing X insead of X,ξ. he proofs of Proposiions V.2.1 and V.2.2 in [15] imply ha, for α N d wih α = 1, he Rd -valued process D α X is he soluion o he SDE m D α X = α + Dσ ν Xs D α Xs dw ν s + DbXs D α Xs ds. ν=1 Here we denoe for x R d by σ ν x R d be he ν-h column vecor of σx R d m, Dσ ν : R d R d d and Db : R d R d d are he oal derivaives of σ ν : R d R d and 11

12 b : R d R d, and W ν is he ν-h componen of W. Using he Burkholder inequaliy we obain for all p 2 and [, ] E sup D α Xr p C p, 1 m + E Dσ ν Xs D α Xs 2 p 2 ds r [,] ν=1 + E DbXs D α Xs p ds C p,,σ,b 1 + E sup D α Xr p ds, r [,s] 4.8 where he consan C p,,σ,b, does no depend on he iniial condiion ξ R d. hus, Gronwall s lemma implies E D α X p C[, ],R = E sup d D α Xr p C p,,σ,b expc p,,σ,b 4.9 r [, ] wih he consan C p,,σ,b from 4.8. aking ino accoun he ime-homogeneiy of Eq. 1.1 his proves he asserion for α = 1. For general α N d \ {}, he proofs of Proposiions V.2.1 and V.2.2 in [15] imply ha he R d -valued process D α X is he soluion o he SDE D α m X = D π σ ν Xs D απ α 1 Xs,..., D π π Xs dwν s π Π{1,..., α } + ν=1 D π bxs D απ 1 Xs,..., D α π π Xs ds, 4.1 where we use Noaion 4.2 and where, for n = 1,..., α, D n σ ν and D n b are he n-h oal derivaives of σ ν : R d R d and b : R d R d, considered as funcions wih values he space of n-fold mulilinear mappings from R d n o R d. Using 4.1, he proof is finished via inducion over α by arguing similarly as in 4.8 and 4.9 and applying he respecive esimaes for E D β X q C[, ],R d, q 2, β N d wih β < α. Passing from s = o general s [, ] is no problem due o he ime-homogeneiy of Eq In he sequel, we always consider coninuous modificaions of he random fields X s,ξ [s, ],ξ R d, s [, ]. Remark 4.3. For n N and ω Ω as in heorem 4.1 i, we consider he n-h Fréche derivaive of he mapping R d ξ X s,ξ, ω C[s, ], R d in ξ R d as usual as an n-fold mulilinear mapping from R d n o C[s, ], R d, D n X s,ξ, ω: R d n C[s, ], R d. Jus as in sandard calculus one sees ha i is given by D n X s,ξ, ωη 1,..., η n = η 1,1... η α1,1 η α1 +1,2... η α1 +α 2,2... α N d α =n where η j = η j,1,..., η j,d R d, j = 1,..., n.... η α α d 1 +1,d... η n,d D α X s,ξ, ω, 12

13 Noaion 4.4. Given a R d -valued random variable Y we se D α X s,y, ω := D α X s,y ω, ω = D α ξ Xs,ξ, ω ξ=y ω C[s, ], R d 4.11 opionally [s, ] or as a C[s, ], R d -valued random for s [, ], α N d \ {} and almos all ω Ω. We consider D α X s,y as a R d -valued process D α X s,y variable, D α X s,y = D α X s,y : Ω C[s, ], R d, ω D α X s,y ω := D α X s,y, ω. We use he analogue noaion for he n-h Fréche derivaives of ξ X s,ξ, ω evaluaed a ξ = Y ω, D n X s,y, ω := D n X s,y ω, ω = D n ξ X s,ξ, ω ξ=y ω L n R d, C[s, ], R d, 4.12 where L n R d, C[s, ], R d is he space of bounded, n-fold mulilinear mappings from R d n o C[s, ], R d. Noe ha he noaion 4.11 is consisen wih our noaion X s,y = X s,y s for he soluion of 1.1 sared a ime s wih F s -measurable iniial condiion Y, since X s,y s,y ω, ω = X, ω for almos all ω Ω. Corollary 4.5. Le s [, ] and Y, Y n, n N, be F s -measurable, R d -valued random variables such ha Y n n Y P-almos surely. hen, for all α N d \ {} and p 1, D α X s,yn n D α X s,y in L p Ω; C[s, ], R d. Proof. Using sandard properies of condiional expecaions, we have D E α X s,y D α X s,yn p D = E E α X s,y C[s, ],R d D α X s,yn p C[s, ],R d F s D = E E α X s,ξ Dα X s,η p C[s, ],R d ξ,η=y,y n Now he asserion follows from he coninuiy of he mapping R d ξ D α X s,ξ C[s, ], R d assered by heorem 4.1i, he esimaes 4.3 and 4.4, and wo applicaions of he dominaed convergence heorem. Corollary 4.6. Le s, ξ R d, α N d \ {} and denoe by Dα X s,ξ [, ] he C[, ]; R d -valued random variable ω D α X s,ξ, ω [, ]. i If α = 1, hen D α X s,ξ [, ] = DX,Xs,ξ D α X s,ξ P-almos surely in C[, ], R d. Noe ha he random variable DX,Xs,ξ values in L R d, C[, ], R d and D α X s,ξ akes values in R d. ii For general α N d \ {} we have D α X s,ξ [, ] = D π X,Xs,ξ D α π1x s,ξ,..., D απ π X s,ξ π Π{1,..., α }. akes P-almos surely in C[, ], R d, where we use Noaion 4.2. Noe ha he random variable D π X,Xs,ξ akes values in L π R d, C[, ], R d and he random variables D απ 1 X s,ξ,..., D απ π X s,ξ ake values in R d. 13

14 Proof. i If α = 1 we have α = e i for some i {1,..., d}. By heorem 4.1i we know ha for almos all ω Ω he derivaive D e i X s,ξ, ω = D e i ξ Xs,ξ, ω exiss as a C[s, ], R d -limi of he corresponding difference quoien. Le h n n N be a sequence of posiive numbers decreasing o zero. hen, P-almos surely, D e i X s,ξ [, ] = C[, ], R d - lim n X s,ξ+hnei [, ] X s,ξ [, ] h n. As a consequence of he unique solvabiliy of Eq. 1.1, we have he ideniies X s,ξ+hne i [, ] = X,Xs,ξ+hne i and X s,ξ [, ] = X,Xs,ξ holding P-almos surely in C[, ], R d. Furher, recall ha X,Xs,ξ+hne i, ω = X,Xs,ξ+hne i,ω, ω and X,Xs,ξ, ω = X,Xs,ξ,ω, ω for P-almos all ω Ω. hus, P-almos surely D e i X s,ξ [, ] = C[, ], R d X s,ξ+hnei [, ] X s,ξ [, ] - lim n h n = C[, ], R d X,Xs,ξ - lim h n n X,Xs,ξ+hnei = DX,Xs,ξ D e i X s,ξ, by he chain rule and using heorem 4.1i. ii he general asserion follows by inducion over α, using similar argumens as in he proof of par i. 5 Regulariy of he funcional F ε Recall he definiion 1.5 of he mappings F ε from D[, ], R d o R, [, ], in Secion 1, i.e., F ε x := Ef ε X,x, x D[, ], R d, where ε > and f ε = f M ε : D[, ], R d R is he regularized version of f : C[, ], R d R defined by 2.2, 2.3, 2.4. Our minimal assumpion on f is ha i is BC[, ], R d /BR-measurable and has polynomial growh. Obviously, under his assumpion, F ε = F ε [, ] is a nonanicipaive funcional on D[, ], R d in he sense of Definiion 3.1. he goal of his secion is o show ha, if f Cp 2C[, ], Rd, R, hen F ε is a regular funcional belonging he class C 1,2 b [, ] inroduced in Definiion 3.4. We divide he proof ino a series of lemmaa. In he proofs we ofen use he fac ha if for some n N he polynomial growh bound holds, hen D n fx L n C[, ],R d,r C 1 + x q C[, ],R d, x C[, ], R d, 5.1 D n f ε x L n D[, ],R d,r C 1 + x q D[, ],R d, x D[, ], R d, wih he same C as in 5.1 independenly of ε. his is he consequence of he chain rule and he equaliy M ε L D[, ],R d,c[, ],R d = 1. 14

15 Lemma 5.1. For f C p C[, ], R d, R and ε >, he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 is lef-coninuous and boundedness-preserving, i.e., F ε C, l [, ] B[, ]. Moreover, F ε x C 1 + x q D[,],R d for all [, ] and x D[, ], R d, where C, q, do no depend on, x or ε. Proof. In order o verify he lef-coninuiy, i suffices o show he following: For every x = x D[, ], R d Λ and every sequence x n n N Λ wih x n = x n D[, n n ], R d, n [, ], and d x n, x n, we have F ε nxn n F ε x. Applying Lemma A.1 wih B = D[, ], R d, S = R, Y = X,x, Y n = X n,x n and ϕ = f ε, i is enough o prove ha X n,x n n X,x in L p Ω; D[, ], R d 5.2 for every p 1. o his end, we sar by esimaing X,x X n,x n n X,x =: A + B D[, ],R d X +,x n n, n D[, ],R d X,xn n, n X n,x n n and deal wih each erm separaely. Concerning he firs erm, noe ha D[, ],R d EA p 2 d p 1 x, x n p + E sup X,x s X,xn n s p s [, ] C d x, x n p, where he second esimae follows from 4.2 and he definiion of he meric d. Since X n,x n n [, ] = X,Xn,x n n P-almos surely as an equaliy in C[, ], R d, he p-h momen of he second erm in 5.3 is bounded by EB p 2 E p 1 sup s [ n,] x n n X n,x n n s p + E sup X,xn n s X,Xn,x n n s p s [, ] C E sup x n n X n,x n n s p, s [ n,] where we used again he esimae 4.2 in he second sep. aking ino accoun he ime-homogeneiy of Eq. 1.1 and using he esimae 4.2 once more, we obain EB p C x n n x p + E sup x X n,x s p s [ n,] + E sup X n,x s X n,x n n s p s [ n,] C x n n x p + E sup x X,x s p x x n n p s [, n ] C d x, x n p + E sup x X,x s p. s [, n ] 15

16 By dominaed convergence, he expecaion in he las line goes o zero as n. he combinaion of 5.3, 5.4 and 5.5 yields 5.2 and hus he lef-coninuiy of F ε. o see ha F ε is boundedness-preserving, we use he polynomial growh of f ε : D[, ], R d R and esimae 4.1 o conclude ha, for all [, ] and x D[, ], R d, F ε x = Ef ε X,x E C 1 + X,x q D[, ],R d C 1 + x q D[,],R d + E X,x q D[, ],R d C 1 + x q D[,],R d where he exponen q 1, and he consan C, do no depend on, x or ε. Lemma 5.2. If f CpC[, 1 ], R d, R and ε >, he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 is verically differeniable. he verical derivaive x F ε = x F ε [, ] is lef-coninuous and boundedness-preserving, i.e., x F ε C, l [, ] B[, ], and is given by x F ε x = E [ Df ε X,x 1 [, ] D e 1 X,x ],..., E [ Df ε X,x 1 [, ] D e d X,x ] R d, 5.6 [, ], x D[, ], R d. Moreover, x F ε x C 1 + x q D[,],R d for all [, ] and x D[, ], R d, where C, q, do no depend on, x or ε. Proof. o show he verical differeniabiliy, we fix [, ], x = x D[, ], R d, i {1,..., d} and apply he differeniaion lemma for parameer-dependen inegrals o he mapping δ, δ Ω h, ω f ε X,xhe i ω R, where δ > and x he i D[, ], R d is he verical perurbaion of x by he i R d. he polynomial growh of Df : C[, ], R d L C[, ], R d, R implies polynomial growh of Df ε : D[, ], R d L D[, ], R d, R. ogeher wih heorem 4.1i his implies ha here exis C, q, such ha, for all h δ, δ, d dh f ε X,x he i Df ε X,xhe i Df = ε X,xhe i 1[, ] D e i X,x+he i L D[, ],R d,r 1 [, ] D e i X,x+he i C 1 + X,xhe i q D[, ],R d D e i X,x+he i C[, ],R d C 1 + x D[,],R d + D[, ],R d sup X,ξ q C[, ],R d sup D e i X,ξ C[, ],R, d ξ B δ x ξ B δ x where he las he upper bound belongs o L p Ω for every p [1, due o 4.3. hus, we can apply he differeniaion lemma for parameer-dependen inegrals and use he chain rule ogeher wih heorem 4.1i o obain d dh E[ f ε X,x he i ] [ d h= = E dh f ε X,x he i 16 h=] = E [ Df ε X,x 1 [, ] D e i X,x ].

17 Nex, we verify he lef-coninuiy of x F ε. o his end, i suffices o prove he following asserion: For every x = x D[, ], R d Λ and every sequence x n n N Λ wih x n = x n D[, n n ], R d, n [, ], and d x n, x n, here exiss a subsequence x n k k N such ha x F ε n k x n k k x F εx. Fix x = x and such a sequence x n n N Λ. For i {1,..., d}, E [ Df ε X,x 1 [, ] D e i X,x ] E [ Df ε X n,x n 1 [ n, ]D e i X n,x n n ] [ E Df ε X,x Df ε X n,x n 1 [, ] D e i X,x ] + [ E Df ε X n,x n 1 [, ] D e i X,x 1 [ n, ]D e i X n,x n n =: A + B. ] 5.7 By he convergence 5.2, by Lemma A.1 wih B = D[, ], R d, S = L D[, ], R d, R, Y = X,x, Y n = X n,x n and ϕ = Df ε, and by he esimae 4.3, he firs erm in 5.7 saisfies A E Df ε X,x Df ε X n,x n E D e i X,x L D[, ],R d,r n C[, ],R d. 5.8 he second erm in 5.7 can be esimaed by B E DfM ε X n,x n 2 M ε L L 1 [, ],R d,r E 1[, ] D e i X,x 1 [ n, ]D e i X n,x n n L 1 [, ],R d M ε L C[, ],R d,l 1 [, ],R d E DfM ε X n,x n 2 L C[, ],R d,r E 1[, ] D e i X,x 1 2 =: B ε B 1 B 2, [ n, ]D e i X n,x n n 2 L 1 [, ],R d where B 1 bounded uniformly in n N due o he polynomial growh of Df : C[, ], R d L C[, ], R d, R, he esimae 4.3, and since x x n n d x, x n n. Noe also ha B ε = M ε L C[, ],R d,l 1 [, ],R d = sup s R η ε s = ε/2 1. We furher have B 2 E 1 [ n,]d e i X n,x n n 2 L 1 [, ];R d C E D e i X,x D e i X n,x n n [, ] 2 =: B 21 + C B 22. C[, ],R d Using he ime-homogeneiy of Eq. 1.1 and he esimae 4.3, one sees ha he erm B 21 in 5.1 ends o zero as n since 1 [ n,]d e i X n,x n n D e i X,xn n L L 1 [, ];R d n 1 [, n ];R d n sup D e i X,ξ 5.11 C[, ],R. d 17 ξ B 1 x

18 for n large enough. Concerning he erm B 22 in 5.1 noe ha D e i X,x D e i X n,x n n C[, D [, ] e i X,x ],R d D e i X,xn n C[, ],R d + D e i X,xn n D e i X n,x n n [, ] C[, ],R d, 5.12 where he L 2 P-norm of he firs erm on he righ hand side goes o zero as n due o Corollary 4.5. For he second erm on he righ hand side of 5.12 we use Remark 4.3 and Corollary 4.6 o obain D e i X,xn n D e i X n,x n n [, ] C[, ],R d = DX,xn n e i DX,X n,x n n n,x n n D e i X n,x n n ei C[, ],R d C[, ],R d,x DX n n DX,X + DX,Xn,x n n ei D e i X n,x n n C[,. ],R d 5.13 Applying Corollary 4.5, arguing as in 5.5, and using he fac ha L p P-convergence implies almos-sure convergence for a subsequence, one sees ha,x DX n k n k DX,X n k,x n k n k k ei C[, ],R d in L 2 P for an increasing sequence n k k N N. Finally, he second erm on he righ hand side of 5.13 ends o zero as n by heorem 4.1ii and a dominaed convergence argumen. hus, in summary, he esimaes yield he lefconinuiy of x F ε. o see ha x F ε is boundedness-preserving, we use he polynomial growh of Df ε : D[, ], R d L D[, ], R d, R, heorem 4.1ii and he esimae 4.1 o conclude ha for all [, ] and x D[, ], R d, E [ Df ε X,x E Df ε X,x 1 [, ] D e i X,x 2 ] L D[, ],R d,r 1 C E1 + X,x p D[, ],R d sup ξ R d C E1 + X,x 2p 1 D[, ],R d 2 C 1 + x q D[,],R d 2 E 1 [, ] D e i X,x 2 D[, ],R d E D e i X,ξ C[, ],R d where he exponens p, q [1, and he consans C, are suiably chosen and do no depend on, x or ε. Lemma 5.3. If f Cp 2C[, ], Rd, R and ε >, he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 is wice verically differeniable. he second verical derivaive 2 xf ε = 2 xf ε [, ] is lef-coninuous and boundedness-preserving, i.e., 2 x F ε C, l [, ] B[, ], and is given by x x F ε i j = 2 xf ε xe i, e j = E [ D 2 f ε X,x 1 [, ] D e i X,x, 1 [, ] D e j X,x 18 + Df ε X,x 1 [, ] D e i+e j X,x ] 5.15

19 [, ], x D[, ], R d, i, j {1,..., d}. Moreover, 2 x F ε x C 1 + x q D[,],R d 5.16 for all [, ] and x D[, ], R d, where C, q, does no depend on, x or ε. Proof. he proof of he saemen follows a line analogous o he proof of Lemma 5.2 and herefore we only give a shor skech. We fix [, ], x = x D[, ], R d, i, j {1,..., d} and apply he differeniaion lemma for parameer-dependen inegrals o he mapping δ, δ Ω h, ω Df ε j X,xhe ω 1 [, ] D e i X,x+he j ω R, where δ > and x he j D[, ], R d is he verical perurbaion of x by he j R d. For fixed ω, we apply he produc rule o a mapping of he form δ, δ h A h f h, where h A h L D[, ], R d, R and h f h D[, ], R d are Fréche differeniable, which akes he usual form wih an analogous proof o he real case d A dh hf h = d A d dh hf h + A h f dh h. Furhermore, A h akes he form A h = Bg h where h g h D[, ], R d and B : D[, ], R d L D[, ], R d, R are also Fréche d differeniable and hence, by he chain rule, A dh hf h = DBg h [ d g d dh h]f h + A h f dh h. hus, d dh Df ε X,x he j = D 2 f ε j X,xhe ω 1 [, ] D e i X,x+he j ω ω 1 [, ] D e j X,x+he j ω, 1 [, ] D e i X,x+he j ω + Df ε j X,xhe ω 1 [, ] D e i D e j X,x+he j ω Using he polynomial growh of Df ε and D 2 f ε ogeher wih heorem 4.1i his implies, as in he proof of Lemma 5.2, ha here exis C, q, such ha, for all h δ, δ d dh Df ε X,x C he j 1[, ] D e i X,x+he j q 1 + x D[,],R d + sup X,ξ C[, ],R d ξ B δ x sup D e i X,ξ ξ B δ x sup C[, ],R d ξ B δ x + sup D e i+e j X,ξ C[, ],R d ξ B δ x, D e j X,ξ C[, ],R d where he las he upper bound belongs o L p Ω for every p [1, by 4.3. herefore, using also he symmery of D 2 f ε, x x F ε i j = d dh [ d = E dh Df ε X,xhe j = E [ D 2 f ε X,x E [Df ε X,xhe j 1[, ] D e i X,x+he ] j 1[, ] D e i X,x+he j h=] 1 [, ] D e i X,x, 1 [, ] D e j X,x h= + Df ε X,x 1 [, ] D e i+e j X,x ]. 19

20 he proof of he lef coninuiy of he second erm is essenially idenical o he proof of he lef coninuiy of x F ε. For he lef coninuiy of he firs erm one uses a elescoping sum and Hölder s inequaliy o ge E [ D 2 f ε X,x 1 [, ] D e i X,x E [ D 2 f ε X n,x n 1 [ n, ]D e i X n,x n, 1 [, ] D e j X,x ], 1 [ n, ]D e j X n,x n EA n u n, v v n + EA A n u n, v + EAu u n, v A n L 4 Ω;L 2 L 1 [, ],R d,r u n L 4 Ω;L 1 [, ],R d v v n L 2 Ω,L 1 [, ],R d + A A n L 2 Ω;L 2 D[, ],R d,r u n L 4 Ω;D[, ],R d v L 4 Ω,D[, ],R d + A L 4 Ω;L 2 L 1 [, ],R d,r u u n L 2 Ω;L 1 [, ],R d v n L 4 Ω;L 1 [, ],R d := a n + b n + c n. Now b n can be reaed as 5.8 using Lemma A.1 wih ] := EAu, v An u n, v n B = D[, ], R d, S = L 2 D[, ], R d, R, Y = X,x, Y n = X n,x n, ϕ = D 2 f ε. he erms a n and c n can be handled analogously o error erm B in 5.9, where we firs selec a subsequence such ha a nk, hen a furher subsequence such ha c nkl. his will finally show ha for every x = x D[, ], R d Λ and every sequence x n n N Λ wih x n = x n D[, n n ], R d, n [, ], and d x n, x n, here exiss a subsequence x n k l l N such ha 2 x F ε n k l x n k l l 2 x F ε x verifying he lefconinuiy of 2 xf. Finally, he esimae 5.16 and hence ha 2 xf is boundedness preserving follows from 5.15 by analogous esimaes as in 5.14, using he polynomial growh of Df ε : D[, ], R d L D[, ], R d, R and D 2 f ε : D[, ], R d L 2 D[, ], R d, R combined wih heorem 4.1ii and he esimae 4.1. Remark 5.4. In a compleely analogous fashion, wih more noaional effor, one can prove ha if f Cp n C[, ], R d, R and ε >, hen he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 is n-imes verically differeniable, n N. he n-h verical derivaive n x F ε = n x F ε [, ] is lef-coninuous and boundedness-preserving, i.e., n x F ε C, l [, ] B[, ], and n x F ε x C 1 + x q D[,],R d for all [, ] and x D[, ], R d, where C, q, does no depend on, x or ε. Lemma 5.5. If f Cp 1C[, ], Rd, R and ε >, he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 is horizonally differeniable. he horizonal derivaive DF ε = D F ε [, is coninuous a fixed imes, and he exension D F ε [, ] of D F ε [, by zero belongs o he class B[, ]. he horizonal derivaive is given by D F ε x = E [ DfM ε X,x xη ε [,, x D[, ], R d. Moreover, D F ε x C ε 1 + x q D[,],R d η ε rx,x r dr ], 5.18 for all [, and x D[, ], R d, where C ε, q, do no depend on or x

21 Proof. Fix [,. In order o verify ha F ε is horizonally differeniable a, we have o show ha for every x = x D[, ], R d he righ derivaive D F ε x = d+ dh Ef ε X +h,x,h 1 = lim h= hց h E[ f ε X +h,x ],h f ε X,x 5.2 exiss. For x D[, ], R d and y D[, ], R d, le x y D[, ], R d denoe he càdlàg funcion defined by xs, s [, x y s := ys, s [, ]. Moreover, for h [, ], le h : D[, ], R d D[, ], R d be he ranslaion operaor defined by y, s [, + h h ys := ys h, s [ + h, ]. Noe ha, due o he ime-homogeneiy of Eq. 1.1, he D[, ], R d -valued random variables X +h,x,h and x h X,x have he same disribuion. As a consequence, we can rewrie 5.2 as D F ε x = d+ dh Ef ε x h X,x = d+ h= dh Ef M ε[ x h X,x ] h= Now, for y D[, ], R d and s [, ], M ε[ x h y ] s = = ε ε η ε s rx r dr + +h η ε s rx r dr + y η ε s ry dr + +h η ε s r dr + +h h η ε s ryr h dr η ε s r hyr dr and herefore, as supp η ε [, ε] and s [, ] and hence he boundary erm vanishes when differeniaing he hird inegral above, d + dh M ε[ x h y ] s = η ε s hy h η ε s r hyr dr = η ε s hy η ε s ryr h dr, s [, ]. +h he above calculaion is also valid uniformly wih respec o s [, ]; ha is, in C[, ], R d, as η is C wih compac suppor. In order o differeniae under he expecaion sign in 5.21, for h [, ], we have he bound d + dh f M ε[ x h X,x ] = Df M ε[ x h X,x ] xηε h η ε rx,x r h dr +h C 1 + x h X,x q 5.23 D[, ],R d xη ε h η ε rx,x r h dr +h C[, ],R d C ε 1 + x D[,],R d + X,x q X,x C[, ],R d C[, ],R d 21

22 where he las upper bounds belongs o L p Ω for every p [1, due o 4.1. herefore, by 5.21, i follows ha D F ε x = d+ dh Ef M ε[ x h X,x ] h= = E [ DfM ε X,x xη ε = E d+ dh f M ε[ x h X,x ] η ε rx,x r dr ], h= 5.24 for [, and x D[, ], R d. he coninuiy of DF ε a fixed imes now follow from he formula 5.18 and he coninuiy of ξ X,ξ assered by heorem 4.1i. Finally, 5.19 follows from 5.23 and 4.1 and herefore he exension D F ε [, ] of D F ε [, by zero belongs o he class B[, ]. Remark 5.6. Using he formulae for x F ε and 2 xf ε from Lemmaa 5.2 and 5.3, respecively, and argumens compleely analogous o he ones in Lemma 5.5 one also has ha x F ε and 2 x F ε are horizonally differeniable, if f Cp 2C[, ], Rd, R and f CpC[, 3 ], R d, R, respecively in fac, n xf ε is horizonally differeniable, if f Cp n+1 C[, ], R d, R for all n N. For n = 1, 2 he horizonal derivaive D n xf ε = D n xf ε [, is coninuous a fixed imes, and he exension D n xf ε [, ] of D n x F ε [, by zero belongs o he class B[, ]. Moreover, D n xf ε x C ε 1 + x q D[,],R d for all [, and x D[, ], R d, where C ε, q, do no depend on or x. For example, using ha he D[, ], R d D[, ], R d -valued random variables +h,x X,h, 1 [+h, ] D e i X +h,x,h and x h X,x, h 1 [, ] D e i X,x have he same disribuion one can calculae, as in Lemma 5.5, D x F ε i x = E [ D 2 fm ε X,x xη ε + E [ DfM ε X,x e i η ε η ε rx,x r dr, M ε [1 [, ] D e i X,x ] ] η ε rd e i X,x r dr ]. Furhermore, using he formula for DF from Lemma 5.5 and argumens analogous o hose in he proof of Lemmaa 5.2 and 5.3 one can explicily check, for n = 1, 2, ha DF is n-imes verically differeniable if f Cp n+1 C[, ], R d, R and n xdf = D n xf in fac his holds for general n N. In summary, he combinaion of Lemmaa 5.1, 5.2, 5.3 and 5.5 implies he desired regulariy of F ε. heorem 5.7. If f Cp 2C[, ], Rd, R and ε >, he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 belongs o he class C 1,2 b [, ]. he verical and horizonal derivaives are given by 5.6, 5.15 and

23 6 Funcional Kolmogorov equaion In his secion we show ha F ε saisfies a backward funcional Kolmogorov equaion. We have already seen in he previous secion ha F ε is regular enough when f is. herefore, in order o apply heorem 3.7 one needs o check wheher F εx [, ] is a maringale w.r.. F [, ]. his is easily done using he following resul. Proposiion 6.1. Le ϕ: D[, ], R d R be a measurable mapping wih polynomial growh and Φ = Φ [, ] be he non-anicipaive funcional defined by Φ x := E ϕx,x, x D[, ], R d. 6.1 hen Φ X [, ] is a maringale w.r.. F [, ]. Proof. he soluion X o Eq. 1.1 is a Markov process w.r.. he filraion F [, ], see, e.g., [28, Secion 19.7]. For s, x R and ψ : D[, ], R d R bounded and measurable we have E ψ X s,x [, ] F = E ψ X,y [, ] y=x s,x, 6.2 compare [16, Proposiion 5.15]. Fix s and assume for a momen ha ϕ is of he form ϕx = ϕ 1 x [,s] ϕ 2 x [s,] ϕ 3 x [, ], x D[, ], R d, wih ϕ 1 : D[, s], R d R, ϕ 2 : D[s, ], R d R and ϕ 3 : D[, ], R d R measurable and bounded. In his case, EΦ X F s = E EϕX,y Fs y=x ] = E [ϕ 1 X [,s] ϕ2 X [s,] E ϕ3 X,y y=x [, ] F s [ = ϕ 1 X [,s] E ϕ 2 X s,x [s,] E ϕ3 X,y y=x [, ] ] s,x = ϕ 1 X [,s] E [ ϕ 2 X s,x [s,] E ϕ3 X s,x [, ] F ] x=xs = ϕ 1 X [,s] E [ ϕ 2 X s,x [s,] ϕ3 X s,x [, ] ] x=xs x=xs 6.3 = E ϕx s,x x=xs = Φ s X s, where we have used he Markov propery 6.2 in he hird and he fourh sep. Le C denoe he collecion of all cylinder ses A B of he form A = {x D[, ], R d : x 1 B 1,..., x n B n } where 1... n, B i BR d, i = 1,..., n, and n N. hen C is closed under finie inersecions, σc = B, and all A C saisfy E E1 A X,y y=x Fs = E 1A X s,x x=xs 6.4 according o 6.3 wih ϕ = 1 A. Since he class of all A B saisfying 6.4 is a Dynkin sysem, we obain ha 6.4 is fulfilled for all ses A B. By approximaion, he indicaor funcion 1 A in 6.4 can be replaced by every measurable ϕ : D[, ], R d R wih polynomial growh. 23

24 Now he backward funcional Kolmogorov equaion for F ε follows almos immediaely. Corollary 6.2. If f Cp 2C[, ], Rd, R and ε >, hen he non-anicipaive funcional F ε = F ε [, ] defined by 1.5 saisfies he funcional parial differenial equaion D F ε x = bx x F ε x 1 2 r 2 xf ε x σx σ x F ε x = fx for all, and all x C[, ], R d wih x = ξ. Proof. I follows from heorem 5.7 ha F ε C 1,2 b [, ] and Proposiion 6.1 shows ha 6.5 F ε X [, ] is a maringale w.r.. F [, ]. As shown in Lemma A.2, he opological suppor of X in C[, ], R d is he se {x C[, ], R d : x = ξ } and hence he resul follows from heorem Error represenaion Here we give an explici formula for he weak error Ef ε X f ε X, where X and X are he soluions o 1.1 and 1.3, respecively, and f ε is he regularized version of a given pah-dependen funcional f as defined in As he following remark shows, we implicily also obain a represenaion of he weak error Ef X fx for he original funcional f. Remark 7.1. Under Assumpions 2.1 and 2.2 and for f C 1 p C[, ], Rd, R, we have E f X fx = lim ε E f ε X f ε X. 7.1 his follows from applying a firs order aylor expansion o f around X, he dominaed convergence heorem, using ha f ε x ε fx for all x C[, ], R and he finieness of E X p C[, ],R + X p C[, ],R for p 1. he proof of our error represenaion formula is based on he funcional Iô formula from heorem 3.6, he regulariy properies of he non-anicipaive funcional F ε and he explici represenaion of is derivaives from heorem 5.7, and he backward funcional Kolmogorov equaion from Corollary 6.2. Recall ha we assume X = X = ξ R d and hence by he definiion 1.5 of F ε, we have E f ε X f ε X = E F ε X F ε X. heorem 7.2. Le Assumpions 2.1 and 2.2 hold, and le X = X and X = X [, ] be he srong soluions o Equaions 1.1 and 1.3, respecively, boh saring from ξ R d. Le f Cp 2C[, ], Rd, R and, for ε >, le f ε and F ε = F ε [, ] be given by and 1.5, respecively. hen, he following weak error formula holds: E f ε X f ε X = E x F ε X b, X b X d { r 2 x F ε X σ, X σ, X σ X σ X } d