Skorohod and Stratonovich integrals for controlled processes
aa r X i v : . [ m a t h . P R ] F e b SKOROHOD AND STRATONOVICH INTEGRALSFOR CONTROLLED PROCESSES
JIAN SONG AND SAMY TINDEL
Abstract.
Given a continuous Gaussian process x which gives rise to a p -geometric roughpath for p P p , q , and a general continuous process y controlled by x , under properconditions we establish the relationship between the Skorohod integral ş t y s d ˛ x s and theStratonovich integral ş t y s d x s . Our strategy is to employ the tools from rough paths theoryand Malliavin calculus to analyze discrete sums of the integrals. Contents
1. Introduction 11.1. Background 21.2. Main result and strategy 31.3. Notation 42. Preliminary material 52.1. Rough path above x x ď p ă Introduction
For sake of clarity, we will divide this introduction in 3 parts. In Section 1.1 we motivateour problem and recall some previous contributions giving Stratonovich-Skorohod correc-tions. Section 1.2 is devoted to a description of our main result, the strategy employed inthe article, and some perspectives for future works. At the end, some notations used in thisarticle are introduced in Section 1.3.
J.S. is partially supported by Shandong University grant 11140089963041.S.T. is partially supported by the National Science Foundation under grant DMS-1952966.
Background.
In recent decades, two approaches for the analysis of dynamical systemsdriven by Gaussian processes have been greatly developed: (i) the “probabilistic” approach,which invokes stochastic analysis tools and leads to Itô-Skorohod integration, and (ii) the“pathwise” approach which employs the theory of rough paths and gives rise to Stratonovichintegration. In general, one gets a more transparent understanding of the system by using thepathwise approach, while it is more convenient to explore probabilistic properties (e.g. com-pute the moments for the solution of a noisy dynamical system driven by a Gaussian noise)via the probabilistic approach. One key ingredient to understand the connection betweenthese two approaches is the relationship between Skorohod and Stratonovich integrals.For a standard Brownian motion, the relationship between Itô and Stratonovich integralsis well-known. It is classically obtained by Itô calculus, although rough paths theory canalso be invoked by observing that both Itô and Stratonovich integrals can be regarded asintegrals against rough paths lifted from a Brownian motion with different second orderterms. For general Gaussian processes (consider fractional Brownian motion as a typicalexample), however, it is non-trivial to obtain the relationship. Indeed, for a general Gaussianprocess Itô calculus (or martingale calculus) is not available, and moreover Skorohod integralscannot be regarded as integrals against rough paths lifted from the corresponding Gaussianprocesses. We briefly recall some results giving Skorohod-Stratonovich corrections below.Let x “ p x , . . . , x d q be a d -dimensional centered Gaussian process with i.i.d componentsgiving rise to a p -geometric rough path, where ă p ă (see Section 2.1 for more detailsabout geometric rough paths). Denote by R the covariance function of x , namely R p s, t q “ E r x t x s s . We also set R t “ R p t, t q . The correction terms between Skorohod and Stratonovichintegrals with respect to x have been considered in the following cases: (i) In [15], the Skorohod-Stratonovich corrections were computed for integrals of the form ş ts ∇ f p x u q d ˛ x u for a smooth function f defined on R d . More specifically, ż t ∇ f p x r q d x r “ ż t ∇ f p x r q d ˛ x r ` ż t ∆ f p x r q d R r , (1) {eq:strato-sko-1}{eq:strato-sko-1} where the integral with respect to x on the left-hand side is a Stratonovich integral whilethe one on the right-hand side is a Skorohod integral. The strategy in [15] relied on the factthat ş t ∇ f p x u q d ˛ x u is obtained by taking limits of Riemann-Wick sums of the form: S Π st , ˛ “ n ´ ÿ i “ N ÿ k “ k ! f p k q p x t i q ˛ ` x t i t i ` ˘ ˛ k , (2) {eq:riemann-wick-sum}{eq:riemann-wick-sum} where ˛ stands for the Wick product. Then the Skorohod-Stratonovich corrections in [15]were analyzed thanks to a computation of the Wick corrections in (2). Notice that anextension of this result to the case of a Gaussian process indexed by r , s with Hölderexponent greater than { was handled in [24]. There some change of variable formulaswere derived for the Stratonovich and Skorohod integrals respectively. As a consequence,the correction terms were computed explicitly. Also note that some preliminary cases for a1-d fractional Brownian motion had been considered in [22]. (ii) The reference [4] is concerned with solutions of rough differential equations driven by x , where x is again a d -dimensional centered Gaussian process with i.i.d components giving KOROHOD AND STRATONOVICH INTEGRALS 3 rise to a p -geometric rough path (recall that ă p ă ). The equation can be written as d y r “ σ p y r q d x r , (3) {eq:rde}{eq:rde} with a smooth enough coefficient σ : R d Ñ R d ˆ d , and we refer to [11] for more details aboutthis object. Below we denote by y the initial condition of (3), and J xu is designated as theJacobian of the flow map y Ñ y u . Then the formula for the correction terms in [4] can beread as ż t y r d x r “ ż t y r d ˛ x r ` ż t tr r σ p y r qs d R r ` ż r ă r ă r ă t s tr “ J xr p J xr q ´ σ p y r q ´ σ p y r q ‰ d R p r , r q . (4) {eq:strato-sko-rde-case}{eq:strato-sko-rde-case} Consider the i -th column σ i p x q of the coefficient matrix σ p x q as a vector field on R d for ď i ď d . If the Lie bracket r σ i , σ j s “ σ i σ j ´ σ j σ i “ for ď i ď j ď d , then the solution y t to (3) is of the form y t “ ϕ p x t , y q with p ∇ x ¨ ϕ qp x, y q “ tr r σ p x, y qs and J xt “ σ p y t q (see [1,Proposition 24]). Clearly in this case, (4) coincides with (1), noting that the last term on theright-hand side of (4) now vanishes. Therefore, relation (4) is indeed compatible with (1).1.2. Main result and strategy.
In this paper, we consider a d -dimensional centered Gauss-ian process x with i.i.d components. Let y be a controlled process relative to x . That is, theincrements δy st : “ y t ´ y s can be decomposed along the increments of x as follows: δy ist “ d ÿ j “ y x ; ijs x ; jst ` r ist , for i “ , . . . , d, (5) {eq:ctrld-proc-intro}{eq:ctrld-proc-intro} where y x has finite p -variation and the remainder r has finite p -variation (one can alterna-tively use Hölder spaces in this definition). Notice that controlled processes are the naturalclass of functions for which a proper rough integration with respect to x can be constructed(see e.g [13]). The following is the main result of this paper (see Theorem 3.1 below for amore precise statement). Under proper conditions on x and y , ż t y r d x r “ ż t y r d ˛ x r ` d ÿ i “ ż t y x ; iir d R r ` d ÿ i “ ż r ă r ă r ă t s ` D ir y ir ´ y x ; iir ˘ d R p r , r q . (6) {eq:main-result}{eq:main-result} On the left-hand side of (6), the integral ş t y r d x r is understood in the rough path sense (seeProposition 2.20 below for further details). On the right-hand side of the same equation, ş t y r d ˛ x r stands for the Skorohod integral, y x is defined by (5), R is the covariance functionalluded to above and D represents the Malliavin derivative (notions of Malliavin calculuswill be recalled in Section 2.4).Note that our formula (6) unifies the previous cases (1) and (4). Indeed, we have arguedthat (4) can be seen as an extension of (1). Furthermore, note that the solution y to the roughdifferential equation (3) with a sufficiently regular coefficient function σ p y q is a controlledprocess with y x ; ijs “ p σ p y s qq ij and D s y t “ J xt p J xs q ´ σ p y s q for s ď t . Therefore it is easy to seethat (6) is an extension of (4), which is the main result of [4]. JIAN SONG AND SAMY TINDEL
Inspired by [4, 15], our proof of the main result is based on the discrete sums methodcombined with tools from rough paths theory and Malliavin calculus, in which some discretetechniques developed in [17] are also invoked. We outline the idea as follows.Consider a controlled process y with a decomposition given by (5), satisfying some pathregularity and Malliavin differentiability conditions. Let π “ π n denote the uniform partitionof r , T s and H be the Hilbert space associated to x . Denote y π p t q “ n ´ ÿ k “ y t k r t k ,t k ` s p t q . We first prove that (see Lemma 3.2) lim n Ñ8 y π “ y in D , p H q . This enables us to show the convergence of the discrete Skorohod integral δ ˛ p y π q to theSkorohod integral δ ˛ p y q “ ş T y r d ˛ x r in L p Ω q , i.e., ż ts y r d ˛ x r “ lim n Ñ8 n ´ ÿ m “ « d ÿ i “ y it m ˛ x ; it m t m ` ff in L p Ω q , (7) {eq:sko-sums-intro}{eq:sko-sums-intro} where we have written π “ t t , . . . , t n u with t m “ s ` m p t ´ s q{ n for m “ , . . . , n. It is alsoknown from the rough paths theory that the following holds true almost surely: ż ts y r d x r “ lim | π |Ñ d ÿ i “ n ´ ÿ k “ ˜ y it k x ; it k t k ` ` d ÿ j “ y x ; ijt k x jit k t k ` ¸ , (8) {eq:strato-sums-intro}{eq:strato-sums-intro} where the left-hand side above stands for the rough paths integral of y with respect to x .The Stratonovich-Skorohod correction terms in (6) now can be obtained by computing thedifference between the right-hand sides in (7) and (8). When computing the difference, onekey ingredient will be the forthcoming Proposition 2.28. This proposition is inspired by theanalogous results in [17] and establishes a general estimate for weighted sums in the secondchaos of the Gaussian process x .To end this subsection, we provide some perspectives for future works. On the one hand,as an application, some central limit theorems for Skorohod integrals could be obtained withthe help of our main result, generalizing the results in [17] and [19]. On the other hand,noting that in this article the Gaussian rough paths with finite p -variation for p P p , q arehandled, we believe that our methodology can be carried out for rougher Gaussian pathswith p ě . It is also interesting to consider the correction terms for the processes arisingfrom delay equations ([18]), Volterra equations ([5, 6, 14]), etc.1.3. Notation.
Let π : 0 “ t ă t ă ¨ ¨ ¨ ă t n “ T be a partition on r , T s . Take s, t P r , T s .We write J s, t K for the discrete interval that consists of t k ’s such that t k P r s, t s . We denoteby S k pr s, t sq the simplex tp t , . . . , t k q P r s, t s k ; t ď ¨ ¨ ¨ ď t k u . In contrast, whenever we dealwith a discrete interval, we set S k p J s, t K q “ tp t , . . . , t k q P J s, t K k ; t ă ¨ ¨ ¨ ă t k u . For t “ t k we denote t ´ : “ t k ´ , t ` : “ t k ` . We also denote by D pr s, t sq the set of all dissections of r s, t s . KOROHOD AND STRATONOVICH INTEGRALS 5
For x “ p x , . . . , x n q and y “ p y , . . . , y n q in R n , we write write xy for their dot product x ¨ y “ ř ni “ x i y i and write | x | for the Euclid norm p ř ni “ x i q { . The L p -norm p E r| ξ | p sq { p ofa random variable ξ is denoted by } ξ } p , for p ě . Generally speaking, we will write C for a generic constant whose exact value can changefrom line to line. 2. Preliminary material
This section contains some basic tools from rough paths theory and Malliavin calculus,as well as some analytical results, which are crucial for the definition and integration ofcontrolled processes.2.1.
Rough path above x . In this subsection we shall recall the notion of a rough pathabove a signal x , and how this applies to Gaussian signals. The interested reader is referredto [8, 11, 13] for further details.As mentioned in Section 1.3, for s ă t and m ě , we consider the simplex S m pr s, t sq “tp u , . . . , u m q P r s, t s m ; u ă ¨ ¨ ¨ ă u m u . For notational sake, we just write S m for S m pr , T sq .The definition of a rough path above a signal x relies on the following notion of increments. Definition 2.1.
Let k ě . Then the space of p k ´ q -increments, denoted by C k pr , T s , R d q or simply C k p R d q , is defined as C k p R d q ” " g P C p S k ; R d q ; lim t i Ñ t i ` g t ¨¨¨ t k “ , i ď k ´ * . We now introduce a finite difference operator called δ , which acts on increments and is usefulto split iterated integrals into simpler pieces. Definition 2.2.
Let g P C p R d q , h P C p R d q . Then for p s, u, t q P S , we set δg st “ g t ´ g s , and δh sut “ h st ´ h su ´ h ut . The regularity of increments in C p R d q will be measured in terms of p -variation as follows. Definition 2.3.
For f P C p R d q and p ą , we define } f } p ´ var “ } f } p ´ var; r ,T s “ sup p t i qP D pr ,T sq ˜ÿ i | f t i t i ` | p ¸ { p . The set of increments in C p R d q with finite p -variation is denoted by C p ´ var2 p R d q . Note that for a continuous function g : r , T s Ñ R d with finite p -variation, if we set } g } p ´ var; r ,T s “ } δg } p ´ var; r ,T s , then we recover its usual p -variation.With these preliminary definitions in hand, we can now introduce the notion of a roughpath. Definition 2.4.
Let x be a continuous R d -valued path with finite p -variation for some p ě .We say that x gives rise to a geometric p -rough path if there exists a family x n ; i ,...,i n st ; p s, t q P S , n ď t p u , i , . . . , i n P t , . . . , d u ( , JIAN SONG AND SAMY TINDEL such that x st “ δx st and (1) Regularity: For all n ď t p u , each component of x n has finite pn -variation in the sense ofDefinition 2.3. (2) Multiplicativity: With δ x n as in Definition 2.2, we have δ x n ; i ,...,i n sut “ n ´ ÿ n “ x n ; i ,...,i n su x n ´ n ; i n ` ,...,i n ut . (9) {eq:multiplicativity}{eq:multiplicativity} (3) Geometricity: Let x ε be a sequence of piecewise smooth approximations of x . For any n ď t p u and any set of indices i , . . . , i n P t , . . . , d u , we assume that x ε,n ; i ,...,i n converges in pn -variation to x n ; i ,...,i n , where x ε,n ; i ,...,i n st is defined for p s, t q P S by x ε,n ; i ,...,i n st “ ż p u ,...,u n qP S n pr s,t sq dx ε,i u ¨ ¨ ¨ dx ε,i n u n . We are now ready to state one of the main assumptions on our standing process x . Hypothesis 2.5.
Throughout the paper, x will designate a continuous R d -valued path withfinite p -variation for p ě . We assume that x gives rise to a geometric rough path in thesense of Definition 2.4. On top of Hypothesis 2.5, we assume that x t “ p x t , . . . , x dt q is a continuous centeredGaussian process with i.i.d. components, defined on a complete probability space p Ω , F , P q .The covariance function of x is given by R p s, t q : “ E “ x js x jt ‰ , (10) {eq:def-covariance-X}{eq:def-covariance-X} for any j P t , . . . , d u . Throughout the paper, we will also set R t : “ R p t, t q .The information on the path regularity of x is mostly contained in the rectangular incre-ments R stuv of its covariance function R , which are defined as R stuv : “ E “ p x jt ´ x js q p x jv ´ x ju q ‰ . (11) {eq:rect-increment-cov-fct}{eq:rect-increment-cov-fct} The regularity of R is expressed thanks to some d-variation type quantities. For sake ofclarity we first recall the definition of the 2d ρ -variation. Definition 2.6.
Let ρ P r , . For a general continuous function R : r , T s Ñ R , its 2d ρ -variation is defined as } R } ρ ´ var; r s,t sˆr u,v s : “ sup p t i qP D pr s,t sqp t j qP D pr u,v sq ¨˝ÿ t j ÿ t i ˇˇˇ R t j t j ` t i t i ` ˇˇˇ ρ ˛‚ ρ . (12) {eq:mixed_var}{eq:mixed_var} where R t j t j ` t i t i ` “ R p t i ` , t j ` q ´ R p t i ` , t j q ´ R p t i , t j ` q ` R p t i , t j q . (13) {eq:R}{eq:R} Observe that, whenever the function R in Definition 2.6 is a covariance function as in (10),the rectangular increment R t j t j ` t i t i ` can also be written as in (11).In the following definition, we consider each element pp s, t q , p u, v qq in S ˆ S as a rectangleand denote it by r s, t s ˆ r u, v s . KOROHOD AND STRATONOVICH INTEGRALS 7
Definition 2.7.
A continuous function ω : S ˆ S Ñ R ` is called a 2d control, if it is zeroon degenerate rectangles, and super-additive in the sense that for all rectangles A, B and C contained in S satisfying A Y B Ă C and A X B “ H , ω p A q ` ω p B q ď ω p C q . With these elementary notions at hand, we next introduce a hypothesis which allows theuse of both rough paths techniques and tools from stochastic analysis for the underlyingprocess x . Hypothesis 2.8.
Let x be a d -dimensional continuous and centered Gaussian process withi.i.d. components, whose initial value is 0 and covariance R is given by (10) . We assumethat for some ρ P r , q , the function R admits a finite 2d ρ -variation. It is well known that for a continuous function g : r , T s Ñ R with finite p -variation, thefunction r a, b s ÞÑ } g } pp ´ var; r a,b s is a control. However, for a continuous function R : r , T s Ñ R with finite 2d ρ -variation, the function r a, b s ˆ r c, d s ÞÑ } R } ρρ ´ var; r a,b sˆr c,d s may fail to be super-additive for ρ ą (see [9, Theorem 1]). To regain this property, here we introduce theso-called controlled 2d ρ -variation for ď ρ ă 8 (this notion is also introduced in [9]). Definition 2.9.
Let ρ P r , . For a continuous function R : r , T s Ñ R , its controlled 2d ρ -variation is defined as ~ R ~ ρ ´ var; r s,t sˆr u,v s : “ sup Π P P pr s,t sˆr u,v sq ¨˝ ÿ r t i ,t i ` sˆr t j ,t j ` sP Π ˇˇˇ R t j t j ` t i t i ` ˇˇˇ ρ ˛‚ ρ , where R t j t j ` t i t i ` is given in (13) , Π is a partition of r s, t sˆr u, v s which is a finite set of essentiallydisjoint rectangles whose union is r s, t s ˆ r u, v s , and P pr s, t s ˆ r u, v sq is the collection of allsuch partitions. The norms } ¨ } and ~ ¨ ~ are comparable thanks to the following property borrowed from[9, Theorem 1]: for all ρ ą ρ there exists a constant C ρ,ρ such that C ρ,ρ ~ f ~ ρ ´ var; r s ,s sˆr t ,t s ď } f } ρ ´ var; r s ,s sˆr t ,t s ď ~ f ~ ρ ´ var; r s ,s sˆr t ,t s . (14) {e:norms-compare}{e:norms-compare} Moreover, the function r a, b s ˆ r c, d s ÞÑ ~ f ~ ρρ ´ var; r a,b sˆr c,d s is a 2d control ([9, Theorem 1]). Remark . Owing to (14), any continuous function R : r , T s Ñ R with finite ρ -variation also has a finite controlled 2d ρ -variation for all ρ ą ρ. Furthermore, for all pp s, t q , p u, v qq P S ˆ S , } R } ρ ρ ´ var; r s,t sˆr u,v s ď ω pr s, t s ˆ r u, v sq , where ω is the 2d control (as introduced in Definition 2.7) given by ω pr s, t s ˆ r u, v sq “ ~ R ~ ρ ρ ´ var; r s,t sˆr u,v s . (15) {e:w}{e:w} Remark . As an example, if the Gaussian process x is a fractional Brownian motionwith Hurst parameter H P p , s , the covariance R of x has finite 2d ρ -variation with ρ “ H and Hypothesis 2.8 is satisfied (see [9, Example 1]). If we choose ρ ą ρ “ H , then JIAN SONG AND SAMY TINDEL the quantity } R } ρ ρ ´ var; r s,t sˆr u,v s is controlled by the 2d control ~ R ~ ρ ρ ´ var; r s,t sˆr u,v s . Note that ~ R ~ ρ ´ var; r ,T s “ 8 if we choose ρ ď H (as shown in [9, Example 2]).In the sequel we will also request the function t ÞÑ R p t, t q to be Hölder continuous. Wenow state an additional assumption which guarantees this Hölder continuity (see e.g [4, 12]for a similar hypothesis). Hypothesis 2.12.
Let ρ P r , q be given in Hypothesis 2.8. We assume that there exists C ă 8 such that for all s, t P r , T s the covariance function R satisfies } R p t, ¨q ´ R p s, ¨q} ρρ ´ var; r ,T s ď C | t ´ s | . (16) {eq:R-var}{eq:R-var} Remark . A direct consequence of Hypothesis 2.12 is that R t : “ R p t, t q has finite ρ -variation, by [4, Lemma 2.14]. Moreover, recall that by Hypothesis 2.8, we have x “ andhence R p , ¨q “ R p¨ , q ” . This together with (16) implies that R p t, ¨q and R p¨ , t q havefinite ρ -variation for each fixed t P r , T s . Remark . Given ρ P r , q , clearly we have, for ď s ď s ď T and ď t ď t ď T , } R } ρ ´ var; r s ,s sˆr t ,t s ď } R } ρ ´ var; r s ,s sˆr ,T s } R } ρ ´ var; r ,T sˆr t ,t s . Furthermore, it is a direct consequence of (16) that for ď s ď t ď T , } R } ρρ ´ var; r s,t sˆr ,T s ď C p t ´ s q . Combining the two inequalities above, we have the following control on the 2d ρ -variationof R : for some positive constant C , } R } ρρ ´ var; r s ,s sˆr t ,t s ď C p s ´ s qp t ´ t q . (17) {e:bound-R-2rho}{e:bound-R-2rho} Remark . Note that for ď γ ď γ ă 8 , } R } γ ´ var; r s,t sˆr u,v s ď } R } γ ´ var; r s,t sˆr u,v s . There-fore, under Hypothesis 2.12, inequality (16) and hence (17) hold with ρ replaced by ρ P p ρ, q and C depending on p ρ, ρ , T q . Remark . Clearly (17) yields the following relations on squares of the form r s, t s , } R } ρρ ´ var; r s,t s ď C p t ´ s q . (18) {e:Holder-control-var-R}{e:Holder-control-var-R} We say that R has finite Hölder-controlled 2d ρ -variation if R satisfies both Hypothesis 2.8and (18). An important consequence of R having finite Hölder controlled 2d ρ -variation isthat x has { p -Hölder continuous sample paths for every p ą ρ . It is also readily checkedthat, whenever x satisfies (18), we have E ”` x ; ist ˘ ı ď c p t ´ s q ρ . (19) {eq:bound-increment-X-L2}{eq:bound-increment-X-L2} Remark . Similarly to the argument in [3, Remark 2.4], for any process x whose covari-ance function R admits a finite ρ -variation one can introduce a deterministic time-change τ : r , T s Ñ r , T s such that ˜ X “ X ˝ τ has finite Hölder-controlled 2d ρ -variation. That isthe time changed process ˜ X satisfies Hypothesis 2.8 and equation (18).The following result (stated e.g. in [11, Theorem 15.33]) relates the 2d ρ -variation of R with the pathwise assumptions allowing to apply the abstract rough paths theory. KOROHOD AND STRATONOVICH INTEGRALS 9
Proposition 2.18.
Let x “ p x , . . . , x d q be a continuous centered Gaussian process withi.i.d. components and covariance function R defined by (10) . If R satisfies Hypothesis 2.8,then x also satisfies Hypothesis 2.5 provided p ą ρ . Proposition 2.18 asserts that under Hypothesis 2.8, the Gaussian process x is amenable torough path analysis. In particular, a rough path integral with respect to x can be constructed.In this context, the natural class of integrand one might want to consider is the family ofcontrolled processes. Its definition is recalled below. Definition 2.19.
Consider a continuous R d -valued path x with finite p -variation for some p ě . We say that a continuous R d -valued path y of finite p -variation is controlled by x , ifthere exist a continuous R d -valued path y x of finite p -variation and a -increment process r P C p ´ var2 p R d q as defined in Definition 2.3, such that δy ist “ d ÿ j “ y x ; ijs x ; jst ` r ist , for i “ , . . . , d. (20) {eq:ctrld-proc-def}{eq:ctrld-proc-def} We are now ready to state the basic integration result for controlled processes, which canbe found e.g. in [8, 11, 13].
Proposition 2.20.
Let T ą be fixed. Let x be a geometric p -rough path lifted from acontinuous R d -valued path with finite p -variation for some p P r , q , and let y be a continuous R d -valued path of finite p -variation that is controlled by x in the sense of Definition 2.19.Then for ď s ă t ď T , one can define the integral ş ts y r d x r as the limit of the followingRiemann sums, ż ts y r d x r “ lim | π n |Ñ n ´ ÿ k “ ˜ d ÿ i “ y it k x ,it k t k ` ` d ÿ i “ d ÿ j “ y x ; ijt k x ijt k t k ` ¸ , (21) {eq:int-sum}{eq:int-sum} where π n “ r s “ t ă t ă ¨ ¨ ¨ ă t n “ t s is a partition of r s, t s and | π n | “ max k Pt ,...,n ´ u | t k ` ´ t k | . In (21) , observe that we have also used the convention on inner products put forward inSection 1.3. Moreover, there exists a constant C “ C p T, p q depending only on p T, p q suchthat for all ď s ă t ď T we have ˇˇˇˇˇż ts y r d x r ´ y s x st ´ d ÿ i “ d ÿ j “ y x ; ijs x ijst ˇˇˇˇˇ ď C ´ } x } p ´ var } r } p ´ var ` } x } p ´ var } y x } p ´ var ¯ | t ´ s | { p , where we recall that r is the increment introduced in (20) . Recall that our main objective is to compute some Skorohod-Stratonovich corrections asin [4]. To this aim we will need a more detailed description of the increments of y thanthe ones given in (20). Namely we will assume that y is a second order controlled processas defined below (for the definition of controlled processes of general order, we refer to [8,Definition 4.17] or [3, Definition 5.1]). Definition 2.21.
Consider a continuous R d -valued path x with finite p -variation for some p ě . We say that a continuous R d -valued path y of finite p -variation is a second-ordercontrolled process with respect to x , if there exist a continuous R d -valued path y x , a contin-uous R d -valued path y xx , both of which are of finite p -variation, and -increment processes r P C p ´ var2 p R d q , r x P C p ´ var2 p R d q as defined in Definition 2.3, such that for i “ , . . . , d and p s, t q P S pr , T sq we have δy ist “ d ÿ j “ y x ; ijs x ; jst ` d ÿ j,k “ y xx ; ijks x ; jkst ` r ist . (22) {e:y-1}{e:y-1} In addition, the increment y x in (22) is a controlled process of order 1, that is for i, j “ , . . . , d and p s, t q P S pr , T sq we have δy x ; ijst “ d ÿ k “ y xx ; ijks x ; kst ` r x ; ijst . (23) {e:y-2}{e:y-2} Higher dimensional Young integrals.
In this subsection, we gather some inequali-ties for Young integrals in R n which will feature in our computations throughout the paper.We start by a relation for integrals in the plane borrowed form [10, 23]. Theorem 2.22.
Let f, R : r , T s Ñ R be continuous functions with finite p -variation and fi-nite q -variation respectively for p ` q ą . Specifically recalling our Definition 2.6, we assume } f } p ´ var; r ,T s ă 8 and } R } q ´ var; r ,T s ă 8 . Moreover, assume that for all s , s P r , T s , both f p s , ¨q and f p¨ , s q have finite 1-dimensional p -variation as given in Definition 2.3. Then the d Young-Stieltjes integral of f with respect to R exists and the following Young’s inequalityholds, for r ¯ s , ¯ s s ˆ r ¯ s , ¯ s s Ă r , T s , ˇˇˇˇˇż r ¯ s , ¯ s sˆr ¯ s , ¯ s s f p s , s q dR p s , s q ˇˇˇˇˇ ď C p,q ˜ | f p ¯ s , ¯ s q| ` } f p ¯ s , ¨q} p ´ var; r ¯ s , ¯ s s ` } f p¨ , ¯ s q} p ´ var; r ¯ s , ¯ s s ` } f } p ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¸ } R } q ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (24) {eq:bound-2d}{eq:bound-2d} We now state a lemma about integration in R which will be invoked in order to analyzediscretization properties for the Malliavin derivative of a controlled process y . Although itsproof might be traced back to [23], we include it here for the sake of clarity since Lemma 2.23is tailored for our specific needs. Lemma 2.23.
Let f, g, R be continuous functions defined on r , T s . Similarly to Theo-rem 2.22, we assume that f, g have finite p -variation, as well as f p s , ¨q , f p¨ , s q , g p s , ¨q and g p¨ , s q for fixed arbitrary s , s , s , s P r , T s . We also suppose that R has finite q -variationon r , T s , with p, q satisfying p ` q ą . Then for ¯ s , ¯ s , . . . , ¯ s , ¯ s P r , T s such that ¯ s j ă ¯ s j for j “ , . . . , , the following Young integral in R is well defined: I f,g,R p ¯ s , ¯ s , . . . , ¯ s , ¯ s q : “ ż r ¯ s , ¯ s sˆr ¯ s , ¯ s sˆr ¯ s , ¯ s sˆr ¯ s , ¯ s s f p s , s q g p s , s q dR p s , s q d R p s , s q . KOROHOD AND STRATONOVICH INTEGRALS 11
Moreover, I f,g,R p ¯ s , ¯ s , . . . , ¯ s , ¯ s q can be bounded as ˇˇ I f,g,R p ¯ s , ¯ s , . . . , ¯ s , ¯ s q ˇˇ ď C p,q } R } q ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s } R } q ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s ˆ ´ | f p ¯ s , ¯ s q| ` } f p¨ , ¯ s q} p ´ var; r ¯ s , ¯ s s ` } f p ¯ s , ¨q} p ´ var; r ¯ s , ¯ s s ` } f } p ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ ˆ ´ | g p ¯ s , ¯ s q| ` } g p¨ , ¯ s q} p ´ var; r ¯ s , ¯ s s ` } g p ¯ s , ¨q} p ´ var; r ¯ s , ¯ s s ` } g } p ´ var; r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ . (25) {eq:bound-4d}{eq:bound-4d} Proof.
We will divide this proof in several steps.
Step 1:
Decomposition of the integral. We can write I f,g,R p ¯ s , ¯ s , . . . , ¯ s , ¯ s q “ ż r ¯ s , ¯ s sˆr ¯ s , ¯ s s F p s , s q d R p s , s q (26) {eq:decom-I}{eq:decom-I} where the function F is defined on r , T s by F p s , s q “ ż r ¯ s , ¯ s sˆr ¯ s , ¯ s s f p s , s q g p s , s q d R p s , s q , (27) {eq:F}{eq:F} and where we observe that the right-hand side of (27) is well defined thanks to Theorem 2.22.Our strategy in order to estimate I f,g,R will rely on some succesive applications of (24).Specifically, with (26) in mind, relation (24) yields ˇˇ I f,g,R p ¯ s , ¯ s , ¯ s , ¯ s , ¯ s , ¯ s , ¯ s , ¯ s q ˇˇ ď C p,q ´ | F p ¯ s , ¯ s q| ` } F p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ` } F p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ` } F } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (28) {eq:bound-I}{eq:bound-I} We will now estimate the terms in right-hand side of (28) separately.
Step 2:
Upper bound for F p ¯ s , ¯ s q . Given p ¯ s , ¯ s q P r , T s and recalling the definition (27)of F , another application of (24) enables to write | F p ¯ s , ¯ s q| ď C p,q ´ | f p ¯ s , ¯ s q g p ¯ s , ¯ s q| ` | f p ¯ s , ¯ s q|} g p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ` | g p ¯ s , ¯ s q|} f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ` } f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s } g p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ¯ } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s and we notice that the above expression can be simplified as | F p ¯ s , ¯ s q| ď C p,q ´ | f p ¯ s , ¯ s q| ` } f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ¯ ˆ ´ | g p ¯ s , ¯ s q| ` } g p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ¯ } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (29) {eq:bound-F1}{eq:bound-F1} Step 3:
Upper bound for } F p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s . Recall the Definition 2.3 of p -variation. Wethus have } F p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s “ sup π ˜ÿ i | F p ¯ s , v i ` q ´ F p ¯ s , v i q| p ¸ { p . Plugging expression (27) into the above relation, we get } F p ¯ s , ¨q} pp ´ var , r ¯ s , ¯ s s “ sup π ÿ i ˇˇˇˇˇż r ¯ s , ¯ s sˆr ¯ s , ¯ s s f p s , ¯ s q ´ g p s , v i ` q ´ g p s , v i q ¯ dR p s , s q ˇˇˇˇˇ p . We now apply (24) again and we end up with } F p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ď C p,q } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ÿ k “ V k , (30) {eq:bound-F2}{eq:bound-F2} where the terms V , V are respectively defined by V “ | f p ¯ s , ¯ s q| sup π ˜ÿ i | g p ¯ s , v i ` q ´ g p ¯ s , v i q| p ¸ { p ; V “ | f p ¯ s , ¯ s q| sup π ˜ÿ i } g p¨ , v i ` q ´ g p¨ , v i q} pp ´ var , r ¯ s , ¯ s s ¸ { p , and similarly the terms V , V are expressed as V “ sup π ˜ÿ i | g p ¯ s , v i ` q ´ g p ¯ s , v i q| p } f p¨ , ¯ s q} pp ´ var , r ¯ s , ¯ s s ¸ { p ; V “ sup π ˜ÿ i } f p¨ , ¯ s qp g p˚ , v i ` q ´ g p˚ , v i qq} pp ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¸ { p . In addition, the terms V , V , V are easily bounded. Indeed, resorting again to Definition 2.3,we get V “ | f p ¯ s , ¯ s q|} g p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s , V ď | f p ¯ s , ¯ s q|} g } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s , (31) {eq:V1}{eq:V1} and V “ } f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s } g p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s . (32) {eq:V3}{eq:V3} For the term V , by Definition 2.6, it is readily checked that V ď } f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s } g } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (33) {eq:V4}{eq:V4} Hence, plugging (31), (32) and (33) into (30), we end up with } F p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ď C p,q ´ | f p ¯ s , ¯ s q| ` } f p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s | ¯´ } g p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ` } g } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (34) {eq:bound-F3}{eq:bound-F3} Furthermore, notice that in a similar way we get } F p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s ď C p,q ´ | g p ¯ s , ¯ s q| ` } g p¨ , ¯ s q} p ´ var , r ¯ s , ¯ s s | ¯´ } f p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ` } f } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . (35) {eq:bound-F4}{eq:bound-F4} Step 4:
Upper bound for } F } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s . According to Definition 2.6, one can write } F } pp ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s “ sup π ÿ t i ,t j ˇˇ F p t i , t j q ` F p t i ` , t j ` q ´ F p t i , t j ` q ´ F p t i ` , t j q ˇˇ p , KOROHOD AND STRATONOVICH INTEGRALS 13 where we recall that π takes the form π P D pr ¯ s , ¯ s sq ˆ D pr ¯ s , ¯ s sq and the notation D pr s, t sq is introduced in Section 1.3. Hence with the expression (27) of F in mind we get } F } pp ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s “ sup π ÿ t i ,t j ˇˇˇˇˇż r ¯ s , ¯ s sˆr ¯ s , ¯ s s p f p s , t i ` q ´ f p s , t i qqp g p s , t j ` ´ g p s , t j qq dR p s , s q ˇˇˇˇˇ p . In this context, relation (24) can thus be read as } F } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ď C } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s sup π ˜ ÿ t i ,t j | Q ij | p ¸ { p , where the term Q ij is defined by Q ij “ |p f p ¯ s , t i ` q ´ f p ¯ s , t i qqp g p ¯ s , t j ` q ´ g p ¯ s , t j qq|` | f p ¯ s , t i ` q ´ f p ¯ s , t i q|} g p¨ , t j ` q ´ g p¨ , t j q} p ´ var , r ¯ s , ¯ s s ` } f p¨ , t i ` q ´ f p¨ , t i q} p ´ var , r ¯ s , ¯ s s | g p ¯ s , t j ` q ´ g p ¯ s , t j q|` } f p¨ , t i ` q ´ f p¨ , t i q} p ´ var , r ¯ s , ¯ s s } g p¨ , t j ` q ´ g p¨ , t j q} p ´ var , r ¯ s , ¯ s s , and we notice that Q ij can easily be simplified as Q ij “ ´ |p f p ¯ s , t i ` q ´ f p ¯ s , t i qq| ` } f p¨ , t i ` q ´ f p¨ , t i q} p ´ var , r ¯ s , ¯ s s ¯ ˆ ´ |p g p ¯ s , t j ` q ´ g p ¯ s , t j qq| ` } g p¨ , t j ` q ´ g p¨ , t j q} p ´ var , r ¯ s , ¯ s s ¯ . Summarizing our computations in this step, we have found that } F } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ď C } R } q ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ´ } f p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ` } f } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ ˆ ´ g p ¯ s , ¨q} p ´ var , r ¯ s , ¯ s s ` } g } p ´ var , r ¯ s , ¯ s sˆr ¯ s , ¯ s s ¯ . (36) {eq:bound-F5}{eq:bound-F5} Step 5:
Conclusion. Let us gather our estimates (29), (34), (35) and (36) into (28). Thenwe let the patient reader check that (25) is achieved. This finishes the proof. (cid:3)
The Hilbert space associated to x . Consider a continuous d-dimensional centeredGaussian process x on r , T s with covariance function R given by (10). Every component of x (say x ) is a 1-dimensional centered Gaussian process with covariance R . In this sectionwe review some basic facts about the related Hilbert space H of functions for which Wienerintegrals with respect to x (see e.g. [20]) are well defined.The Hilbert space H is the completion of the set of step functions E “ n ÿ i “ a i r ,t i s : a i P R , t i P r , T s , i “ , . . . , n for n P N + , with respect to the inner product C n ÿ i “ a i r ,t i s , m ÿ j “ b j r ,s j s G H “ n ÿ i “ m ÿ j “ a i b j R p t i , s j q . Observe that this inner product can also be written as C n ÿ i “ a i r ,t i s , m ÿ j “ b j r ,s j s G H “ ż T ż T ˜ n ÿ i “ a i r ,t i s p t q ¸ ˜ m ÿ j “ b j r ,s j s p s q ¸ dR p t, s q . (37) {eq:def-inner-pdt-H}{eq:def-inner-pdt-H} One can further relate H to our driving process x in the following way: let H be the closureof the set E “ !ÿ ni “ a i x t i : a i P R , t i P r , T s , i “ , . . . , n for n P N ) , in L p Ω , F , P q . Then the linear map x : E Ñ E defined by x p r ,t s q “ x t extends to alinear isometry between H and H . Hence, H “ t x p h q , h P H u and this family is known asthe isonormal Gaussian process related to x (see [20, Definition 1.1.1]). Note that x p h q for h P H is called the Wiener integral of h with respect to x and is usually denoted by ş T h p s q dx s . Remark . Recall that we have assumed x “ and thus R p , q “ . Thus relation (37)suggests x h , h y H “ ż T ż T h p s q h p t q dR p s, t q for h , h P H , (38) {rep H norm}{rep H norm} whenever the 2D Young’s integral on the right-hand side is well-defined (see, e.g., [2, Propo-sition 4] for details). Remark . Denoting by E pr a, b sq the set of step functions in E restricted on r a, b s Ă r , T s ,the closure H pr a, b sq of E pr a, b sq with respect to the inner product (37) then coincides with H restricted on r a, b s , and for f, g P H , @ f r a,b s , g r a,b s D H “ x f, g y H pr a,b sq . (39) {eq:norm-H-as-2d-young}{eq:norm-H-as-2d-young} Malliavin calculus for Gaussian processes.
In this subsection, we collect somebasic concepts of Malliavin calculus, and we refer to [20] for more details.Recall that x t is a continuous centered d -dimensional Gaussian process with i.i.d. com-ponents, defined on a complete probability space p Ω , F , P q . For the sake of simplicity,we assume that F coincides with the σ -algebra generated by t x t ; t P r , T su . For the d -dimensional process x , we define an extension of the Wiener integral defined as follows:let ϕ “ p ϕ , . . . , ϕ d q be an element of H d where we recall that H has been introduced inSection 2.3. Then we set x p ϕ q “ d ÿ j “ x j p ϕ j q , (40) {e:Wiener-int}{e:Wiener-int} where each term x j p ϕ j q is a 1-d Wiener integral as in Section 2.3.A smooth functional of x is a random variable of the form F “ f p x p ϕ q , . . . , x p ϕ n qq , where n ě , t ϕ , . . . , ϕ n u is a family of elements of H d and each x p ϕ i q is understood as in (40).Moreover, we assume that the function f : R n Ñ R is smooth and its partial derivativesgrow at most polynomially fast. Then, the Malliavin derivative D F of F is the H d -valuedrandom variable defined by D F “ n ÿ k “ B f B x k p x p ϕ q , . . . , x p ϕ n qq ϕ k . (41) {e:DF}{e:DF} KOROHOD AND STRATONOVICH INTEGRALS 15
One can show that D is closable from L p Ω q to L p Ω; H q , and thus one may span the spaceof the smooth and cylindrical random variables under the norm } F } , “ ` E r F s ` E r} D F } H s ˘ . The resulting closure is called Sobolev space D , . Remark . As seen in (41), the Malliavin derivative D F of a functional F is a R d -valuedprocess. The i -th coordinate of D F corresponds to the Malliavin derivative of F with respectto the randomness in x i only. It will be denoted by D i F in the sequel.The divergence operator δ ˛ (also known as the Skorohod integral) is the adjoint operatorof the Malliavin derivative operator D defined by the duality relation E r F δ ˛ p u qs “ E rx D F, u y H d s , for all F P D , and for all u P Dom δ ˛ . Here Dom δ ˛ is the domain of the divergence operator δ ˛ , which is the space of H -valuedrandom variables u P L p Ω; H d q such that | E rx D F, u y H d s| ď c F } F } with some constant c F depending on F , for all F P D , . In particular, D , p H d q Ă Dom δ ˛ . Note that for u P Dom δ ˛ , we have δ ˛ p u q P L p Ω q and E r δ ˛ p u qs “ . By convention, we also take thefollowing notation, for u P Dom δ ˛ , ż T u t d ˛ x t : “ δ ˛ p u q . (42) {e:divergence}{e:divergence} For our main computations below we shall invoke the following relation taken from [22]:for any G P D , p R d q and ď a ă b ď T we have δ ˛ p G r a,b s q “ d ÿ i “ ż ba G i d ˛ x it “ d ÿ i G i ˛ δx iab , (43) {e:relation}{e:relation} where ˛ stands for the Wick product (see [15] for a brief account on Wick products). More-over, according to [16, Proposition 4.7], relation (43) can be simplified as δ ˛ p G r a,b s q “ d ÿ i “ G i δx iab ´ x D i G i , r a,b s y H . (44) {e:relation’}{e:relation’} Discrete rough paths techniques.
In this subsection, we develop some inequalitiesabout discrete sums in a rough paths context. This kind of sum will feature prominently inthe analysis of our Skorohod-Stratonovich corrections.We state a crucial lemma about convergence of discrete sums in the second chaos of x .It generalizes [17, Lemma 3.4] to a generic Gaussian process (as opposed to the fractionalBrownian motion case handled in [17]). Proposition 2.27.
Let x be a R d -valued Gaussian process satisfying Hypotheses 2.8 and 2.12.For n ě we consider the uniform partition on r , T s , namely t k “ kn T . We define a process F “ t F ijt ; t P J , T K , i, j “ , . . . , d u by F ij “ , and for all t ą , F ijt “ t ´ ÿ t k “ ´ x ijt k t k ` ´ E r x ijt k t k ` s ¯ “ $’’&’’% t ´ ř t k “ x ijt k t k ` , i ‰ j, t ´ ř t k “ ´ x iit k t k ` ´ E r x iit k t k ` s ¯ , i “ j, (45) {eq:def-F}{eq:def-F} where we recall the notation t ´ from Section 1.3 and where x is introduced in Definition 2.4.Then for all q ě , ρ P p ρ, q , p s, t q P S p J , T K q and n ě the following inequality holdstrue ´ E ”ˇˇ δF ijst ˇˇ q ı¯ { q ď C p t ´ s q n β ´ , (46) {e:estimation-dF}{e:estimation-dF} where C “ C p q, ρ, T q , β “ ρ P p { , s for i ‰ j , and C “ C p q, ρ, ρ , T q , β “ ρ P p { , q for i “ j .Proof. Due to the hyper-contractivity property of the second Wiener chaos, it suffices to showthe case q “ . In addition, we assume (without loss of generality) that s “ t m ă t “ t m for ď m ă m ď n . Case 1: i “ j . In this case, due to the definition (45) of F and the geometric nature of x assumed in Definition 2.4, we have E ”` δF iist ˘ ı “ E ˜ m ´ ÿ k “ m “ p x ; it k t k ` q ´ E rp x ; it k t k ` q s ‰¸ , and expanding the square on the right hand side above we get E ”` δF iist ˘ ı “ m ´ ÿ k,l “ m E rp x ; it k t k ` q p x ; it l t l ` q s ´ E rp x ; it k t k ` q s E rp x ; it l t l ` q s ( . (47) {a2}{a2} In order to evaluate the right-hand side of (47) we apply a particular case of Wick’s formulafor centered Gaussian random variables X and Y , which can be stated as: E r X Y s ´ E r X s E r Y s “ p E r X Y sq . Plugging this result into (47) and recalling the definition (11) of R uvst we obtain E ”` δF iist ˘ ı “ m ´ ÿ k,l “ m ` E “ x ; it k t k ` x ; it l t l ` ‰˘ “ m ´ ÿ k,l “ m ´ R t k t k ` t l t l ` ¯ . (48) {a21}{a21} Therefore invoking elementary properties of p -variations we end up with E ”` δF iist ˘ ı ď k,l ´ R t k t k ` t l t l ` ¯ ´ ρ } R } ρρ ´ var ; r s,t s . (49) {e:delta-F2}{e:delta-F2} On the right-hand side of (49), notice that under Hypothesis 2.12, } R } ρρ ´ var ; r s,t s can be upperbounded by C p t ´ s q thanks to (18). Moreover, a simple use of Cauchy-Schwarz inequality,together with (19), shows that | R t k t k ` t l t l ` | “ ˇˇ E r x ,it l t l ` x ,it k t k ` s ˇˇ ď ´ E ”ˇˇ x ,it l t l ` ˇˇ ı E ”ˇˇ x ,it k t k ` ˇˇ ı¯ ď C T n ρ . KOROHOD AND STRATONOVICH INTEGRALS 17
Reporting this information into (49) and recalling that β “ ρ , it is seen that E ”` δF iist ˘ ı ď CT β ´ p t ´ s q n β ´ . (50) {e:estimation-dF1}{e:estimation-dF1} This ends our proof for the case i “ j . Case 2: i ‰ j . According to our definition (45), if i ‰ j we have E ”` δF ijst ˘ ı “ E »–˜ m ´ ÿ k “ m x ; ijt k t k ` ¸ fifl “ m ´ ÿ k,l “ m E “ x ; ijt k t k ` x ; ijt l t l ` ‰ . Therefore, invoking the proofs of [11, Theorem 15.33 and Proposition 15.28] for the compu-tation of E r x ; ijt k t k ` x ; ijt l t l ` s , we end up with E ”` δF ijst ˘ ı “ m ´ ÿ k,l “ m ż t k ` t k ż t l ` t l R t l v t k v d R p v , v q . (51) {a3}{a3} We now fix p k, l q and denote G p v , v q “ R t l v t k v . Then G p t k , ¨q “ G p¨ , t l q “ . For any ρ Pp ρ, q , Hypothesis 2.8 implies R has finite 2d ρ -variation, and Hypothesis 2.12 implies both R p t, ¨q and R p¨ , t q have finite ρ -variation for all t P r , T s . Hence resorting to Theorem 2.22,we have for some fixed ρ P p ρ, q , ˇˇˇˇż t k ` t k ż t l ` t l R t l v t k v d R p v , v q ˇˇˇˇ ď C } R } ρ -var ; r t k ,t k ` sˆr t l ,t l ` s , for some constant C “ C p ρ , T q depending on p ρ , T q only. Plugging this inequality into (51)we obtain E ”` δF ijst ˘ ı ď C p ρ , T q m ´ ÿ k,l “ m } R } ρ -var ; r t k ,t k ` sˆr t l ,t l ` s ď C p ρ , T q sup k,l } R } ´ ρ ρ -var ; r t k ,t k ` sˆr t l ,t l ` s m ´ ÿ k,l “ m } R } ρ ρ -var ; r t k ,t k ` sˆr t l ,t l ` s . Therefore, thanks to Remark 2.10, we have E ”` δF ijst ˘ ı ď C p ρ , T q sup k,l } R } ´ ρ ρ -var ; r t k ,t k ` sˆr t l ,t l ` s m ´ ÿ k,l “ m ω pr t k , t k ` s ˆ r t l , t l ` sq , where ω is a control given in (15). Furthermore, the super-additivity of ω yields E ”` δF ijst ˘ ı ď C p ρ , T q sup k,l } R } ´ ρ ρ -var ; r t k ,t k ` sˆr t l ,t l ` s ω pr s, t s qď C p ρ, ρ , T q sup k,l } R } ´ ρ ρ -var ; r t k ,t k ` sˆr t l ,t l ` s p t ´ s q , (52) {e:estimation-dF2}{e:estimation-dF2} where the last inequality is due to (15), (14), and (17). Finally, by Hypothesis 2.12 (andRemark 2.15), we have } R } ρ ρ -var ; r s,t sˆr u,v s ď C p ρ, ρ , T qp t ´ s qp u ´ v q , and therefore setting β “ { ρ , inequality (52) becomes E ”` δF ijst ˘ ı ď C p ρ, ρ , T q ˆ Tn ˙ β ´ p t ´ s q . (53) {e:estimation-dF3}{e:estimation-dF3} With (50) and (53) in hand our claim (46) is now easily achieved, which concludes theproof. (cid:3)
Note that (46) is still valid for both cases of i “ j and i ‰ j , if we choose β “ ρ for any ρ P p ρ, q . We now give a weighted version of Proposition 2.27, which plays an importantrole in our correction computations. Proposition 2.28.
Let x be a R d -valued Gaussian process satisfying Hypotheses 2.8 and 2.12.Let ρ P p ρ, q be fixed. For n ě we consider the uniform partition on r , T s , namely t k “ kn T , as well as the process F defined by (45) . Let now f be a controlled process in the L q p Ω q sense, namely such that there exists a process g fulfilling (in the matrix sense), forsome γ P p , ρ q and for all q ě , } f t } q ` } g t } q ď C, } δf st ´ g s x st } q ď C p t ´ s q γ , } δg st } q ď C p t ´ s q γ . (54) {eq:controlled-f}{eq:controlled-f} Then the following estimate holds true for p s, t q P S p J , T K q : ››› t ´ ÿ t k “ s f t k b δF t k t k ` ››› q ď C p t ´ s q n β ´ , where C “ C p q, ρ, ρ , T q and β “ ρ P p { , q .Proof. This proposition was proved in [17, Corollary 4.9] when x is a fractional Brownianmotion. Although we generalize this result to a wider class of Gaussian processes, our proofgoes along the same lines. Therefore we shall omit the details for sake of conciseness. (cid:3) Correction terms in the case ď p ă In this section we derive a correction formula for controlled processes which are also in thedomain of the Skorohod integral. As mentioned in the introduction, we have restricted ouranalysis to the case p ă . Although we believe that our methodology could be extendedto p ă , this generalisation would require a cumbersome study of third order integrals andrelated weighted sums. Theorem 3.1.
Let x be a Gaussian rough path with covariance given by (10) satisfyingHypotheses 2.8 and 2.12 with ρ P r , q . This implies that x has finite p -variation for p ą ρ .We can assume p ` ρ ą , noting that ρ ă . Let y be a second-order controlled process in the sense of Definition 2.21, and we assume E r} y } p ´ var; r ,T s s ă 8 . In particular, the rough integral ş t y r d x r is defined as in Proposi-tion 2.20, resorting to the convention on inner products of Section 1.3. We also assume that y P D , p H d q , so that the Skorohod integral of y given in (42) is well defined. Furthermore, KOROHOD AND STRATONOVICH INTEGRALS 19 we suppose that D y has finite p -variation with E r} D y } p ´ var; r ,T s s ă 8 , and D y has finite2d p -variation with E r} D y } p ´ var; r ,T s s ă 8 . Then for all t P r , T s we have almost surely ż t y r d x r “ ż t y r d ˛ x r ` d ÿ i “ ż t y x ; iir d R r ` d ÿ i “ ż S pr ,t sq ` D ir y ir ´ y x ; iir ˘ d R p r , r q , (55) {eq:strato-sko-first}{eq:strato-sko-first} where we recall from Section 2.1 that R r : “ R p r, r q and where the Malliavin derivative D i isintroduced in Remark 2.26.Proof. Let π “ π n be the uniform partition of order n of r , t s , whose generic element is stilldenoted by t k “ kn t. A natural discretization of y along π is given by y π p r q “ n ´ ÿ k “ y t k r t k ,t k ` q p r q , r P r , t s . (56) {eq:y-pi}{eq:y-pi} Notice that we have assumed that y P D , p H d q . Hence both divergence integrals δ ˛ p y π q and δ ˛ p y q , as given in (42), are well defined. Moreover, according to (43), we have ż t y πr d ˛ x r “ d ÿ i “ n ´ ÿ k “ y it k ˛ x ; it k t k ` , and owing to (44) this can be recast as ż t y πr d ˛ x r “ d ÿ i “ n ´ ÿ k “ y it k x it k t k ` ´ x D i y it k , r t k ,t k ` s y H . (57) {e:discrete-int}{e:discrete-int} In addition, we will prove in the forthcoming Lemma 3.2 that δ ˛ p y π q converges in L p Ω q to δ ˛ p y q . Otherwise stated, for t P r , T s we have ż t y r d ˛ x r “ lim n Ñ8 ż t y πr d ˛ x r . (58) {e:discrete-int’}{e:discrete-int’} Therefore combining (57) and (58), we get the following limit in L p Ω q : ż t y r d ˛ x r “ lim n Ñ8 d ÿ i “ n ´ ÿ k “ ` y it k x ; it k t k ` ´ x D i y it k , r t k ,t k ` s y H ˘ . (59)On the other hand, owing to the fact that y is a controlled process in the sense of Defi-nition 2.19, Proposition 2.20 asserts that ş t y r d x r is defined as a rough paths integral andhence almost surely we have ż t y r d x r “ lim n Ñ8 ˜ d ÿ i “ n ´ ÿ k “ y it k x ; it k t k ` ` d ÿ i,j “ n ´ ÿ k “ y x ; ijt k x ; ijt k t k ` ¸ . (60) {b2}{b2} Gathering relations (59) and (60), we get the following expression for the Stratonovich-Skorohod correction term: ż t y r d x r ´ ż t y r d ˛ x r “ lim n Ñ8 d ÿ i “ n ´ ÿ k “ ˜ d ÿ j “ y x ; ijt k x ijt k t k ` ` x D i y it k , r t k ,t k ` s y H ¸ , (61) {b3}{b3} where the limit on the right-hand side above is understood in probability. In (61), notice thatthe left-hand side is well defined thanks to the standing assumptions of our Theorem. Hencethe right-hand side of (61) also makes sense, and we will now identify the limits therein.In order to compute the limit for the terms y x ; ijt k x ijt k t k ` in (61), observe that y is a sec-ond order controlled process according to Definition 2.21. Hence y x is a controlled processsatisfying relation (23). Since we have assumed that Hypotheses 2.8 and 2.12 are fulfilled,Proposition 2.28 for the increment F can be applied with f “ y x . Recalling (see (45)) that δF ijt k t k ` “ x ijt k t k ` ´ E r x ijt k t k ` s , we end up with the following relation, valid for i, j “ , . . . , d , where the limit has to beconsidered in the L p Ω q sense: lim n Ñ8 n ´ ÿ k “ y x ; ijt k ´ x ijt k t k ` ´ E “ x ijt k t k ` ‰ ¯ “ . (62) {e:60}{e:60} In particular, going back to (61), we get that for i ‰ j we have lim n Ñ8 d ÿ i ‰ j n ´ ÿ k “ y x ; ijt k x ijt k t k ` “ . (63) {e:60’}{e:60’} Let us deal with the left-hand side of (62) when i “ j . Specifically, we will expressthe limit of the sums ř n ´ k “ y x ; iit k E “ x iit k t k ` ‰ as a Young integral. To this aim, notice that x ,iit k t k ` “ ` x ,it k t k ` ˘ due to the geometric assumption in Definition 2.4. Hence invoking thefact that R t k “ R p t k , t k q we have E “ x iit k t k ` ‰ “ E ” p x it k ` ´ x it k q ı “ R t k ` ´ R p t k ` , t k q ` R t k “ ` R t k ` ´ R t k ˘ ´ p R p t k ` , t k q ´ R p t k , t k qq . Therefore for all i “ , . . . , d , we obtain a decomposition of the form n ´ ÿ k “ y x ; iit k E “ x iit k t k ` ‰ “ I in ´ J in (64) {e:61}{e:61} where I in , J in are respectively defined by I in “ n ´ ÿ k “ y x ; iit k δR t k t k ` , and J in “ n ´ ÿ k “ y x ; iit k ` R p t k ` , t k q ´ R p t k , t k q ˘ . (65) {e:62}{e:62} The limit of for the term I in in (64) can be computed easily. Indeed, thanks to Remark 2.13we know that t Ñ R t has finite ρ -variation. Furthermore, since y is a second order controlledprocess, Definition 2.21 entails that y x has finite p -variation. We have also mentioned inTheorem 3.1 that p ´ ` ρ ´ ą . Hence classical Young integration arguments reveal thatfor i “ , . . . , d we have almost surely, lim n Ñ8 I in “ ż t y x ; iir dR r . (66) {e:63}{e:63} KOROHOD AND STRATONOVICH INTEGRALS 21
As far as the term J in in (65) is concerned, let us recast this expression in terms of a 2-dRiemann sum. Namely we define another uniform partition t v l ; 0 ď l ď n ´ u of r , t s , with v l “ ln t. Then we start by writing J in “ n ´ ÿ k “ y x ; iit k ` R p t k ` , v k q ´ R p t k , v k q ˘ . (67) {e:64}{e:64} In addition, notice that thanks to Remark 2.13 we have R p¨ , q “ . Thus an immediatetelescoping sum argument yields the following relation, valid for k “ , . . . , n ´ R p t k ` , v k q ´ R p t k , v k q “ k ´ ÿ l “ R t k t k ` v l v l ` . Reporting this identity into (67), we get J in “ n ´ ÿ k “ y x ; iit k k ´ ÿ l “ R t k t k ` v l v l ` “ ÿ ď l ă k ď n ´ y x ; iit k R t k t k ` v l v l ` . (68) {e:65’}{e:65’} This decomposition prompts us to define a degenerate function f in the plane as f i p u, v q “ y x ; iiu r ă v ă u ă t s . With this notation in hand, relation (68) reads J in “ n ´ ÿ k,l “ f i p t k , v l q R t k t k ` v l v l ` . In order to analyze the convergence of J in , we now argue as follows: first R has a finite2-dimensional ρ -variation. The function f i p u, v q “ y x,iiu r ă v ă u ă t s is also easily seen to have afinite 2-dimensional p -variation (owing to the fact that y x,ii has finite p -variation), and recallthat p ´ ` ρ ´ ą . Hence standard convergence procedures for 2d-Young integrals showthat almost surely lim n Ñ8 J in “ ż t ż t f i p u, v q d R p u, v q “ ż S pr ,t sq y x ; iir d R p r , r q . (69) {e:65}{e:65} Summarizing our considerations for the case i “ j , we gather (66) and (69) into thedecomposition (64). We conclude that almost surely, lim n Ñ8 d ÿ i “ n ´ ÿ k “ y x ; iit k E r x iit k t k ` s “ d ÿ i “ ż t y x ; iir d R r ´ d ÿ i “ ż S pr ,t sq y x ; iir d R p r , r q . (70) {b4}{b4} We now go back to (61), and handle the terms x D i y it k , r t k ,t k ` s y H therein. We write theinner product in H in an explicit way thanks to (38), which yields x D i y it k , r t k ,t k ` s y H “ ż t ż t D ir y it k r ,t k s p r q r t k ,t k ` s p r q d R p r , r q . We thus have lim n Ñ8 d ÿ i “ n ´ ÿ k “ x D i y it k , r t k ,t k ` s y H “ lim n Ñ8 d ÿ i “ n ´ ÿ k “ ż t ż t D ir y it k r ,t k s p r q r t k ,t k ` s p r q d R p r , r q . We now argue similarly to what we did for (69). Namely one of our standing assumptionsis that p r , r q Ñ D ir y ir S p r , r q has a finite 2-dimensional p -variation. Since R admits afinite ρ -variation and p ´ ` ρ ´ ą , standard results concerning convergence of Riemannsums to Young integrals show that almost surely we have lim n Ñ8 d ÿ i “ n ´ ÿ k “ x D i y it k , r t k ,t k ` s y H “ d ÿ i “ ż S pr ,t sq D ir y ir d R p r , r q . (71) {b5}{b5} We can now conclude our proof easily. That is plugging (62), (63), (70) and (71) into (61),we end up with, almost surely, ż t y r d x r ´ ż t y r d ˛ x r “ d ÿ i “ ż t y x ; iir d R r ´ d ÿ i “ ż S pr ,t sq y x ; iir d R p r , r q ` d ÿ i “ ż S pr ,t sq D ir y ir d R p r , r q , from which the claim (55) is immediately deduced. This concludes the proof. (cid:3) We close this section by proving a technical result which has been used in order to deriverelation (58).
Lemma 3.2.
Assume the same conditions as in Theorem 3.1. Then y π defined in (56) converges to y in D , p H d q , i.e. lim | π |Ñ E r} y π ´ y } H d ` } D y π ´ D y } p H d q b s “ .Proof. According to (38), we have } y π ´ y } H d “ n ´ ÿ i,j “ ż r t i ,t i ` sˆr t j ,t j ` s x y t i ´ y s , y t j ´ y t y d R p s, t q , where we recall that π “ t “ t ă t ă ¨ ¨ ¨ ă t n “ t u . On each rectangle r t i , t i ` s ˆ r t j , t j ` s we apply Theorem 2.22 to the function f ij p s, t q “ x y s ´ y t i , y t ´ y t j y , which is allowed since f ij is easily seen to be a function in C p ´ var .Recall that we have assumed p ´ ` ρ ´ ą . Throughout the proof, we choose p ą p and ρ ą ρ ą ρ satisfying p p q ´ ` p ρ q ´ ą and p p q ´ ` p ρ q ´ ą . Since we also have f ij p t i , ¨q “ and f ij p¨ , t j q “ , we get } y π ´ y } H d ď C n ´ ÿ i,j “ ´ } y } p ´ var; r t i ,t i ` s } y } p ´ var; r t j ,t j ` s ¯ } R } ρ ´ var; r t i ,t i ` sˆr t j ,t j ` s . (72) {e:y-y’}{e:y-y’} In order to bound the right-hand side of (72), we introduce a new function ω , defined by ω pr a, b s ˆ r c, d sq “ } y } p p ´ var; r a,b s } y } p p ´ var; r c,d s . (73) {e:w1}{e:w1} KOROHOD AND STRATONOVICH INTEGRALS 23
Then it is readily checked that ω is also a 2d-control in the sense of Definition 2.7. Thefollowing is easily deduced from (72): } y π ´ y } H d ď C sup i,j ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ ρ ´ ρ ˆ n ´ ÿ i,j “ ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ p ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ ρ , (74) {e:y-y”}{e:y-y”} where ω is the control defined in (15). Now both ω and ω above are 2d-controls. Hence aneasy extension of [11, Exercise 1.9] to a 2d setting shows that ω { p ω { ρ is also a 2d-control.Hence one can resort to the super-additivity property of ω { p ω { ρ in order to deduce thefollowing from (74): } y π ´ y } H d ď C sup i,j ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ ρ ´ ρ ´ ω pr , t s q ¯ p ´ ω pr , t s q ¯ ρ . (75) {e:y-y}{e:y-y} We now turn to an upper bound on } D y π ´ D y } p H d q b . To this aim we first express thisquantity using the norm in p H d q b induced by (38). This yields } D y π ´ D y } p H d q b “ n ´ ÿ i,j “ ż r ,t i ` sˆr ,t j ` sˆr t i ,t i ` sˆr t j ,t j ` s x D u y t i ´ D u y s , D v y t j ´ D v y t y d R p u, v q d R p s, t q . (76) {e:dy-dy1}{e:dy-dy1} We apply Lemma 2.23 to the right-hand side of (76) and get } D y π ´ D y } p H d q b ď C n ´ ÿ i,j “ } R } ρ ´ var; r ,t i ` sˆr ,t j ` s } R } ρ ´ var; r t i ,t i ` sˆr t j ,t j ` s ˆ ´ } D y t i ´ D y ¨ } p ´ var; r t i ,t i ` s ` } D ¨ y t i ´ D ¨ y ¨ } p ´ var; r ,t i ` sˆr t i ,t i ` s ¯ ˆ ´ } D y t j ´ D y ¨ } p ´ var; r t j ,t j ` s ` } D ¨ y t j ´ D ¨ y ¨ } p ´ var; r ,t j ` sˆr t j ,t j ` s ¯ . As a preliminary step, we also bound the variations on intervals of the form r , t j s by varia-tions on r , T s . Thus one can bound } D y π ´ D y } p H d q b by C } R } ρ ´ var; r ,T s n ´ ÿ i,j “ } R } ρ ´ var; r t i ,t i ` sˆr t j ,t j ` s ´ } D y } p ´ var; r t i ,t i ` s ` } D y } p ´ var; r ,T sˆr t i ,t i ` s ¯ ˆ ´ } D y } p ´ var; r t j ,t j ` s ` } D y } p ´ var; r ,T sˆr t j ,t j ` s ¯ . (77) {e:dy-dy’}{e:dy-dy’} We now wish to apply super-additivity properties of the p -variations, as we did for (75).However, note that the function r a, b s ˆ r c, d s ÞÑ } D y } p p ´ var; r a,b sˆr c,d s may fail to be super-additive (see [9, Theorem 1]). Hence we need to resort to the controlled 2d variation asintroduced in Definition 2.9. Specifically, it follows from (77) that } D y π ´ D y } p H d q b can be upper bounded by C } R } ρ ´ var; r ,T s n ´ ÿ i,j “ } R } ρ ´ var; r t i ,t i ` sˆr t j ,t j ` s ´ } D y } p ´ var; r t i ,t i ` s ` ~ D y ~ p ´ var; r ,T sˆr t i ,t i ` s ¯ ˆ ´ } D y } p ´ var; r t j ,t j ` s ` ~ D y ~ p ´ var; r ,T sˆr t j ,t j ` s ¯ . (78) {e:dy-dy2}{e:dy-dy2} Notice that the right-hand side of (78) is finite, noting that p ă p and owing to (14).Furthermore, noting that the function r c, d s ÞÑ ~ D y ~ p p ´ var; r ,T sˆr c,d s is a control, we candefine the following 2d controls (where we use [11, Exercise 1.9] again): ω pr a, b s ˆ r c, d sq “ } D y } p p ´ var; r a,b s } D y } p p ´ var; r c,d s (79) ω pr a, b s ˆ r c, d sq “ ~ D y ~ p p ´ var; r ,T sˆr a,b s ~ D y ~ p p ´ var; r ,T sˆr c,d s (80) ω pr a, b s ˆ r c, d sq “ } D y } p p ´ var; r a,b s ~ D y ~ p p ´ var; r ,T sˆr c,d s . (81)Now, similarly to (75), relation (78) entails } D y π ´ D y } p H d q b (82) {e:Dy-Dy}{e:Dy-Dy} ď C } R } ρ ´ var; r ,T s sup i,j ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ ρ ´ ρ ´ ω pr , t s q ¯ ρ ÿ k “ ´ ω k pr , t s q ¯ p , where ω is the control given in (15). This is our desired bound for the difference D y π ´ D y .Let us summarize our considerations so far. Gathering inequalities (75) and (82), we haveproved that } y π ´ y } H d ` } D y π ´ D y } p H q b (83) {e:y-y-dy-dy}{e:y-y-dy-dy} ď C ` ` } R } ρ ´ var; r ,T s ˘ ´ ω pr , t s q ¯ ρ ÿ k “ ´ ω k pr , t s q ¯ p sup i,j ´ ω pr t i , t i ` s ˆ r t j , t j ` sq ¯ ρ ´ ρ , where the controls ω, ω , ω , ω , ω are respectively defined by (15), (73), (79), (80) and (81).We can now argue as follows: first, according to (17), (14) and (15) we have lim n Ñ8 sup i,j ω pr t i , t i ` s ˆ r t j , t j ` sq “ . Next we have assumed in Theorem 3.1 that E ” } y } p ´ var; r ,T s ` } D y } p ´ var; r ,T s ` } D y } p ´ var; r ,T s ı ă 8 . Therefore one can take expected valued in (83) in order to get lim n Ñ8 E ” } y π ´ y } H d ` } D y π ´ D y } p H d q b ı “ , which is our claim. This concludes our proof. (cid:3) We close this section by showing that our main Theorem 3.1 generalizes previous Skorohod-Stratonovich integral correction formulae.
KOROHOD AND STRATONOVICH INTEGRALS 25
Remark . In [15], the relationship (1) is obtained for a γ -HölderGaussian process x with γ P p , q . The process y considered in [15] is of the special form y “ f p x q for f P C N with N “ t γ u .Our main result Theorem 3.1 holds for Gaussian processes possessing finite p -variationwith p P p , q (or ρ P r , q ). Noting that Propositions 2.27 and 2.28 hold under Hypotheses2.8 and 2.12 for p P p , q , we believe that our approach could be extended to p P p , q . Akey difference between p P p , q and p P r , q is that a weighted sum in the third chaos of x will be involved in the rough integral (60) for p P r , q . Thus for p P p , q , to calculatethe Skorohod-Stratonovich correction term, we also need develop some estimation for theweighted sum in third chaos of x which is parallel to Proposition 2.28.Note that the condition p P p , q ( ρ P r , q ) is also used to define the Young integralsappearing in (66), (69) and (71). However, if we further assume that R ¨ and R p¨ , t q for each t P r , T s are absolutely continuous as in [15, Hypothesis 3.1], which is satisfied by fractionalBrownian motion B H with Hurst parameter H P p , q , then the integrals in (66), (69) and(71) are automatically well-defined as Riemann integrals. Remark . In [4], the relationship between ş t y r d x r and ş t y r d ˛ x r is studied, where x satisfies Hypotheses 2.8 and 2.12, and y is the solution to (3) with σ being sufficiently regular. Indeed, under the conditions assumed in the main Theorem in[4], our main result Theorem 3.1 also holds. More specifically, E r} y } p ´ var; r ,T s s ă 8 is aconsequence of [4, Theorem 2.25]; E r} D y } p ´ var; r ,T s s ` E r} D y } p ´ var; r ,T s s ă 8 follows from D s y t “ r ,t q p s q J X t p J X x q ´ σ p Y s q and Theorem [4, Theorem 2.27] (see also the end of the proof[4, Proposition 4.10]). With those relations in mind, our main Theorem 3.1 also covers theanalysis performed in [4]. Acknowledgment . We would like to thank Tom Cass for some interesting discussions aboutthis project.
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Email address : [email protected] Samy Tindel: Department of Mathematics, Purdue University, 150 N. University StreetWest Lafayette, Indiana 47907, USA
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