ON STRATONOVICH AND SKOROHOD STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES

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1 The Annal of Probability 3, Vol. 4, No. 3A, DOI:.4/-AOP75 Intitute of Mathematical Statitic, 3 ON STRATONOVICH AND SKOROHOD STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES BY YAOZHONG HU,MARIA JOLIS AND SAMY TINDEL 3 Univerity of Kana, Univeritat Autònoma de Barcelona and Intitut Élie Cartan Nancy In thi article, we derive a Stratonovich and Skorohod-type change of variable formula for a multidimenional Gauian proce with low Hölder regularity γ (typically γ /4). To thi aim, we combine tool from rough path theory and tochatic analyi.. Introduction. Starting from the eminal paper [7], the tochatic calculu for Gauian procee ha been thoroughly tudied during the lat decade, fractional Brownian motion being the main example of application of the general reult. The literature on the topic include the cae of Volterra procee correponding to a fbm with Hurt parameter H>/4 (ee[, ]), with ome extenion to the whole range H (, ), ain[, 6, ]. It hould be noticed that all thoe contribution concern the cae of real valued procee, thi feature being an important apect of the computation. In a parallel and omewhat different way, the rough path analyi open the poibility of a pathwie type tochatic calculu for general (including Gauian) tochatic procee. Let u recall that thi theory, initiated by Lyon in [] (ee alo [9, 3, 9] for introduction to the topic), tate that if a γ -Hölder proce x allow to define ufficient number of iterated integral, then: () One get a Stratonovich-type change of variable for f(x)when f i mooth enough. () Differential equation driven by x can be reaonably defined and olved. In particular, the rough path method i till the only way to olve differential equation driven by Gauian procee with Hölder regularity exponent le than /, except for ome very particular (e.g., Brownian, linear or one-dimenional) ituation. More pecifically, the rough path theory relie on the following et of aumption: Received January ; revied February. Supported in part by a Grant from the Simon Foundation #96 and a General Reearch Fund from Univerity of Kana. Supported in part by Grant MTM Miniterio de Ciencia e Innovación and FEDER. 3 Supported in part by the (French) ANR grant ECRU. MSC ubject claification. Primary 6H35; econdary 6H7, 6H, 65C3. Key word and phrae. Gauian procee, rough path, Malliavin calculu, Itô formula. 656

2 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 657 HYPOTHESIS.. Let γ (, ) and x : [,T] R d be a γ -Hölder proce. Conider alo the nth order implex S n,t ={(u,...,u n ) : u < <u n T } on [,T]. The proce x i uppoed to generate a rough path, which can be undertood a a tack {x n ; n /γ } of function of two variable atifying the following three propertie: () Regularity: Each component of x n i nγ -Hölder continuou [in the ene of the Hölder norm introduced in (.)] for all n /γ, and x t = x t x. () Multiplicativity: Letting (δx n ) ut := x n t xn u xn ut for (,u,t) S 3,T, one require (.) ( δx n ) ut (i,...,i n ) = n n = x n u (i,...,i n )x n n ut (i n +,...,i n ). (3) Geometricity: For any n, m uch that n + m /γ and (, t) S,T, we have (.) x n t (i,...,i n )x m t (j,...,j m ) = x n+m t (k,...,k n+m ), k Sh(ī, j) where, for two tuple ī, j, (ī, j) tand for the et of permutation of the indice contained in (ī, j), and Sh(ī, j) i a ubet of (ī, j) defined by Sh(ī, j)={σ (ī, j) ; σ doe not change the ordering of ī and j}. With thi et of abtract aumption in hand, one can define integral like f(x)dx in a natural way (a recalled later in the article), and more generally et up the bai of a differential calculu with repect to x. Notice that according to Lyon terminology [9], the family {x n ; n /γ } i aid to be a weakly geometric rough path above x. Without any urprie, ome ubtantial effort have been made in recent year to contruct rough path above a wide cla of Gauian procee, from which emerge the cae of fractional Brownian motion. Let u recall that a fractional Brownian motion B with Hurt parameter H (, ), defined on a complete probability pace (, F, P), iad-dimenional centered Gauian proce. It law i thu characterized by it covariance function, which i given by (.3) E [ B t (i)b (i) ] = ( t H + H t H ) (i=j),,t R +. The variance of the increment of B i then given by E [( B t (i) B (i) ) ] = (t ) H, (,t) S,T,i =,...,d, and thi implie that almot urely the trajectorie of the fbm are γ -Hölder continuou for any γ<h. Furthermore, for H = /, fbm coincide with the uual Brownian motion, converting the family {B = B H ; H (, )} into the mot natural generalization of thi claical proce. Thi i why B can be conidered a one of the canonical example of application of the abtract rough path theory.

3 658 Y. HU, M. JOLIS AND S. TINDEL Until very recently, the rough path contruction for fbm were baed on pathwie-type approximation of B, ain[4, 3, 8]. Namely, thee reference all ue an approximation of B by a regularization B ε, conider the aociated (Riemann) iterated integral B n,ε and how their convergence, yielding the exitence of a geometric rough path above B. Thee approximation all fail for H /4. Indeed, the ocillation of B are then too heavy to define even B, following thi kind of argument, a illutrated by [5]. Neverthele, the article [] aert that a rough path exit above any γ -Hölder function, and recent progre [6, 8]how that different concrete rough path above fbm (and more general procee) can be exhibited, even if thoe rough path do not correpond to a regularization of the proce at take. Summarizing what ha been aid up to now, there are (at leat) two way to handle tochatic calculu for Gauian procee: (i) Stochatic analyi tool, mainly leading to a Skorohod-type integral and (ii) rough path analyi, baed on the pathwie convergence of ome Riemann um, giving rie to a Stratonovich-type integral. Though ome effort have been made in [3] to relate the two approache (eentially for a fbm with Hurt parameter H>/4), the current article propoe to delve deeper in thi direction. Namely, we hall addre the following iue: () We recall that, tarting from a given rough path of order N above a d- dimenional proce x, one can derive a Stratonovich change of variable of the form d t ( ) (.4) f(x t ) f(x ) = i f(x u )dx u (i) := J t f(xu )dx u i= for any f C N+ (R d ; R), andwhere i f tand for f/ x i. Thi formula i at the core of rough path theory, and i explained at large, for example, in [9]. Furthermore, it i well-known that the following repreentation for the integral J t ( f(x u )dx u ) hold true: conider a family of partition t ={ = t,...,t n = t} of [,t], whoe meh tend to. Then, denoting by N = γ, (.5) J t ( f(xu )dx u ) = lim n N t q= k= ( k! k+ i k,...,i i f(x t q ) ) x t q t q+ (i k ) x t q t q+ (i )x t q t q+ (i). Thee modified Riemann um will alo be eential in the analyi of Skorohodtype integral. () We then pecialize our conideration to a Gauian etting, and ue Malliavin calculu tool (in particular ome elaboration of [, 6]). Namely, uppoing that x i a Gauian proce, plu mild additional aumption on it covariance function, we are able to prove the following aertion:

4 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 659 (i) Conider a C (R d ; R) function f with exponential growth, and < t T. Then the function u [,t) (u) f(x u ) lie in the domain of an extenion of the divergence operator (in the Malliavin calculu ene) called δ. (ii) The following Skorohod-type formula hold true: (.6) f(x t ) f(x ) = δ ( [,t) f(x) ) + f (x u )R u du, where tand for the Laplace operator, u R u := E[ x u () ] iaumedtobe a differentiable function, and R tand for it derivative. It hould be emphaized here that formula (.6) i obtained by mean of tochatic analyi method only, independently of the Hölder regularity of x. Otherwie tated, a in many intance of Gauian analyi, pathwie regularity can be replaced by a regularity on the underlying Wiener pace. When both regularity of the path and regularity of the underlying Wiener pace are atified, we obtain the relation between the Stratonovich-type integral and the extended divergence operator. Let u mention at thi point the recent work [8] that conider problem imilar to our. In that article, the author define alo an extended divergence-type operator for Gauian procee (in the one-dimenional cae only) with very irregular covariance and tudy it relation with a Stratonovich-type integral. For the definition of the extended divergence, ome condition on the ditributional derivative of the covariance function R are impoed, one of them being that t R t atifie that μ(d, dt) := t R t(t ) (i.e., well defined) i the difference of two Radon meaure. Our condition on R are of a different nature, and we uppoe more regularity, but only for the firt partial derivative of R and the variance function. On the other hand, the definition of the Stratonovich-type integral in [8] i obtained through a regularization approach intead of rough path theory. A a conequence, ome additional regularity condition on the Gauian proce have to be impoed, while we jut rely on the exitence of a rough path above x. (3) Finally, one can relate the two tochatic integral introduced o far by mean of modified Wick Riemann um. Indeed, we hall how that the integral δ ( [,t) f(x))introduced at relation (.6) can alo be expreed a δ ( [,t) f(x) ) (.7) = lim n N t q= k= ( k! k+ i k,...,i i f(x t q ) t ) x t q t q+ (i k ) x t q t q+ (i ) x t q t q+ (i), where the (almot ure) limit i till taken along a family of partition t ={ = t,...,t n = t} of [,t] whoe meh tend to, and where tand for the uual

5 66 Y. HU, M. JOLIS AND S. TINDEL Wick product of Gauian analyi. Thi reult can be een a the main contribution of our paper, and i obtained by a combination of rough path and tochatic analyi method. Specifically, we have mentioned that the modified Riemann um in (.5) can be proved to be convergent by mean of rough path analyi. Our main additional technical tak will thu conit of computing the correction term between thoe Riemann um and the Wick Riemann um which appear in (.7). Thi i the aim of the general Propoition 4.7 on Wick product, which ha an interet in it own right, and i the key ingredient of our proof. It i worth mentioning at thi point that Wick product are uually introduced within the landmark of white noie analyi. We rather rely here on the introduction given in [6], uing the framework of Gauian pace. Let u alo mention that Riemann Wick um have been ued in [8] to tudy Skorohod tochatic calculu with repect to (one-dimenional) fbm for H greater than /, the cae of /4 <H / being treated in [5]. We go beyond thee cae in Theorem 4.8, and will go back to the link between our formula and the one produced in [5] at Section 4.3. In concluion, thi article i devoted to howing that Stratonovich and Skorohod tochatic calculu are poible for a wide range of Gauian procee. A link between the integral correponding to thoe tochatic calculu i made through the introduction of Riemann Wick modified um. On the other hand, the reader might have noticed that the integrand conidered in our tochatic integral are retricted to procee of the form f(x). The ymmetrie of thi kind of integrand implify the analyi of the Stratonovich Skorohod correction, reducing all the calculation to correction involving x only. An extenion to more general integrand would obviouly require a lot more in term of Wick-type computation, epecially for the term involving x k for k, and i deferred to a ubequent publication. Here i how our paper i organized: Section recall ome baic element of rough path theory which will be ueful in the equel. We obtain a Skorohod change of variable with Malliavin calculu tool only in Section 3. Finally, the repreentation of thi Skorohod integral by Wick Riemann um i performed in Section 4.. Some element of rough path theory. We recall here the minimal amount of rough path conideration allowing to obtain a Itô Stratonovich-type change of variable for a general R d -valued path. Thee preliminarie will be preented uing terminology taken from the o-called algebraic integration theory, which i a variant of the rough path theory introduced in [3]. We alo refer to [4] fora detailed introduction to the topic... Increment. The extended pathwie integration introduced in [3] i baed on the notion of increment, together with an elementary operator δ acting on them. The algebraic tructure they generate i decribed in [3, 4], but here we

6 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 66 preent directly the definition of interet for u, for ake of conciene. Firt of all, for an arbitrary real number T >, a vector pace V and an integer k, we denote by C k (V ) the et of function g : [,T] k V uch that g t,...,t k = whenever t i = t i+ for ome i k. Such a function i called a (k )-increment. The operator δ alluded to above act on general increment, but we only need it definition on C and C for our purpoe. Indeed, let g C and h C. Then, for any,u,t [,T],wehave (.) (δg) t = g t g and (δh) ut = h t h u h ut. Analytic type aumption are alo eential in the equel, and we meaure the ize of -increment by Hölder norm defined in the following way: for f C (V ), let f t (.) f μ = up t μ and C μ (V ) = { f C (V ); f μ < }.,t [,T ] Obviouly, the uual Hölder pace C μ (V ) will be determined in the following way: for a continuou function g C (V ),weimplyet (.3) g μ = δg μ, andwewillaythatg C μ (V ) if and only if g μ i finite. Notice that μ i only a emi-norm on C (V ), but we will generally work on pace of the type (.4) C μ,a (V ) = { g : [,T] V; g = a, g μ < } for a given a V,onwhich g μ define a ditance in the uual way... Itô Stratonovich formula. One of the important byproduct of the rough path theory i that it allow u to integrate function of the form ϕ(x) with repect to the path x. We refer to the aforementioned reference [9, 3, 9] for more detail on the topic, but let u tre informally the following fact: Whenever x C γ give raie to a rough path of order N = /γ, the integral of ϕ(x) with repect to x i defined for any function ϕ C N (R d ; R d ).For < t T, the integral from to t i denoted by J t (ϕ(x) dx) or t g(x u ), dx u R d. The integral J t (ϕ(x) dx) coincide with the uual Lebegue Stielje integral of ϕ(x) with repect to x whenever x i a mooth function. One hould be aware of the fact that the quantity J t (ϕ(x) dx) depend on the whole tack {x n ; n /γ }. It hould thu be undertood a a function of the rough path x above x. Some uitable converging Riemann um can be related to J t (ϕ(x) dx). They will be recalled in Theorem. for the pecial cae g = f with f C N+ (R d ; R).

7 66 Y. HU, M. JOLIS AND S. TINDEL The generalized integration theory induced by rough path analyi ha many conequence in term of differential calculu with repect to an irregular path x. In thi article, we hall focu on an important apect, namely the change of variable formula for f(x). Thi formula can then be read a follow (ee, e.g., [9], Chapter 3 and ): THEOREM.. For a given γ>with /γ =N, let x be a proce atifying the regularity, multiplicative and geometric aumption of Hypothei.. Let f be a C N+ (R d ; R) function. Then (.5) [ δ ( f(x) )] t = J t ( f(x)dx ) = t f(xu ), dx u R d, where the integral above ha to be undertood in the rough path ene. Moreover, ( ) J t f(x)dx [ (.6) = lim n t q= [ n = lim t q= i f(x tq )x t q,t q+ (i) + i f(x tq )x t q t q+ (i) N k= ] i k+ k,...,i i f(x t q )x k+ t q t q+ (i k,...,i,i) ] N + k! k+ i k,...,i i f(x t q )x t q t q+ (i k ) x t q t q+ (i )x t q t q+ (i) k= for any <t T, where the limit i taken over all partition t ={ = t,...,t n = t} of [,t], a the meh of the partition goe to zero. 3. Skorohod-type formula via Malliavin calculu. We take now a completely different direction in our conideration: the pointwie point of view, which had been adopted previouly, i abandoned in thi ection, and we try to contruct an integral with repect to a (Gauian) proce x by mean of tochatic analyi tool. We then prove that for any <t T, the function [,t) f(x)i in the domain of an extended divergence operator with repect to x, and prove an aociated Skorohod-type formula. A we hall ee, thi mainly tem from an extenion of [] to the d-dimenional cae, which i allowed thank to the ymmetrie of f(x). 3.. Preliminarie on Gauian procee. From now on, we pecialize our etting to a centered Gauian proce x = (x(),...,x(d))with i.i.d. coordinate, and covariance function (3.) R t := E [ x ()x t () ] and R t := E [ x t () ] = R tt,,t [,T].

8 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 663 We will add later ome hypothee on thee function. We can alo aume that x (j) =. The Gauian integration theory i baed on a completion [in L ( )] ofelementary integral with repect to x, which can be ummarized a follow (ee [4] for more detail): conider the pace of d-dimenional elementary function { S = f = (f,...,f d ); f j = n j i= a j i [t j i,tj i+ ), = t <t j < <tj n j = T } for j =,...,d. For any element f in S, we define the integral of firt order of f with repect to x a I (f ) := n d j a j i j= i= ( xt j i+ (j) x j t (j) ). i For θ : [,T] R, andj {,...,d}, denote by θ [j] the function with value in R d having all the coordinate equal to zero, except the jth coordinate which i equal to θ. It i readily een that E [ ( [j] ) ( I [,) I [k] [,t))] = (j=k) R t. So, we can define for ome indicator function of S, the following ymmetric and emi-definite form: [j] [,), [k] [,t) S = (j=k)r t, and we can extend it to all element of S by linearity. If we identify two function f and g in S when f g,f g S =, then, S become an inner product on S (actually, on the quotient pace obtained by thi identification). Therefore, for f and g in S, wehavethat E [ I (f )I (g) ] = f,g S and I define an iometric map from S, endowed with the inner product, S into a ubpace of L ( ). Thi map can be extended in the tandard way to an iometric map, denoted alo a I, from a real Hilbert pace that we will denote by H into a cloed ubpace of L ( ). From now on, denote the inner product of thi extended iometry by, H. We will aume that H i a eparable Hilbert pace (which i atified whenever R t i continuou). Let {e,e,...} be an orthonormal bai of H, andlet ˆ denote the ymmetric tenor product. Then (3.) f n = finite f i,...,i n e i ˆ ˆ e in, f i,...,i n R

9 664 Y. HU, M. JOLIS AND S. TINDEL i an element of H ˆ n with the Hilbert norm (3.3) f n H ˆ n = finite f i,...,i n. Moreover, H ˆ n i the completion of all the element like (3.) with repect to the norm (3.3). For an element f n H ˆ n, the multiple Itô integral of order n i well defined. Firt, any element of the form given by (3.) can be rewritten a (3.4) f n = finite f j,...,j m e ˆ k j ˆ ˆ e ˆ k m j m, where the j,...,j m are different and k + +k m = n. Then, if f n H ˆ n i given under the form (3.4), define it multiple integral a (3.5) I n (f n ) = finite f j,...,j m H k ( I (e j ) ) H km ( I (e jm ) ), where H k denote the kth normalized Hermite polynomial given by H k (x) = ( ) k e x / dk dx k e x / = j k/ ( ) j k! j j!(k j)! xk j. It hold that the multiple integral of different order are orthogonal and that E I n (f n ) = n! f n H ˆ n. Thi lat iometric property allow u to extend the multiple integral for a general f n H ˆ n by L ( ) convergence (notice once again that thi kind of cloure i different in pirit from the pathwie convergence conidered at Section ). Finally, one can define the integral of f n H n by putting I n (f n ) := I n ( f n ), where f n H ˆ n denote the ymmetrized verion of f n. Moreover, the chao expanion theorem tate that any quare integrable random variable F L (, G,P), where G i the σ -field generated by x, can be written a F = I n (f n ) with E [ F ] (3.6) = n! f n H ˆ n. n= We will introduce now the (iterated) derivative and divergence operator of the Malliavin calculu. We denote by C p (Rn ) the et of infinitely continuouly differentiable function f : R n R uch that f and all it partial derivative have polynomial growth. Let S denote the cla of mooth random variable of the form n= (3.7) F = f ( I (h ),...,I (h n ) ),

10 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 665 where f Cp (Rn ), h,...,h n are in H, andn. The derivative of a mooth random variable F S of the form (3.7)itheH-valued random variable given by n (3.8) DF = i f ( I (h ),...,I (h n ) ) h i, i= where i denote a uual x i. One can alo define for h H and F S the derivative of F in the direction of h a D h F = F,h H. The iteration of the operator D i defined in uch a way that for a mooth random variable F S the iterated derivative D k F i a random variable with value in H k. We alo conider for h k H k the kth derivative of F in the direction of h k, defined a D k h k F = D k F,h k H k. Let u fix now a notation for the domain of the iterated derivative D k :forevery p and any natural number k we introduce the eminorm on S given by [ F k,p = E ( F p) k + E ( D j F p ) ] /p. H j j= It i well-known that the operator D k i cloable from S into L p ( ; H k ). We will denote by D k,p the completion of the family of mooth random variable S with repect to the norm k,p. We will alo refer the pace D k, a the domain of the operator D k and denote it by Dom D k.iff ha the chaotic repreentation (3.6), we have that E ( D k F ) H = k n(n ) (n k + )n! f n H ˆ n n=k and a ueful characterization of Dom D k i the following: F Dom D k if and only if n k n! f n H ˆ n <. n= We will denote by δ the adjoint of the operator D (thi operator i alo referred a the divergence operator) and more generally, we denote by δ k the adjoint of D k. The operator δ k i cloed and it domain, denoted by Dom δ k,ithe et of H k -valued quare integrable random variable u L ( ; H k ) uch that E ( D k F,u H k ) C F for all F Dom D k,wherec i ome contant depending on u. Moreover, for u Dom δ k, δ k (u) i the element of L ( ) characterized by the duality relationhip E ( Fδ k (u) ) = E ( D k F,u ) (3.9) H k

11 666 Y. HU, M. JOLIS AND S. TINDEL for any F Dom D k.foru Dom δ, the random variable δ (u) i uually called Skorohod integral of u, becaue it coincide with the uual integral of u with repect to x for a large cla of elementary procee u;ee[4] for further detail. 3.. An operator aociated to x. In thi ection, we will conider a d- dimenional continuou proce atifying the following et of aumption: HYPOTHESIS 3.. The proce x = (x(),..., x(d)) i a centered Gauian proce with i.i.d. coordinate and null at. Letting R t and R t be defined a in (3.), we uppoe that thoe two function are continuou and the following two condition hold: () The variance function R t := R tt i differentiable at any point t (,T)and atifie that (3.) T R t dt <. () For each t [,T], the function R t i abolutely continuou. Moreover, the firt partial derivative R t of R t i well defined a.e. on [,T] and verifie (3.) T T R y ddy <. We will try now to identify a ueful operator for our future Gauian computation. Let ϕ = (ϕ(),...,ϕ(d)) (D T ) d,whered T i the pace of C function with compact upport contained in (,T). We have (ee, e.g., [7], where the - dimenional cae i conidered) that ϕ H and that I (ϕ) = T x,ϕ d, where, denote the ordinary Euclidean product in R d. Moreover, (D T ) d i a dene ubet of H. From now on, we ue alo the notation x(f) for I (f ). Given a function h : [,T] R, recall that h [j] denote the function with value in R d in which all the coordinate except the jth one are equal to and the jth coordinate equal to h. Therefore, for β D T and a<b T,wehavethat (3.) [j] [a,b),β[j] H = E[ I ( [l] ) ( [a,b) I β [j] )] [ (xb = E (l) x a (l) ) T T = (j=l) (R bt R at )β t dt xt, [ β [j]] ] t dt

12 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 667 T ( b = (j=l) = (j=l) b a a ( T ) R t d β t dt R y β y dy ) d. We will conider the firt iterated integral appearing on the right-hand ide of (3.) a a linear operator defined on D T. That i, we conider for [,T] and β D T, the following function: T Aβ() := R y β y dy. We will uppoe from now on that the following hypothei hold. HYPOTHESIS 3.. For any β D T, Aβ L ([,T]). REMARK 3.3. Condition (3.) onr t, tated in Hypothei 3., implie that Aβ belong to L ([,T]) whenever β D T. We have impoed the additional condition Aβ L ([,T]) in order to guarantee the integrability of many term appearing in the equel. Although one can weaken Hypothei 3., thi would complicate ome of the next tatement. We have thu choen to impoe it for the ake of implicity. EXAMPLE 3.4. It i eay to verify that the covariance function R y of the fractional Brownian motion atifie Hypothei 3.. It alo atifie Hypothei 3.. In fact, T Aβ() = R y β (y) dy = T H ( H y H ign( y) ) β (y) dy T = H ( y H ign( y) ) β (y) dy, becaue β D T. And from thi, it i eaily een that Aβ L ([,T]) if β D T. EXAMPLE 3.5. Conider now the ubfractional Brownian motion whoe covariance function i given by R t = H + t H ( ( + t) H + t H ), with H (, ). Argument imilar to thoe ued in the above example allow u to ee that thi covariance verifie Hypothee 3. and 3..

13 668 Y. HU, M. JOLIS AND S. TINDEL EXAMPLE 3.6. The fractional Brownian motion i, in ome ene, a canonical example of the cla of procee with tationary increment. Namely, for a centered proce with tationary increment and null at zero, the covariance function i expreed a R t = ( )) ( R() + R(t) R t, [we ue here the notation R( ) for R ]. In thi particular cae, we can how that relation (3.) yield the other relation in 3. andthatwealohaveaβ L ([,T]). Indeed, condition () of Hypothei 3. i a conequence of the fact that, for any y [,T], R(,y) = ( R () ign( y)r ( y )) i well defined for any (,T)\{y} and atifie T R(,y) d <. Thi implie, in particular, that R y i abolutely continuou. Moreover T T R(,y) ddy i finite. On the other hand, for >, T Aβ() = = T R y β (y) dy ( R () R ( y ) ign( y) ) β (y) dy T = R ( y ) ign( y)β (y) dy, and thu Aβ i clearly bounded by a contant, due to (3.). EXAMPLE 3.7. Now let R y be the covariance function of the bi-fractional Brownian motion with parameter H,K (, ),o R y = (( H K + y H ) K y HK ). It i traightforward to verify that R y atifie Hypothei 3.. Inordertoee that R alo atifie Hypothei 3. when H>/4, we compute R y = HK (( H K + y H ) K H y HK ign( y) ). If β ha the upport contained in [t β,t], with t β >, then Aβ() = T t β R y β (y) dy [ T ( C H + y H ) K H dy + t β C ( H + HK + (T ) HK), T ] y HK dy

14 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 669 from which it i eaily verified that the covariance function of the bi-fractional Brownian motion atifie Hypothei 3. if H>/4. Moreover, one can check that the condition H>/4 i alo neceary in order to have that A(β) L ([,T]) for any β D T. In fact, one readily ee that A(β) L ([,T]) if and only if T ( U β () := y H + H ) K H β (y) dy L ( [,T] ). Suppoe now that β i a poitive function (β and not null), and by the integration by part formula, we have that for >, T U β () = ( K) H ( y H + H ) K y H β(y)dy bβ ( K)C β H ( y H + T H ) K y H dy = C H,K,β H, a β where we have ued that there exit a trictly poitive contant C β andaninterval [a β,b β ] (,T)uch that β C β on [a β,b β ]. Let u now relate our operator A to the inner product in H: equation (3.) tell u that for any elementary function g = (g(),...,g(d)) S and β D T,wehave that β [j],g(l) [l] T H = (j=l) g (l)aβ()d. Since for ϕ (D T ) d,g S, wehaveϕ = d j= ϕ(j) [j] and g = d l= g(l) [l],we obtain that d T T (3.3) ϕ,g H = g (j)aϕ(j)() d = g, Aϕ() d, j= whereweuethenotationaϕ = (Aϕ(),...,Aϕ(d)). Extending thi lat relation by continuity, the following ueful repreentation for the inner product in H i readily obtained: LEMMA 3.8. (3.4) For any g H (L ([,T])) d and ϕ (D T ) d, one can write T ϕ,g H = g, Aϕ() d, where, tand for the inner product in R d. Going back to our example 3.4, notice that expreion (3.4) i imilar to the following one pointed out in [] for the one-dimenional fractional Brownian motion with Hurt parameter H</: ϕ,g H = c H T g()d α + Dα ϕ()d,

15 67 Y. HU, M. JOLIS AND S. TINDEL where α = H ; Dα + and Dα are the Marchaud fractional derivative (ee [7] for more detail about thee object), and c H i a certain poitive contant Extended divergence operator. Let u take up here the notation of Section 3.. Having noticed that fbm give rie to an operator D α + Dα which i a particular cae of our operator A (ee Example 3.3), one can naturally try to define an extenion of the operator δ uing imilar argument to thoe of []. The idea i to conider firt u Dom δ (L ( [,T])) d and F = H n (x(ϕ)) where H n i the nth normalized Hermite polynomial, and ϕ (D T ) d.since(l ( [,T])) d L ( ; L ([,T]; R d )) and Dom δ L ( ; H), wehavethatu (L ( [,T])) d H almot urely. Moreover, DH n (x(ϕ)) = H n (x(ϕ))ϕ (D T ) d, a.. So, uing (3.4), the uual duality relationhip between D and δ can be written in the following way: E [ δ ( )] [ ( ] [ ( ) ] (u)h n x(ϕ) = E u, DHn x(ϕ) ) H = E Hn x(ϕ) u, ϕ H ( ) = E [H T n x(ϕ) u, Aϕ() ] (3.5) d = T and thi motivate the following definition. E [ Hn ( x(ϕ) ) u ], Aϕ() d, DEFINITION 3.9. We ay that u Dom δ if u (L ( [,T])) d and there exit an element of L ( ), that will be denoted by δ (u), uch that for any ϕ (D T ) d and any n, the following i atified: (3.6) E [ δ ( )] T [ ( ) ] (u)h n x(ϕ) = E Hn x(ϕ) u, Aϕ() d. REMARK 3.. Since the linear pan of the et {H n (ϕ) : n,ϕ (D T ) d } i dene in L ( ), the element δ (u), if it exit, i uniquely defined. REMARK 3.. One can eaily ee from our definition of the extended divergence that it i a cloed operator in the following ene: if {u k } k N Dom δ and atifie () u k u in (L ( [,T])) d and () δ (u k ) X in L ( ),then u Dom δ and δ (u) = X. We how in the following propoition that the extended operator δ defined above i actually an extenion of the divergence operator of the Malliavin calculu. PROPOSITION 3.. following ene: The domain Dom δ i an extenion of Dom δ in the Dom δ ( L ( [,T] )) d = Dom δ L ( ; H).

16 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 67 Furthermore, the extended operator δ retricted to Dom δ (L ( [,T])) d coincide with the tandard divergence operator. PROOF. If u Dom δ (L ( [,T])) d,thenu (L ([,T])) d H almot urely. Thu (3.4) can be applied to u, and(3.5) hold true for δ (u) (the tandard divergence operator). Thi prove that Dom δ (L ( [,T])) d Dom δ L ( ; H) and that δ i an extenion of the tandard divergence operator on Dom δ (L ( [,T])) d. To ee the other incluion, take u Dom δ L ( ; H). By our Definition 3.9 of Dom δ, u belong alo to (L ( [,T])) d. We will how that u Dom δ. Firt, we will prove that the element δ (u) defined by equality (3.6) atifie, for any ϕ (D T ) d and any n, that (3.7) E [ δ (u)h n ( x(ϕ) )] = E [ u, DHn ( x(ϕ) ) H]. Indeed, ince u (L ([,T])) d H a.. by aumption, we can apply again identity (3.4), and o ( ) u, DHn x(ϕ) H = H ( ) T n x(ϕ) u, Aϕ() d. Hence, uing Fubini theorem and (3.6) we end up with E ( ) T u, DH n x(ϕ) H = [ ( ) ] E Hn x(ϕ) u, Aϕ() d = E [ δ (u)h n ( x(ϕ) )], which i exactly (3.7). By uing denity argument [the linear pace generated by the element of the form H n (x(ϕ)), with ϕ (D T ) d, n, i dene in Dom D] we obtain that E [ u, DF H ] = E [ δ (u)f ] for any F Dom D, and thi finihe the proof. EXAMPLE 3.3. Go back to our fbm Example 3.4, and let u compare the extended divergence operator introduced above with the one defined in []. Firt of all, we mut point out that in [], the (tandard) divergence operator i preented in a more general etting than our: the divergence can belong to any L p ( ), for p>. In our paper, we will only conider thi divergence over L pace for ake of conciene. According to the computation carried out in [5] (ee identity (5.3) of that work), for any element ψ in the pace of tet function D T, one ha T ch Dα + Dα ψ()= H y H ign( y)ψ (y) dy.

17 67 Y. HU, M. JOLIS AND S. TINDEL On the other hand, we have already een in example 3.4 that Aψ() = T H y H ign( y)ψ (y) dy.thati,ond T, we have the following identity of operator: ch Dα + Dα = A. Moreover, thee operator can be extended (and coincide) by denity argument to I α(e H );ee[] for the definition of thi pace. Finally, in thi cae, H = I α(l ([,T]) i a ubet of L ([,T]). Uing thee obervation, it i readily checked that the extended divergence operator defined above coincide with the extended divergence given in [], retricted to L pace Change of variable formula for Skorohod integral. We can now turn to the main aim of thi ection, namely the proof of a change of variable formula for f(x)baed on our extended divergence operator δ. Interetingly enough, thi will be achieved under ome nonretrictive exponential growth condition on f. DEFINITION 3.4. We will ay that a function f : R d R atifie the growth condition (GC) if there exit poitive contant C and λ uch that (3.8) λ< and 4d max t [,T ] R f(x) Ce λ x for all x R d. t Notice that max t [,T ] R t = max t [,T ] E[ x t ]. Thu the growth condition above implie that [ E up f(x t ) r] C r E ( e rλmax t [,T ] x t ), t [,T ] and thi lat expectation i finite (ee, e.g. [], Corollary 5.4.6) if and only if rλ < max t [,T ] E ( x r ) = d max t [,T ] R t. So, if condition (3.8) i atified, there exit r>uch that [ (3.9) E f(x t ) r] <. up t [,T ] With thee preliminarie in hand, we firt tate a Skorohod-type change of variable formula for a very regular function f. PROPOSITION 3.5. Let f C (R d ) uch that f and all it derivative atify the growth condition (GC) (with poibly different λ and C ). Then, for any <t T, and [,t) ( ) f(x ) Dom δ δ [ [,t) ( ) f(x ) ] = f(x t ) f(x ) t f (x ρ )R ρ dρ.

18 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 673 PROOF. Since f and all it derivative atify growth condition (GC), the proce [,t) f(x)i an element of (L ( [,T])) d, and we alo have f(x t ) f(x ) t f (x ρ )R ρ dρ L ( ). So, we only need to how that for any n andanyϕ (D T ) d, the following equality i atified: [( E f(x t ) f(x ) t ) f (x ρ )R ρ dρ ( ) ] H n x(ϕ) (3.) t [ ( ) = E Hn x(ϕ) f(xρ ) ], Aϕ(ρ) dρ. The proof of thi fact i imilar to that of [], Lemma 4.3, although ome technical complication arie from the fact that here we deal with the multidimenional cae. Conider thu the Gauian kernel ( p(σ,y) = (πσ) d/ exp y ) (3.) for σ>,y R d. σ It i a well-known fact that σ p = p. Moreover, we have that E[g(x t)] = R d p(r t, y)g(y) dy for any regular function g : R d R uch that g and all it derivative atify (GC). Uing thee identitie, we can perform the following computation: d dt E[ g(x t ) ] = d dt p(r t, y)g(y) dy = R d R d σ p(r t,y)r t g(y)dy = R t p(r t, y)g(y) dy = (3.) R d R t p(r t,y) g(y)dy R d = R t E[ g(x t ) ]. Thi how that the function dt d E[g(x t)] i defined in all t (,T)and i integrable on [,T]. A a conequence, E[g(x t )] i abolutely continuou. Uing thi fact and identity (3.), we can now prove (3.) whenn =. Indeed, oberve that in thi cae H (x) and, by definition, H (x). Hence, the right-hand ide of (3.) i equal to while the left-hand ide give [ E f(x t ) f(x ) t ] f (x ρ )R ρ dρ t d = dρ E[ f(x ρ ) ] dρ t E [ f (x ρ ) ] R ρ dρ, and thi lat quantity vanihe due to (3.).

19 674 Y. HU, M. JOLIS AND S. TINDEL Let now n. Define for j {,...,d} and t [,T], ρ (3.3) G j ϕ (ρ) = Aϕ(j)() d = [j] [,ρ),ϕ(j)[j] H. Clearly, G j ϕ i abolutely continuou, and (G j ϕ) = Aϕ(j) (a.e). Moreover, for a regular function g atifying (GC) together with all it derivative and for any multiindex (j,...,j n ) {,...,d} n,wehave d ( [ E g(xρ ) ] G j ϕ dρ (ρ) Gj n ϕ (ρ) ) (3.4) = E[ g(x ρ ) ] R ρ Gj ϕ (ρ) Gj n ϕ (ρ) + E [ g(x ρ ) ] n r= [ Aϕ(j r )(ρ) G j l ϕ ], (ρ) l:l r wherewehaveued(3.). Set M j,...,j n t E[ j n,...,j n f(x ρ )] n l= G j l ϕ (ρ) for ρ [,t], where we are writing j n,...,j n f for n y j y jn f. By integrating (3.4) from to t and taking g = j n,...,j n f, we obtain that M j,...,j n t M j,...,j n = ( t E [ j n,...,j n f(x ρ ) ] ) n R ρ G j l ϕ (ρ) dρ + ( t l= E [ j n,...,j n f(x ρ ) ]( n Aϕ(j r )(ρ) r= l:l r G j l ϕ (ρ) )) dρ. Summing thee expreion over all the multiindice (j,...,j n ) belonging to {,...,d} n and owing to the fact that j,...,j n t = n E [ j n,...,j n f(x ρ ) ]( n Aϕ(j r ) t r= E [ j n,...,j n f(x ρ ) ] n j,...,j n l= we end up with an expreion of the form [ j M,...,j n t M j,...,j n ] j,...,j n = t (3.5) E [ j n,...,j n f(x ρ ) ] R ρ j,...,j n + n j,...,j n t E [ j n,...,j n f(x ρ ) ] n l= l:l r G j l ϕ (ρ) ) dρ G j l ϕ (ρ)aϕ(j n)(ρ) dρ, n l= G j l ϕ (ρ) dρ G j l ϕ (ρ)aϕ(j n)(ρ) dρ.

20 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 675 It hould be oberved at thi point that, a in identity (.6), the ymmetrie of the partial derivative of f play a crucial role in the proof of the current propoition. Thi ymmetry property appear preciely in the computation above. We will ee now how to obtain the deired identity (3.) from (3.5). Indeed, it i a well-known fact (ee [4] again)thath n (x(ϕ))ϕ Dom δ and δ [H n (x(ϕ))ϕ] =nh n (x(ϕ)). Uing thee lat two fact, the duality relationhip between D and δ and the definition (3.3)ofG ϕ j we have that, for g atifying (GC) a well a it derivative, E [ H n ( x(ϕ) ) g(xt ) ] = n E H n ( x(ϕ) ) ϕ,dg(xt ) H = n E ( ) H n x(ϕ) ϕ,[,t) g(x t ) H = [ n E ( ) d ] t H n x(ϕ) j g(x t )Aϕ(j)(ρ) dρ j= = d E [ ( ) H n x(ϕ) j g(x t ) ] G j ϕ n (t). j= Iterating thi procedure n time, one end up with the identity E [ H n ( x(ϕ) ) g(xt ) ] = E [ j n n!,...,j n g(x t ) ] G j ϕ (t) Gj n ϕ (t). j,...j n A an application of thi general calculation, we can deduce the following equalitie: E [ ( ) H n x(ϕ) f(xt ) ] = E [ j n n!,...,j n f(x t ) ] G j ϕ (t) Gj n (3.6) ϕ (t), j,...j n E [ ( ) H n x(ϕ) f(x ) ] = E [ j n n!,...,j n f(x ) ] G j ϕ () Gj n (3.7) ϕ (), j,...j n E [ ( ) H n x(ϕ) f (xρ ) ] = E [ j n n!,...,j n f (x ρ ) ] G j ϕ (ρ) Gj n (3.8) ϕ (ρ) j,...j n and E [ ( ) H n x(ϕ) j f(x ρ ) ] (3.9) = E [ j,...,j (n )! n j f(x ρ ) ] G j ϕ (ρ) Gj n ϕ (ρ). j,...j n

21 676 Y. HU, M. JOLIS AND S. TINDEL Subtituting now (3.6) (3.9) in(3.5), we obtain n!e [ H n ( x(ϕ) ) f(xt ) ] n!e [ H n ( x(ϕ) ) f(x ) ] = t n! E [ ( ) H n x(ϕ) f (xρ ) ] R ρ dρ t + n(n )! d E [ ( ) H n x(ϕ) jn f(x ρ ) ] Aϕ(j n )(ρ) dρ, j n = and thi i actually equality (3.). The proof i now complete. Since the Skorohod divergence operator i cloable, we can now generalize our change of variable formula: THEOREM 3.6. The concluion of Propoition 3.5 till hold true whenever f i an element of C (R d ) uch that f and it partial derivative up to econd order verify the growth condition (GC). PROOF. Let λ be the contant appearing in the growth condition (GC). Given k>λ, denote by p k (y) = p( k,y) the Gauian kernel defined in (3.), and introduce f k (y) = (f p k )(y) where, a uual, denote the convolution product. We firt claim that there exit k N, C > andλ atifying λ<λ < 4d max t [,T ] R t, uch that up f k (y) (3.3) C e λ y. k k Indeed, condition (GC) eaily yield f k (y) ( ) k d/ f(y z) e k z / dz π R d ( ) k d/ C π R d eλ y z e k z / dz d ( k = C e λ(y i z i ) e kz i / dz i ). π R i= On the other hand, {( ) } k e λ(y i z i ) k λk e kz i / dz i = π k λ exp yi, k λ R λk and lim k k λ = λ. Hence, given λ (λ, 4d max t [,T ] R t ), there exit k N uch that for any k k, the following inequalitie are atified: λ< λk k λ <λ <. 4d max t [,T ] R t

22 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 677 Our claim (3.3) i now eaily deduced. Notice that (3.3) mean that for k k, f k alo atifie the growth condition (GC) (with C and λ ubtituting C and λ, rep.). Moreover, we have that [ E up up f k (x ρ ) ] <. k k ρ [,T ] Thank to thi inequality, a well a imilar one involving the derivative of f, one can eaily ee that: () f k (x ) f(x ) and f k (x t ) f(x t ) in L ( ), () t f k (x ρ )R ρ dρ t f (x ρ )R ρ dρ in L ( ) and (3) [,t) f k (x ) [,t) f(x ) in (L ( [,T])) d. The reult i finally obtained by applying Propoition 3.5 and the cloene of the extended operator δ alluded to at Remark Repreentation of the Skorohod integral. Up to now, we have given two unrelated change of variable formula for f(x): one baed on pathwie conideration (Theorem.) and the other one by mean of Malliavin calculu (Theorem 3.6). We propoe now to make a link between the two formula and integral by mean of Riemann um. Namely, let x be a proce generating a rough path of order N. Wehave een in equation (.6) that the Stratonovich integral J t ( f(x)dx) i given by lim t S t, where S n t := N q= k= k! k i k,...,i f(x tq )x t q,t q+ (i )x t q,t q+ (i ) x t q,t q+ (i k ). In a Gauian etting, it i thu natural to think that a natural candidate for the Skorohod integral δ ( f(x))i alo given by lim t S t,, with (4.) S t, := n N q= k= k! k i k,...,i f(x tq ) x t q,t q+ (i ) x t q,t q+ (i k ), where denote the Wick product. We hall ee that thi i indeed the cae, with the following trategy: (i) One hould thu firt check that lim t S t, exit. In order to check thi convergence, we hall ue extenively Wick calculu, in order to write k i k,...,i f(x tq ) x t q,t q+ (i ) x t q,t q+ (i k ) = k i k,...,i f(x tq )x t q,t q+ (i ) x t q,t q+ (i k ) + ρ tq,t q+,

23 678 Y. HU, M. JOLIS AND S. TINDEL where ρ i a certain correction increment which can be computed explicitly. Plugging thi relation into (4.), we obtain (4.) S t, = S n t + ρ tq,t q+. (ii) Manipulating the exact expreion of the remainder ρ, we will be able to prove that lim n t q= ρ t q,t q+ = t f (x v )R v dv. Hence, going back to (4.) and invoking the fact that S t converge to J t ( f(x)dx), we obtain q= lim t S t, ( ) = J t f(x)dx = [ δf (x) ] t t t f (x v )R v dv f (x v )R v dv. Thi give both the convergence of S t, and an Itô Skorohod formula of the form [ ] δf (x) t = lim t S t, + t f (x v )R v dv. (iii) Putting together thi lat equality and Theorem 3.6, it can be deduced that under Hypothee 3. and 3., the limit of S t, coincide with the Skorohod integral δ ( f(x)). Thi give our link relating the Stratonovich integral J ( f(x)dx) and the Skorohod integral δ ( f(x)). Thi relatively traightforward trategy being et, we turn now to the technical detail of it realization. To thi end, the main iue i obviouly the computation of the correction between Wick and ordinary product in um like S t,.we thu tart by recalling ome baic fact of Wick computation. 4.. Notion of Wick calculu. We preent here the notion of Wick calculu needed later on, baically following [6]. We alo ue extenively the notation introduced in Section 3.. One way to introduce Wick product on a Wiener pace i to impoe the relation I n (f n ) I m (g m ) = I n+m (f n ˆ g m ) for any f n H ˆ n and g m H ˆ m, where the multiple integral I n (f n ) and I m (g m ) are defined by (3.5). If F = N n= I n(f n ) and G = N m= I m(g m ),wedefinef G by F G = N N n= m= I n+m (f n ˆ g m ). By a limit argument, we can then extend the Wick product to more general random variable; ee [6] for further detail. In thi paper, we will take the limit in the L ( ) topology.

24 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 679 For f H, we define it exponential vector E(f ) by ( E(f ) := e I (f ) = exp I (f ) f ) ( H = exp I (f ) E( I (f ) ) ) = n! I ( n f n ) = n! I (f ) n. n= n= In a imilar way we can define the complex exponential vector of f by ( e ıi (f ) = exp ıi (f ) + f ) H ı n (4.3) = n! I (f ) n, where ı denote the imaginary unity. With thee notation in hand, an important property of a Wick product i the following relation: for any two element f and g in H, wehave (4.4) E(f ) E(g) = E(f + g), an analogou property for the complex exponential vector being alo atified. We now tate a reult which i a generalization of [6], Propoition 4.8. n= PROPOSITION 4.. Let F Dom D k and g H k. Then: () F I k (g) i well defined in L ( ); () Fg Dom δ k ; (3) F I k (g) = δ k (Fg). PROOF. Let F Dom D k.thiimpliethatf admit the chao decompoition F = n= I n (f n ), with (4.5) n= n k n! f n H ˆ n <. Define then F N = N n= I n (f n ). Conider alo g H k. In order to check (), we hall ee that the limit in L ( ) of F N I k (g) exit, a N.ButF N I k (g) = Nn= I n+k (f n ˆ g), and the limit in L ( ) of thi expreion exit if and only if (n + k)! f n ˆ g H ˆ n+k <. n= Thi lat condition i clearly atified, thank to (4.5). Now we will prove our claim () and (3) for F with a finite chao decompoition and g = g g k, with g i H, fori =,...,k. More preciely, we will ee that for any G S the following relationhip hold: (4.6) E [( F I k (g) ) G ] = E [ Fg,D k G H k ].

25 68 Y. HU, M. JOLIS AND S. TINDEL Thi will be done by an induction argument. For k =, thi i a conequence of [6], Propoition 4.8, ince F I (g) = δ (Fg). Suppoe now that (4.6) i atified for k = K. Therefore, E [[ F I K+ (g) ] G ] = E [( F I (g ) I k (g g K+ ) ) G ] = E [ ( F I (g ) ) g g K+,D K G H K ], where in the lat equality, we have ued that F I (g ) ha a finite chao expanion and the induction hypothei. The lat expreion can be rewritten a E [( F I (g ) ) D K g g K+ G ]. Since Dg K g K+ G S, we can apply the cae k = to the above expreion, and we obtain E [[ F I K+ (g) ] G ] = E [ Fg,Dg K g G ] K+ H = E [ FD ( g D K g g G )] = E [ FD K+ K+ g g g G ] K+ = E [ Fg g K+,D K+ G H K+ ], which finihe our induction procedure. Thu, (4.6) i atified for F with a finite chao expanion and g a tenor product of element of H. To extend the reult to a general F Dom(D k ) and g H k, we firt conider the cae F Dom(D k ) and g = g g k. In thi ituation, identity (4.6) ia conequence of the fact that thi relationhip hold for F N = N n= I n (f n ) defined above and of the part () of the propoition. Finally, for a general g H k,uing the fact that both ide of (4.6) are linear in g, we can generalize thi identity to g belonging to the linear pan of element of the form g g k, which i a dene ubpace of H k.soifg H k and {g M } M N i a equence of element of thi linear pan of tenor product uch that g M g in H k, one can eaily ee [by uing (4.5)] that a M in L ( ).Since F I k ( g M ) F I k (g) E [( F I k ( g M )) G ] = E [ Fg M,D k G H k ], the proof i completed by a limiting argument. 4.. One-dimenional cae. In order to implify a little our preentation, we firt how the identification of δ ( f(x))with lim t S t, when d =, that i, when x i a one-dimenional Gauian proce atifying Hypothei 3.. The firt tep in thi direction i a general formula for Wick product of the form G(X) Y p,wherex and Y are element of the firt chao; ee Section 3. for a definition. Notice that the proof of thi propoition i deferred to the Appendix for ake of clarity.

26 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 68 PROPOSITION 4.. Let g,h H and let G : R R be differentiable up to order p uch that all it derivative G (j) are element of L r (μ g ) for any j =,...,p and for ome r>, with μ g = N (, g H ). Define X = I (g) and Y = I (h). Then the Wick product G(X) Y p can be expreed in term of ordinary product a (4.7) G(X) Y p = G(X)Y p + <l+m p ( ) m+l p! m m!(p m l)! G (l) (X) [ E(XY ) ] l [ E ( Y )] m Y p m l. EXAMPLE 4.3. In order to illutrate the kind of correction term we obtain, let u write formula (4.7) forp =,, 3. G(X) Y = G(X)Y G (X)E(XY ), G(X) Y = G(X)Y G(X)E ( Y ) G (X)E(XY )Y + G (X) [ E(XY ) ], G(X) Y 3 = G(X)Y 3 3G(X)E ( Y ) Y + 3G (X)E(XY )E ( Y ) 3G (X)E(XY )Y + 3G (X) [ E(XY ) ] Y G (X) [ E(XY ) ] 3. We are now ready to tate our repreentation of the Skorohod integral by Riemann Wick um: THEOREM 4.4. Let x be a -dimenional centered Gauian proce with covariance function fulfilling Hypothee 3. and 3., and aume that x alo atifie Hypothee.. Let f be a function in C N (R) uch that f (k) verifie the growth condition (GC) for k =,...,N. Then the Skorohod integral δ ( [,t) f (x)) (whoe exitence i enured by Theorem 3.6) can be repreented a a.. lim t S t,, where S t, i defined by Moreover, we have (4.8) S t, n = δ ( [,t) f (x) ) = N i= k= t k! f (k) (x ti ) ( x t i t i+ ) k. t f (x ρ )dx ρ f (x ρ )R ρ dρ. PROOF. A mentioned at the beginning of the ection, our main tak i to compute S t, in term of ordinary product. Thi will be achieved by applying

27 68 Y. HU, M. JOLIS AND S. TINDEL Propoition 4. to each term in the above um, with G = f, X = x ti and Y = x t i t i+ = x ti+ x ti. To thi end, notice firt that the integrability condition on f required at Propoition 4. are fulfilled a oon a condition (GC) (ee Definition 3.4) i met. Fix then i {,...,n }, recall that we et X = x ti and Y = x t i t i+ and conider the quantity S i k := k! f (k) (x ti ) (x t i t i+ ),k. A direct application of Propoition 4. yield N Sk i = N ( ) l+m k= k= l+m k m m!l!(k m l)! f (k+l) (X) [ E(XY ) ] l [ E ( Y )] m Y k m l. Making a ubtitution q = k + l and l + m = u, thi expreion can be implified into (4.9) N Sk i = N f (q) ( ) l+m (X) k= q= l+m q/ m m!l!(q l m)! = = N q= N q= [q/] f (q) ( ) u (X) (q u)! u= [q/] f (q) (X) u= [ E(XY ) ] l [ E ( Y )] m Y q l m l+m=u ( ) u [ (q u)!u! [ [ ] l E(Y ] ) m E(XY ) Y q u m!l! E(XY ) + E(Y ) ] u Y q u. Moreover, recalling again that X = x ti and Y = x ti+ x ti, it i eaily een that (4.) E(XY ) + E(Y ) = [ E ( x ti+ ) E ( x ti )] = δr t i t i+, where we recall that δr ti t i+ tand for R ti+ R ti. Therefore, umming now over i {i,...,n }, weget (4.) S t, n = where the quantity T q,u i (4.) N Sk i = N [q/] i= k= q= u= T q,u i i defined by ( ) u n (q u)!u! u i= = f (q) (x ti )(δr ti t i+ ) u( x t i,t i+ ) q u. We now eparate the tudy into different cae. T q,u i,

28 STOCHASTIC CALCULUS FOR GAUSSIAN PROCESSES 683 Cae : If u = andq u = (namelyq = ), then n i= T q,u i = n = f (x ti )[R ti+ R ti ] i= ( t n i= f (x ti ) [ti,t i+ )(ρ) ) R ρ dρ, where in the lat equality we reort to the fact that R ρ i abolutely continuou; ee Hypothei 3.. From thi expreion, by a dominated convergence argument one eaily get lim n n i= T q,u i = t f (x ρ )R ρ dρ. Cae : If u oru =, q u, then lim n n i= T q,u i =. Indeed, recalling definition (4.) oft q,u i, we oberve that n i= T q,u { i max x q u t i n i t i+, δrti t i+ u } ( t n i= f (q) (x ti ) [ti,t i+ )(ρ)) R (ρ) dρ. On the right-hand ide of the above inequality, it i now eaily een that lim max { x q u t i t i+, δrti t i+ u } =, n i n while the integral term remain bounded by C t aumption. Thi complete the proof of our claim. Cae 3: If u =, then n i= T q,u n i = N i= q= q! f (q) (x ti ) ( x n ) q t i t i+ + R ρ N i= q=n+ dρ, which i bounded by q! f q (x ti ) ( x t i t i+ ) q. Thu Theorem. aert that the firt um above converge to t f (x u )dx u, while it i eay to ee that the econd um converge to, thank to the regularity propertie of x. Plugging now the tudy of our 3 cae into equation (4.), the proof of our theorem i eaily completed Relationhip with exiting reult. Several reult exit on the convergence of Riemann Wick um, among which emerge [5], dealing with a ituation which i imilar to our in the cae of a one-dimenional proce. In order to compare our reult with thoe of [5], let u pecialize our ituation to the cae of a dyadic partition of an interval [,t] with <t(the cae of a general partition i handled in [5], but thi retriction will be more convenient for our