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i) Our key assumption is a regularity assumption of the elements in the Cameron-Martin space of X in terms of their quadratic variation. The crucial role, which regularity in the Cameron-Martin space plays for deriving the Gaussian Itˆoformula, has apparently been largely unnoticed in the literature. On the one hand, it opens the door to extend the Gaussian Itˆoformula to processes with jumps. On the other hand, it can easily be checked in all references (which we are aware of), in which the Itˆoformula (1.1) is proved for stochastically continuous processes. Hence, our approach has a unifying and generalizing character at the same time. ii) The second derivative term in the jump component of our Itˆoformula depends on the covariance structure of the Gaussian process X via the expression E[XsXs ]. This is in contrast to the stochastically continuous case, in which only the variance function− appears in the corresponding formula (1.1). If X is a Gaussian martingale with jumps, then this extra second derivative term clearly vanishes, and the Itˆoformula (1.2) reduces to its classiscal form:
T T 1 c F (XT ) = F (X0)+ F ′(Xs )dXs + F ′′(Xs )d[X]s − 2 − Z0+ Z0+ + F (Xs) F (Xs ) F ′(Xs )(Xs Xs ), (1.3) − − − − − − s (0,T ] ∈X (where we exploit that the continuous part V c of the variance coincides with the continuous part of the quadratic variation [X] in the Gaussian case and that, by [14], all the jumps of a Gaussian martingale occur at the deterministic times of stochastic discontinuities). (iii) The Skorokhod type integration, which we develop in this paper, and Itˆo’s formula can be extended to conditionally Gaussian processes. We explain this construction in detail for fractional Brownian motion subordinated by an increasing L´evy process. This class of processes shares many important properties of fractional Brownian motion like stationary increments and short/long memory (in dependence of the Hurst parameter and the L´evy subordinator). Additionally, subordinated frac- tional Brownian motion features jumps and heavier tails than Gaussian, which makes it a promising model for real-world phenomena, see e.g. Ganti et al. [12], Meerschaert et al. [25], Debicki et al. [8]. To the best of our knowledge, this paper constitutes the first approach to a Skorokhod type stochastic calculus for subordinated fractional Brownian motion (or, more generally, conditionally Gaussian processes with jumps). The paper is organized as follows. In Section 2, we first recall some preliminaries on Gaussian processes and then introduce the concept of a weakly regulated process. It allows giving a meaning to the one- sided limits Xs without imposing any path regularity assumption on X, provided that the functions in the Cameron-Martin± space of X are regulated. These one-sided limits appear in our general version of Itˆo’s formula in Theorem 5.1. In Section 2, we also study the set of stochastic discontinuities for weakly regulated Gaussian processes. In Section 3, we recall the definition of the Malliavin derivative and of the divergence operator, with an emphasis on the role of the S-transform and of regularity in the Cameron-Martin space. The observations in Section 3 serve as a main motivation for our notion of Wick-Skorokhod integration, which we define in Section 4 in terms of the S-transform and the Henstock- Kurzweil integral. This S-transform approach has already been adopted in [4, 5], and actually, can be viewed as a main tool for studying Hitsuda-Skorokhod integration in a white noise framework as in [22] and the references therein. We also show, that our integrals extends the classical stochastic Itˆointegral for predictable integrands to anticipating integrands in the case that X is a Gaussian martingale. After these preparations we can state and prove our Itˆoformula in its general form in Section 5. Section 6 is devoted to subordinated fractional Brownian motion. We finally explain how to check the required structural assumption on the Cameron-Martin space in Section 7 and compare our assumptions to the ones imposed in the existing literature in Section 8. In the appendix, we provide, following ideas of [27], a chain rule for the Henstock-Kurzweil integral, which is required in our proof of Itˆo’s formula.
2. Weakly regulated processes
The main purpose of this section is to give a meaning to the one-sided limits Xs , which occur in the Itˆoformula (Theorem 5.1), without imposing path regularity assumptions on X. Before± doing so, let us recall some facts on Gaussian processes for ready reference. Suppose (Xt)0 t T is a centered Gaussian process on a complete probability space (Ω, , P ) with ≤ ≤ 2 F variance function V (t) := E[X(t) ]. We denote by HX the first chaos of X, i.e. the Gaussian Hilbert C. Bender /Itˆo’s Formula for Gaussian Processes 3
2 space, which is obtained by taking the closure of the linear span of Xt; t [0,T ] in L (Ω, , P ). To each element h H , one can associate a function { ∈ } F ∈ X h : [0,T ] R, t E[X h]. → 7→ t The space of functions CM := h; h H X { ∈ X } is called the Cameron-Martin space associated to X. As, by definition, the set X ; t [0,T ] is total { t ∈ } in HX , the map H CM , h h X → X 7→ is bijective. It becomes an isometry, if one equips the Cameron-Martin space with the inner product
h,g := E[hg]. h iCMX The Wick exponential of h H is defined to be ∈ X 2 exp⋄(h) := exp h E[h ]/2 . { − } If is a dense subset of H , then, by Corollary 3.40 in [15], the set A X exp⋄(h); h { ∈ A} 2 2 X X is total in (LX ) := L (Ω, , P ), where is the completion by the P -null sets of the σ-field generated F F 2 by X. Hence, every random variable ξ (LX ) is uniquely determined by its S-transform restricted to , ∈ A (Sξ)(h) := E[exp⋄(h)ξ], h . ∈ A 2 This means, the identity ξ = η is valid in (LX ), if and only if (Sξ)(h) = (Sη)(h) for every h . We also recall the following straightforward identities (g,h H , t [0,T ]) ∈ A ∈ X ∈ E[gh] E[gh] exp⋄(h)exp⋄(g) = e exp⋄(g + h), (S exp⋄(g))(h)= e , (2.1)
(Sg)(h) = E[gh], (SXt)(h)= h(t).
More, generally, we note that, by a classical result on Gaussian change of measure,
(SG(g1,...,gD))(h)= E[G(g1 + E[g1h],...,gD + E[gDh])] (2.2)
D for every g1,...,gD,h HX and measurable G : R R, provided that the expectation on the right- hand side exists, see e.g.∈ Janson [15, Theorem 14.1]. → After these preliminaries, we now define the notion of a weakly regulated process. 2 Definition 2.1. A stochastic process Y : [0,T ] (LX ) is called weakly regulated, if for every s (0,T ] 2 → ∈ there is a random variable Ys (LX ) such that for every sequence (sn) which converges to s from the left − ∈ 2 lim Ysn = Ys weakly in (LX ), n − →∞ 2 and, if for every s [0,T ) there is a random variable Ys+ (LX ) such that for every sequence (sn) which converges to s∈from the right ∈
2 lim Ys = Ys+ weakly in (L ), n n X →∞
For a weakly regulated process, we shall apply the convention Y0 := Y0 and YT + := YT . Moreover, we write − + ∆Ys = Ys+ Ys , ∆ Ys = Ys+ Ys, ∆−Ys = Ys Ys . − − − − − The analogous notation is used for the jumps of deterministic regulated functions. Weakly regulated processes can be characterized via the S-transform as follows: Proposition 2.2. Suppose Y : [0,T ] (L2 ). If Y is weakly regulated, then the map t E[Y 2] is → X 7→ t bounded and, for every h HX , the map t (SYt)(h) is regulated (i.e., has limits from the left and from the right). ∈ 7→ 2 Conversely, assume that is a dense subset of HX . If the map t E[Yt ] is bounded and, for every h , the map t (SY )(hA) is regulated, then Y is weakly regulated.7→ ∈ A 7→ t C. Bender /Itˆo’s Formula for Gaussian Processes 4
Before we prove this proposition, let us state the following corollary concerning the Gaussian process X.
Corollary 2.3. The Gaussian process (Xt)t [0,T ] is weakly regulated, if and only if its variance function ∈ V is bounded and the Cameron-Martin space CMX has a dense subset consisting of regulated functions. 2 In this case the weak (LX )-limits Xs belong to the first chaos HX . ± Proof. In view of Proposition 2.2, the first statement is an immediate consequence of (2.1). For the second one, let s [0,T ) and denote by π the orthogonal projection from (L2 ) on H . Then, ∈ HX X X E[ X π (X ) 2] = E[X (X π (X ))] | s+ − HX s+ | s+ s+ − HX s+ 1 = lim E[Xs+ (Xs+ πHX (Xs+))]. n n →∞ −
The right-hand side equals 0, because Xs+1/n HX . Hence, Xs+ HX , and the left-sided limits can be handled analogously. ∈ ∈ Proof of Proposition 2.2. Suppose first that Y is weakly regulated. Fix h H and s (0,T ]. Then, for ∈ X ∈ every sequence (sn) which converges to s from the left,
lim (SYs )(h) = lim E[Ys exp⋄(h)] = E[Ys exp⋄(h)] n n n n − →∞ →∞
Hence, t (SYt)(h) has a limit from the left at s. In the same way, one observes that it has a limit from the right7→ at every s [0,T ). We next prove the boundedness of t E[Y 2] by contradiction. Thus, ∈ 7→ t suppose to the contrary that this function is unbounded, and choose a sequence (tn) in [0,T ] such that E[Y 2 ] converges to infinity. We can extract a subsequence (t ) which converges to some t [0,T ] and tn nk ∈ satisfies (by passing to another subsequence, if necessary) tnk
E[Y exp⋄(h)] = h˜(t ) h˜(s+) = E[Y exp⋄(h)]. tn n → s+ 2 As, moreover, supn N E[ Ytn ] < , we conclude thanks to Theorem V.1.3 in [36] that ∈ | | ∞ Y Y weakly in (L2 ). tn → s+ X 2 An analogous argument shows, for every s (0,T ], existence of a weak limit Xs in (LX ) (as u approaches s from the left). ∈ − The following example is instructive and will be applied in the proof of Itˆo’s formula. Example 2.4. Suppose that the variance function V of X is regulated and that the Cameron-Martin space CMX has a dense subset consisting of regulated functions. Then, by Corollary 2.3, X is weakly regulated. We consider, for some continuous function F : R R, the process Yt := F (Xt). Assume that F satisfies the following subexponential growth condition →
2 F (x) Ceax , x R, (2.3) | |≤ ∈ 1 for constants C 0 and 0 a< (4λ)− with λ := supt [0,T ] V (t). This standard assumption guarantees ≥ ≤ ∈ that t E[Y 2] is bounded. By (2.2), the S-transform of Y is given by 7→ t t (SF (X ))(h)= E[F (X + h(t))] = ψ (V (t),h(t)), h H , (2.4) t t F ∈ X C. Bender /Itˆo’s Formula for Gaussian Processes 5
where 1 y2/2 ψF : [0, λ] R R, (t, x) F (x + √ty) e− dy. (2.5) × → 7→ R √2π Z Note that ψF (0, x)= F (x). By (2.4) and Proposition 2.2, Y is weakly regulated and the weak limits Yt+, Yt satisfy − (SY )(h)= ψ (V (t ),h(t )), h H . t± F ± ± ∈ X 2 Define V ±(t)= E[Xt ]. By Theorem V.1.1 in [36], V ±(t) V (t ), and the inequality can be strict, as 2 ± ≤ ± Xt are weak (LX )-limits only. Then, ± + Yt+ = ψF (V (t+) V (t),Xt+), Yt = ψF (V (t ) V −(t),Xt ), (2.6) − − − − − since, e.g., for every h H ∈ X S ψ (V (t+) V +(t),X ) (h) F − t+ 2 2 + + 1 (u +y )/2 = F (h(t+) + V (t+) V (t)u + V (t)y) e− d(u,y) R2 − 2π Z = ψF (V (t+),h(t+))p . p
Thus, the identity Yt+ = F (Xt+) can, in general, only be expected to be valid at those t [0,T ), for + ∈ which V (t+) = V (t) (which is equivalent to saying that the convergence from the right to Xt+ takes place in probability). We next study the set of stochastic discontinuities of X. Definition 2.5. The process X is said to be stochastically continuous at t [0,T ], if for every sequence ∈ (tn) in [0,T ] t t X X in probability. n → ⇒ tn → t We denote by CX the set of points, at which X is stochastically continuous, and by DX = [0,T ] CX the set of stochastic discontinuities of X. \ Proposition 2.6. Suppose that the variance function V of X is regulated, X is weakly regulated and HX is separable. Then the set DX of stochastic discontinuities of X is at most countable.
Proof. We first apply the separability of H and choose a countable dense subset ′ of H . Let X A X D = s [0,T ]; V is discontinuous at s { ∈ } s [0,T ]; h is discontinuous at s . ∪ { ∈ } h ′ ! [∈A As V and all the h’s are regulated functions thanks to Proposition 2.2, the set D is at most countable. If t [0,T ] D, then we obtain, for every h ′ ∈ \ ∈ A
lim E[Xsh] = lim h(s)= h(t)= E[Xth] s t s t → → 2 2 lim E[ Xs ] = lim V (s)= V (t)= E[ Xt ]. s t s t → | | → | | Now, Theorem V.1.3 in [36] implies that
2 lim E[ Xs Xt ]=0, s t → | − | and, hence, X is stochastically continuous at t. In particular, D D is at most countable. X ⊂ Remark 2.7. Note that the separability of HX cannot be dispensed with. Indeed, if DX is countable, then the rational span of X ; t ([0,T ] Q) D { t ∈ ∩ ∪ X } is a countable dense subset of HX . C. Bender /Itˆo’s Formula for Gaussian Processes 6
3. Divergence operator and Skorokhod integral
In this section, we briefly recall the definition of the Malliavin derivative and the divergence operator, and discuss the notion of Skorokhod integration. Our presentation differs from the usual account in the literature, in that we dwell upon the role of the S-transform and on regularity properties in the Cameron-Martin space. Let us first re-interpret the first chaos of X as an isonormal Gaussian process in the sense of Definition 1.1.1 in [28], parametrized by the Cameron-Martin space of X. Then, the Malliavin derivative D is a 2 2 2 X densely defined closed operator from (LX ) to LX (CMX ) := L (Ω, , P ; CMX ), which is constructed as follows: F 1 K For a smooth random variable F = f(h1,...,hK ), where h1,...hK HX and F b (R , R), one defines ∈ ∈ C K ∂F DF = (h1,...hK ) hk. ∂xk Xk=1 2 2 The Malliavin derivative for smooth random variables is closable from (LX ) to LX (CMX ) by Proposition 1,2 1.2.1 in [28]. Its domain is usually denoted by DX . The adjoint operator is known as the divergence 1,2 operator. Note that the linear span of the Wick exponentials is dense in DX , see Theorem 15.110 in [15], and that D exp⋄(h) = exp⋄(h) h, h H . ∈ X 2 Hence, u LX (CMX ) belongs to the domain Dom(δ) of the divergence operator δ, if and only if there is a random∈ variable δ(u) (L2 ) such that, for every h H , ∈ X ∈ X E[ D exp⋄(h),u ]= E[exp⋄(h)δ(u)], h iCMX or, equivalently, in terms of the S-transform, S ( u,h ) (h)= S(δ(u))(h), h iCMX for every h HX (or, by Corollary 3.40 in [15], from a dense subset of HX ). The divergence operator ∈ 2 2 δ, defined as above, then is a densely defined closed linear operator from LX (CMX ) to (LX ). As the Cameron-Martin space is a function space, our definition of the divergence operator maps stochastic processes to random variables, but it does not yet qualify as an integral operator with respect to X. Indeed, the minimal property, which such an integral should satisfy is that it maps indicator functions on increments, namely 1 X X , t [0,T ]. (0,t] → t+ − 0+ ∈ However, in terms of the covariance function R(t,s) := E[Xt Xs], we get δ(R(t+, )) = X , · t+ because S ( R(t+, ),h ) (h) = R(t+, ),h = E[X h]= h(t+) h · iCMX h · iCMX t+ = (SXt+)(h). Hence, one can define an integral operator in the following way: Suppose is a Banach space of (equiv- R alence classes of) functions from [0,T ] to R, such that the indicator functions 1(0,t], t [0,T ], constitute a total subset of E. Suppose ι : CM is a continuous linear map such that ∈ R → X ι(1 )= R(t+, ) R(0+, ). (3.1) (0,t] · − · In case of existence, the map ι is uniquely determined and extends in a canonical way to a continuous 2 2 X 2 linear mapping from the Bochner space LX ( ) := L (Ω, , P ; ) to LX (CMX ). We may then say that 2 R F R Z LX ( ) belongs to the domain of the Skorokhod integral, if ι(Z) Dom(δ), and define δ (Z) := δ ∈ι(Z). IfR we introduce a continuous bilinear map on CM via ∈ R ◦ R× X r, h ,CM := ι(r),h CM , (r, h) CMX , h iR X h i X ∈R× 2 then we obtain the following S-transform characterization of the Skorokhod integral: Z LX ( ) belongs 2 ∈ R to the domain of δ , if and only if there is a δ (Z) (LX ) such that for every h from a dense subset of R R ∈ HX , S ( Z,h ,CM ) (h)= S(δ (Z))(h). h iR X R Hence, it is crucial to understand the bilinear map r, h ,CM . Here are some examples. h iR X C. Bender /Itˆo’s Formula for Gaussian Processes 7
Example 3.1. (i) Suppose that X is an Gaussian martingale with RCLL paths. Then, the variance function V is RCLL and nondecreasing and
· 2 CM = h = h˙ (s)dV (s); h˙ L ([0,T ], dV ) , h = h˙ 2 . X ∈ k kCMX k kL (dV ) Z0
2 2 ˙ · ˙ We can now choose = L ([0,T ], dV ) and ι : L ([0,T ], dV ) CMX , h 0 h(s)dV (s), which is an isometric isomorphism.R Consequently, → 7→ R T T r, h ,CM = r(s)h˙ (s)dV (s)= r(s)dh(s). h iR X Z0 Z0
The operator δL2(dV ) then coincides with the classical notion of the Skorokhod integral on general measure spaces, see e.g. Janson [15], Chapters 7.2, 15.11, and 16.4. (ii) Recall that fractional Brownian motion is a centered Gaussian process X which has, in dependence of the Hurst parameter H (0, 1), the covariance function ∈ 1 R(t,s)= t2H + s2H t s 2H . 2 − | − | If H = 1/2, then X is a Brownian motion and we are back in item (i). We now consider the case H > 1/2. Then every element h of the Cameron-Martin space is absolutely continuous with respect to 1/(1 H) the Lebesgue measure with a density h˙ L − ([0,T ], dt), see Decreusefond and Ust¨unel¨ [9]. There is, however, no known Hilbert space of∈ functions on [0,T ] such that ι in (3.1) provides an isometric R isomorphism to CMX , see the discussion in Pipiras and Taqqu [31]. In this case, by standard results on fractional integrals, one can choose = L1/H ([0,T ], dt) and obtains, R T T r, h ,CM = r(s)h˙ (s)ds = r(s)dh(s). h iR X Z0 Z0 (iii) We now assume that X is a fractional Brownian motion with Hurst parameter H < 1/2. Then, there is a Hilbert space of functions on [0,T ] such that ι provides an isometric isomorphism on CMX , see Pipiras and Taqqu [31]. In thisR case, certain elements of the Cameron-Martin space may fail to be absolutely continuous. However, Corollary 2.8 in Bender [3] or the discussion preceding Definition 3.3 in Cheridito and Nualart [7] can be reformulated in the following way: The subspace of elements h of the Cameron-Martin space such that h has a square-integrable density h˙ with respect to the Lebesgue measure and such that for every r ∈ R T T r, h ,CM = r(s)h˙ (s)ds = r(s)dh(s) h iR X Z0 Z0 is dense. Unfortunately, the space is unfavorably small and does, with probability 1, not contain the paths of a fractional Brownian motion,R if H 1/4, see Cheridito and Nualart [7]. This observation led to the notion of extended Skorokhod integration,≤ see Remark 4.5, (ii), below. The bottom line of these examples is, that in typical cases, the Skorokhod integral can be characterized by the identity T (SZs)(h)dh(s)= S(δ (Z))(h), (3.2) R Z0 for elements h from a dense subspace of HX such that the corresponding element in the Cameron-Martin space h is sufficiently smooth. See Mocioalca and Viens [26] for more examples.
4. Wick-Skorokhod integration
In this section, we introduce a class of generalized Skorokhod integrals, which is applied throughout the paper. It contains the class of extended Skorokhod integrals, see e.g. Cheridito and Nualart [7], Mocioalca and Viens [26], Lei and Nualart [24], as special case. The definition of the integral will be relative to smoothness properties of the elements of the Cameron- Martin space. To this end, let denote a vector space of real-valued functions on [0,T ]. R C. Bender /Itˆo’s Formula for Gaussian Processes 8
Definition 4.1. We say that the Gaussian process X is -dense, if CM is a dense subset of R X ∩ R CMX . In this case, we denote := h HX ; h , AR { ∈ ∈ R} which defines a dense subset of the first chaos of X. If CMX (and, consequently, = HX ), we call X -regular. ⊂ R AR R Example 4.2. We denote by L˙ p([0,T ],µ) the space of functions which are absolutely continuous with respect to the measure µ with a density in Lp([0,T ],µ), p [1, ]. Then, Example 3.1 entails the following: Any RCLL Gaussian martingale X is L˙ 2([0,T ], dV )∈-regular,∞ fractional Brownian motion with 1/(1 H) Hurst parameter H > 1/2 is L˙ − ([0,T ], dt)-regular, and fractional Brownian motion with Hurst parameter H < 1/2 is L˙ 2([0,T ], dt)-dense. Moreover, if X is weakly regulated, then, by Proposition 2.2, X is R([0,T ])-regular, where R([0,T ]) stands for the space of all regulated functions on [0,T ]. Trivially, any Gaussian process X is CMX -regular. In view of (3.2), we propose the following definition: 2 Definition 4.3. Suppose X is -dense. A process Z : [0,T ] (LX ) is said to be -Wick-Skorokhod R → T R integrable with respect to X, if for every h the integral (SZs)(h) dh(s) exists (in the sense R 0 ∈ A T 2 of Henstock and Kurzweil) and if there is a random variable 0 RZs d⋄ Xs (LX ) such that for every h R ∈ ∈ AR T T R S Zs d⋄ Xs (h)= (SZs)(h) dh(s). R Z0 ! Z0 T We then call 0 Zs d⋄ Xs the -Wick-Skorokhod integral of Z with respect to X. R R Note that,R if X is -regular, then the S-transform characterization of the -Wick-Skorokhod integral in the above definitionR holds for every element h in the first chaos, and not onlyR on a dense subspace. T T We sometimes write 0+ Zs d⋄ Xs := 0 1(0,T ](s)Zs d⋄ Xs, provided the integral on the right-hand side exists. The same notation willR be applied for other typesR of integrals, when integration is understood R R over (0,T ] rather than [0,T ]. Remark 4.4 (Henstock-Kurzweil integral). The Henstock-Kurzweil integral can be applied to give a T R meaning to integrals of the form 0 u(s)dr(s) for suitably pairs of functions u, r : [0,T ] , without assuming that the integrator r is of bounded variation. We briefly recall the construction: A gauge→ function R is any function δ : [0,T ] (0, ). A tagged partition τ := ([si 1,si],yi); i = 1,...,n of the interval → ∞ { − } [0,T ] is called δ-fine, if yi δ(yi) si 1 yi si yi + δ(yi) for every i = 1,...,n. The Riemann T − ≤ − ≤ ≤ ≤ sum for the integral 0 u(s)dr(s) with respect to the tagged partition τ is given by R n SRS (u, r, τ) := u(yi)(r(si) r(si 1)). − − i=1 X If there is an I R such that for every ǫ> 0 there is a gauge function δ such that SRS(u, r, τ) I <ǫ ∈ T | − | for every δ-fine tagged partition τ, then I is uniquely determined, denoted by 0 u(s)dr(s) and called the Henstock-Kurzweil integral of u with respect to r. A brief review of the relation between the Henstock- Kurzweil integral and other Stieltjes-type integrals can be found in Appendix FR of Part I in [10]. We just note the following important relation to the Lebesgue-Stieltjes integral: If r is of bounded variation, then r uniquely determines a signed measure µr via the relation
µ ([0,t]) := r(t+) r(0), 0 t Indeed, in all these cases, can be chosen as a suitable subspace of L˙ p([0,T ], dt) for some fixed p (1, ). It is important to note thatR some of these references impose integrability conditions on the paths∈ of∞ the 2 2 integrand Z, e.g., Z LX (L ([0,T ])) in [26] (where p = 2). In our definition, we merely require that T ∈ 0 (SZs)(h), dh(s) exists for h . As we will work under a much weaker regularity condition than absolute continuity for the elements∈ AR of the Cameron-Martin space for most of the paper, this relaxation Rof the integrability conditions turns out to be crucial to ensure e.g. integrability of X with respect to itself. The next example, which discusses the integrability of step processes, clarifies the relation of our Skorokhod type integral to Wick products: Example 4.6. Suppose X is weakly regulated, and, thus, every element h CMX is a regulated function. We say, Z is a simple integrand, if it is of the form ∈ n Z := F01 0 + Gi1(t − ,t ) + Fi1 t , { } i 1 i { i} i=1 X 0 = t < t < < t = T, F , G D1,2. Then, t (SZ )(h) is a real-valued step function and, 0 1 ··· n i i ∈ X 7→ t because h is a regulated function for every h H , the Henstock-Kurweil integral T (SZ )(h)dh(s) ∈ X 0 t exists. Indeed, we get, for every h H (applying the jump convention ∆h(T ) = ∆h−(T )), ∈ X R T + (SZs)(h) dh(s) = (SF0)(h) ∆ h(0) (4.1) 0 · Zn + (SGi)(h) (h(ti ) h(ti 1+)) + (SFi)(h) ∆h(ti). − − − · i=1 X 2 Let us now recall that two random variables η, ξ (LX ) are said to have a Wick product, if there is a random variable η ξ (L2 ) such that ∈ ⋄ ∈ X S(η ξ)(h) = (Sη)(h)(Sξ)(h) ⋄ 1,2 for every h HX . By Proposition 1.3.3 in [28], F g exists for every F DX , g HX and relates to the ordinary∈ product via ⋄ ∈ ∈ F g = Fg DF,g , F D1,2, g H . ⋄ − h iCMX ∈ X ∈ X In view of (2.1) and (4.1), we, thus, obtain for every h H , ∈ X T (SZs)(h) dh(s) Z0 n + = S F0 (∆ X0)+ Gi (Xt Xt − +)+ Fi (∆Xt ) (h) ⋄ ⋄ i− − i 1 ⋄ i i=1 ! X Hence, T n + Zs d⋄ Xs = F0 (∆ X0)+ Gi (Xti Xti−1+)+ Fi (∆Xti ) , R ⋄ ⋄ − − ⋄ Z0 i=1 X for any function space such that X is -dense, e.g. = R([0,T ]). Hence, the Wick-Skorokhod integral of a simple integrand isR a Young-StieltjesR sum involvingR Wick products, as expected for Skorokhod type integrals. We next relate our notion of Wick-Skorokhod integration to the usual stochastic Itˆointegral [see e.g. 32] in the martingale case: X X Theorem 4.7. Let F = ( t )t [0,T ] denote the augmentation of the filtration generated by X. Suppose X is an FX -martingale withF RCLL∈ paths and variance function V . If Z is FX -predictable and satisfies T E Z 2dV (s) < , | s| ∞ "Z0 # T ˙ 2 then the Wick-Skorokhod integral 0 Zs d⋄ Xs exists for = L ([0,T ], dV ) and coincides with the T R R stochastic Itˆointegral 0+ ZsdXs. R R C. Bender /Itˆo’s Formula for Gaussian Processes 10 Remark 4.8. The statement of the previous theorem cannot be extended to semimartingales, because Skorokhod integration is based on the Wick product. Indeed, suppose that X is a Gaussian RCLL semi- martingale. If X is not an FX -martingale, then we find 0 a T Z d⋄ X = (X X ) (X X ) = (X X )(X X ) C s R([0,T ]) s b − a ⋄ d − c b − a d − c − Z0 = (X X )(X X ), 6 b − a d − c where the right-hand side equals the Itˆointegral of Z with respect to X. Proof of Theorem 4.7. Note first that, by the martingale property of X, the set DX of stochastic dis- continuties of X consists of the jump times of the nondecreasing RCLL variance function V of X, and is thus at most countable. By Theorem 1.8 in [14], c Xt = Xt + (Xs Xs ), − − s D (0,t] ∈ XX ∩ where Xc is a Gaussian martingale with continuous paths. Of course, in this decomposition, the pathwise 2 left limits of the RCLL martingale X coincide, for every s (0,T ], P -almost surely with the weak (LX )- c d ∈ c limits Xs . We write V and V for the continuous and the discrete part of V . Then, V is the quadratic − c d 2 variation of X , V is the predictable compensator of (Xs Xs ) , and, hence, V is the s DX (0, ] − − predictable compensator of the quadratic variation [X] of X∈. Thus,∩ · if Z is predictable and satisfies the P T assumed integrability condition, then, by the isometry of the stochastic Itˆointegral, 0+ ZsdXs exists in (L2 ) and satisfies X R 2 T T T 2 2 E ZsdXs = E Zs d[X]s = E Zs dV (s) . 0+ " 0+ | | # " 0+ | | # Z Z Z In particular, the first chaos of X can be represented as T H = aX + h(s)dX ; a R, h L2([0,T ], dV ) . X 0 s ∈ ∈ ( Z0+ ) T We now fix a generic element h = aX0 + 0+ h(s)dXs of the first chaos of X and define R t t a2 1 := exp aX V (0) exp h(s)dX h(s) 2dV (s) , Et 0 − 2 s − 2 | | Z0+ Z0+ t Yt := ZsdXs. Z0+ t Note that h(t)= aV (0) + 0+ h(s)dV (s). Thus, in view of Remark 4.4, all we need to show is that R T E[ Y ]= E[ Z ]h(s)dV (s), (4.2) ET T ET s Z0+ where the integral on the right-hand side is a Lebesgue-Stieltjes integral. Applying Itˆo’s formula for semimartingales [32, Theorem II.32], we obtain after some elementary manipulations t 1 2 c h(s)∆Xs 2 h(s) ∆V (s) t = 0 + s h(s)dXs + s e − 1 . E E 0+ E − E − − Z s DX (0,t] ∈ X∩ Integration by parts (for semimartingales) thus yields t t tYt = s dYs + Ys d s + [ , Y ]t (4.3) E E − − E E Z0+ Z0+ C. Bender /Itˆo’s Formula for Gaussian Processes 11 where the quadratic covariation of and Y is given by E t c [ , Y ]t = Zs s h(s)dV (s) E E − Z0+ 1 2 h(s)∆Xs h(s) ∆V (s) + Zs s e − 2 1 ∆Xs. (4.4) E − − s D (0,t] ∈ XX ∩ Since and Y are square-integrable RCLL martingales, we may conclude from Emery’s inequality [32, TheoremE V.3] that both stochastic integrals in (4.3) are martingales (of class 1) and, consequently, have H zero expectation. Taking into account, that, for every s DX , the jumps sizes ∆Xs are independent of X X ∈ s and Zs is s -measurable by predictability, we obtain, thanks to (4.3)-(4.4), F − F − T c E[ T YT ] = E[Zs s ]h(s)dV (s) E E − Z0+ 1 2 h(s)∆Xs h(s) ∆V (s) + E[Zs s ]E e − 2 1 ∆Xs E − − s D (0,T ] ∈ X∩ h i As, by (2.1), h(s)∆X 1 h(s)2∆V (s) E e s− 2 1 ∆X = h(s)∆V (s), − s we arrive at h i T E[ T YT ]= E[Zs s ]h(s)dV (s). E E − Z0+ X Now, since Zs is s -measurable and is a martingale, we finally obtain F − E X [Zs s ]= E[ZsE[ T s ]] = E[Zs T ], E − E |F − E which yields (4.2), and finishes the proof. For the statement and proof of the Itˆoformula, we formulate the regularity requirement of elements in the Cameron-Martin space in terms of the quadratic variation as follows: We denote by W2∗ = W2∗([0,T ]) the subspace of regulated functions u : [0,T ] R such that → 2 + 2 σ (u) := ∆−u(s) + ∆ u(s) < 2 | | | | ∞ s (0,T ] s [0,T ) ∈X ∈X and such that, for every ǫ> 0, there is a partition λ of [0,T ] such that for all refinements κ = 0= s < { 0 s1 < (H) The Cameron-Martin space CMX of X is separable and X is W2∗-dense. Under condition (H), we can thus study the Wick-Skorokhod integral for = W2∗. In order to lighten the notation, we skip the subscript from the notation of the Wick-SkorokhodR integral in this case and simply write T T Zs d⋄Xs := Zs d⋄ ∗ Xs. W2 Z0 Z0 Remark 4.9. (i) Recall that, for p 1, the p-variation of a regulated function u : [0,T ] R is defined to be ≥ → m p vp(u) := sup u(tj ) u(tj 1) ; | − − | j=1 nX C. Bender /Itˆo’s Formula for Gaussian Processes 12 m N, 0= t0 5. Itˆo’s formula After these preparations on Wick-Skorokhod integration and weakly regulated processes, we can now state and prove a general Gaussian Itˆoformula in the presence of stochastic discontinuities. Let us first recall that V ±(t) denotes the variance of Xt and that, for a sufficiently integrable function ± F , ψF is given by the convolution 1 y2/2 ψF (t, x) := F (x + √ty) e− dy, R √2π Z see (2.5). Theorem 5.1. Suppose X is a centered Gaussian process satisfying (H), the variance function V of X is of bounded variation and E[(∆+X )2] + (V (s+) V +(s)) (5.1) s − s D [0,T ) ∈ X∩ 2 + E[(∆−X ) ] + (V (s ) V −(s)) < . s − − ∞ s D (0,T ] ∈ X∩ 2 Assume F (R) and F, F ′, F ′′ satisfy the growth condition (2.3) with λ = supt [0,T ] V (t). Then, ∈ C ∈ T 2 0 F ′(Xs)d⋄Xs exists and the following Itˆoformula holds in (LX ): R F (X ) F (X ) T − 0 T 1 T = F ′(X )d⋄X + F ′′(X )dV (s) s s 2 s Z0 Z0 + F (Xs) ψF (V (s ) V −(s),Xs ) F ′(Xs)∆−Xs − − − − − s D (0,T ] ∈ X∩ 1 2 + F ′′(X )(E[(∆−X ) ]+ V (s ) V −(s)) 2 s s − − + + + ψ (V (s+) V (s),X ) F (X ) F ′(X )∆ X F − s+ − s − s s s D [0,T ) ∈ X∩ 1 + 2 + F ′′(X )(E[(∆ X ) ]+ V (s+) V (s)) . −2 s s − Here, the set DX of stochastic discontinuities of X is at most countable and both sums converge absolutely 2 in (LX ). Note first that, under the assumptions of Theorem 5.1, X is weakly regulated by Corollary 2.3 and has at most countably many stochastic discontinuities by Proposition 2.6. C. Bender /Itˆo’s Formula for Gaussian Processes 13 We prove Theorem 5.1 by an S-transform approach, which originates in the work by Kubo [18] on Itˆo’s formula for generalized functionals of a Brownian motion in the setting of white noise analysis. It has since then be succesfully applied to wider classes of (stochastically) continuous Gaussian processes, see e.g. Bender [3, 4], Alpay and Kipnis [2], Lebovits and Vehel [23], Lebovits [22]. In a closely related development, the Malliavin calculus approach to Itˆo’s formula, replaces the S-transform (and, thus, the pairing with Wick exponentials) by pairings with Hermite polynomials, see Al`os et al. [1], Mocioalca and Viens [26], Kruk et al. [17], Lei and Nualart [24]. The key idea of the S-transform approach to Itˆo’s formula is the following: While the process F (Xt) may lack good path regularity, its S-transform t S(F (Xt))(η) is typically more regular and may be expanded via a ‘classical’ chain rule. In a second step,7→ the resulting terms, which appear after application of the chain rule, must be identified as the S-transforms of the different terms in the Gaussian Itˆoformula. The main contribution to this technique of proof in the present paper is to deal with the jumps that occur at the times of stochastic discontinuities and to make this technique applicable under the very weak regularity assumption (H) on the Cameron-Martin space. Expanding the S-transform of F (Xt) via the chain rule in Theorem A.1 for the Henstock-Kurzweil integral leads to the following result. T Proposition 5.2. Let all assumptions of Theorem 5.1 be in force. Then, for every h ∗ , (S F (X ))(h) dh(s) W2 0 ′ s exists as Henstock-Kurzweil integral and ∈ A R (S F (XT ))(h) T 1 T = (S F (X ))(h)+ (S F ′(X ))(h) dh(s)+ (S F ′′(X ))(h) dV (s) 0 s 2 s Z0 Z0 + (S F (Xs))(h) S( ψF (V (s ) V −(s),Xs ))(h) − − − − s D (0,T ] ∈ X∩ 1 (S F ′(X ))∆−h(s) (S F ′′(X ))∆−V (s) − s − 2 s + S( ψ (V (s+) V +(s),X ))(h) (S F (X ))(h) F − s+ − s s D [0,T ) ∈ X∩ + 1 + (S F ′(X ))∆ h(s) (S F ′′(X ))∆ V (s) , − s − 2 s where the two sums converge absolutely. Proof. Recall that, by (2.4), (SF (Xt))(h)= ψF (V (t),h(t)). So, we first check that ψF satisfies the regularity requirements of Theorem A.1. By the subexponential growth condition (2.3), we can interchange differentiation and integration and obtain, for every x R, t [0, λ] ∈ ∈ ∂ ∂2 ψ (t, x)= ψ ′ (t, x), ψ (t, x)= ψ ′′ (t, x). (5.2) ∂x F F ∂x2 F F As ψF ′′ is continuous on [0, λ] R, the Lipschitz condition in the first line of (A.1) is clearly satisfied. Moreover, for t (0, λ] and x ×R, integration by parts yields ∈ ∈ ∂ 1 y y2/2 1 ψF (t, x)= F ′(x + √ty) e− dy = ψF ′′ (t, x). (5.3) ∂t 2 R √2πt 2 Z This identity is also valid at t = 0, because for every ǫ> 0, by a Taylor expansion, ψ (ǫ, x) ψ (0, x) F − F ǫ 2 y y2/2 1 y y2/2 = F ′(x) e− dy + F ′′(x) e− dy + Rx(ǫ) R √2πǫ 2 R √2π Z Z 1 = F ′′(x)+ R (ǫ) 2 x with remainder term 1 2 y y2/2 Rx(ǫ)= (1 v)(F ′′(x + v√ǫy) F ′′(x))dv e− dy, R − − √2π Z Z0 C. Bender /Itˆo’s Formula for Gaussian Processes 14 and by dominated convergence Rx(ǫ) tends to zero, as ǫ goes to zero, for every x R. In order to show the H¨older-type condition in the second line of (A.1), we define ∈ ψ ′ (t,x) ψ ′ (s,x) | F − F | t s 1/2 , t = s, K : R [0, λ] [0, λ] R 0, (x,t,s) | − | 6 × × → ≥ 7→ ( 0, t = s. Then, by (5.2), ∂ ∂ ψ (t, x) ψ (s, x) = K(x,t,s) t s 1/2, ∂x F − ∂x F | − | and it suffices to show that K is continuous. However, ψ ′ (t, x) ψ ′ (s, x) F − F x+√ty 1 y2/2 = F ′′(r)dr e− dy = (√t √s) R √2π − Z Zx+√sy 1 y y2/2 F ′′(x + y(√s(1 v)+ v√t)) F ′′(x) dv e− dy. × R 0 − − √2π Z Z Thus, for t = s, 6 t s 1/2 K(x,t,s)= | − | √t + √s 1 y y2/2 F ′′(x + y(√s(1 v)+ v√t)) F ′′(x) dv e− dy , × R − − √2π Z Z0 2 which, by dominated convergence, implies that K is continuous at every (x0,t0,s0) R ([0, λ] (0, 0) ). ∈ × \{ } In order to show continuity at (x0, 0, 0) for x0 R, let (xn,tn,sn) be as sequence converging to (x0, 0, 0). Then, ∈ K(xn,tn,sn) | 1 | y y2/2 F ′′(xn + y(√sn(1 v)+ v√tn)) F ′′(xn) dv | | e− dy, ≤ R − − √2π Z Z0 which tends to zero by dominated convergence. ∗ We can, thus, apply the chain rule in Theorem A.1 to ψF (V (t),h(t)) for h W2 , and obtain, in view of (5.2)–(5.3), ∈ A ψ (V (T ),h(T )) ψ (V (0),h(0)) F − F T 1 T = ψ ′ (V (s),h(s)) dh(s)+ ψ ′′ (V (s),h(s)) dV (s) F 2 F Z0 Z0 + ψ (V (s),h(s)) ψ (V (s ),h(s )) ψ ′ (V (s),h(s))∆−h(s) F − F − − − F s (0,T ] ∈X 1 ψ ′′ (V (s),h(s))∆−V (s) −2 F + + ψ (V (s+),h(s+)) ψ (V (s),h(s)) ψ ′ (V (s),h(s))∆ h(s) F − F − F s [0,T ) ∈X 1 + ψ ′′ (V (s),h(s))∆ V (s), −2 F including the existence of the integral with respect to h as Henstock-Kurzweil integral and the absolute convergence of the two sums. As h and V are continuous at s, if X is stochastically continuous at s, the two sums can be restricted to (0,T ] DX and [0,T ) DX , respectively, without changing their values. Finally, each term in the above identity∩ can be rewritten∩ in terms of the S-transform by an application of Example 2.4, which yields the assertion. We are now in the position to prove the Itˆoformula in Theorem 5.1. C. Bender /Itˆo’s Formula for Gaussian Processes 15 Proof of Theorem 5.1. We first treat the term, which involves the jumps from the right. For s DX [0,T ), we consider ∈ ∩ + + + J := ψ (V (s+) V (s),X ) F (X ) F ′(X )∆ X s F − s+ − s − s s 1 + 2 + F ′′(X )(E[(∆ X ) ]+ V (s+) V (s)). −2 s s − + 2 Note that the subexponential growth condition (2.3) ensures that each Js belongs to (LX ). In order to compute the S-transform of J +, we note that, for every h H , s ∈ X + + + S(F ′(Xs)∆ Xs)(h)= E[F ′(Xs + h(s))(∆ Xs + ∆ h(s))] (5.4) + + = E[F ′(Xs + h(s))Xs]E[Xs∆ Xs]/V (s)+ E[F ′(Xs + h(s))]∆ h(s) + + = E[F ′′(Xs + h(s))]E[Xs∆ Xs]+ E[F ′(Xs + h(s))]∆ h(s) + + = S(F ′′(Xs))(h)E[Xs∆ Xs]+ S(F ′(Xs))(h)∆ h(s). + Here, the first equality is due to (2.2), the second one follows by projecting ∆ Xs on yXs; y R , the third one by rewriting the expectation as integral with respect to the Gaussian density{ and by integration∈ } by parts. Note that this well-known identity can alternatively be derived from the relation between Wick product and Malliavin derivative, see Nualart [28, Proposition 1.3.4]. As 2E[X ∆+X ]+ E[(∆+X )2]+ V (s+) V +(s)= V (s+) V (s) = ∆+V (s), (5.5) s s s − − we obtain, for every h H ∈ X S(J +)(h) = S( ψ (V (s+) V +(s),X ))(h) (S F (X ))(h) (5.6) s F − s+ − s + 1 + (S F ′(X ))∆ h(s) (S F ′′(X ))∆ V (s). − s − 2 s + 2 We next show that the sum of Js converges absolutely in (LX ) as s runs through DX [0,T ). To this end, we decompose J + = J +,1 J +,2, where ∩ s s − s +,1 + + J := ψ (V (s+) V (s),X ) F (X ) F ′(X )∆ X , s F − s+ − s − s s +,2 1 + 2 + J := F ′′(X )(E[(∆ X ) ]+ V (s+) V (s)). s 2 s s − Taylor’s theorem yields for s D [0,T ) ∈ X ∩ +,1 Js + = F (Xs+ + V (s+) V (s)y) F (Xs) R − − Z p y2/2 + + e− F ′(Xs)(∆ Xs + V (s+) V (s)y) dy − − √2π 1 p + + 2 = (∆ Xs + V (s+) V (s)y) R − Z Z0 p y2/2 + e− (1 u)F ′′((1 u)Xs + u(Xs+ + V (s+) V (s)y))du dy. × − − − √2π p 1 We now define ǫ := ((4aλ)− 1)/2, where a is the constant in the subexponential growth condition, let − ǫ∗ := 1/ǫ, and abbreviate l(s,y,u) := (1 u)X + u(X + V (s+) V +(s)y). − s s+ − Then, by Jensen’s inequality, Fubini’s theorem, and H¨older’sp inequality, E[ J +,1 2]1/2 | s | 1/2 1 y2/2 + + 4 2 e− E[(∆ Xs + V (s+) V (s)y) F ′′(l(s,y,u)) ]du dy ≤ R 0 − | | √2π ! Z Z p C. Bender /Itˆo’s Formula for Gaussian Processes 16 1 1 y2/2 2(1+ǫ) 2(1+ǫ) e− E[ F ′′(l(s,y,u)) ]du dy ≤ R | | √2π Z Z0 ! 1 2 2(1+ǫ∗) ∗ y /2 + + 4(1+ǫ ) e− E[ ∆ Xs + V (s+) V (s)y ] dy × R | − | √2π ! Z p ∗ The second factor is the square of the L4(1+ǫ )-norm of a centered Gaussian random variable with variance + 2 + E[(∆ X ) ]+ V (s+) V (s). Hence, there is a constant d ∗ such that s − ǫ 2 1 ∗ y /2 ∗ + + 4(1+ǫ ) e− 2(1+ǫ ) E ∆ Xs + V (s+) V (s)y dy R | − | √2π Z h + 2 p + i d ∗ (E[(∆ X ) ]+ V (s+) V (s)). ≤ ǫ s − The first factor is bounded by a constant Kǫ independent of s by the subexponential growth condition. Indeed, by convexity, 1 y2/2 2(1+ǫ) e− E[ F ′′(l(s,y,u)) ]du dy R | | √2π Z Z0 y2/2 2(1+ǫ) 2(1+ǫ)a l(s,y,u) 2 e− C max E e | | dy ≤ R u 0,1 √2π Z ∈{ } h i 2(1+ǫ) 2(1+ǫ)a X 2 C E e | s| ≤ h i y2/2 + 2 e 2(1+ǫ)a Xs++√V (s+) V (s)y − + E e | − | dy R √2π Z h i z2/2 2 2 e C2(1+ǫ) sup e2(1+ǫ)aV (t)z + e2(1+ǫ)aV (t+)z − dz ≤ t [0,T ) R √2π ∈ Z =: K2(1+ǫ) < . ǫ ∞ Thus, +,1 2 1/2 + 2 + E[ J ] K d ∗ (E[(∆ X ) ]+ V (s+) V (s)). (5.7) | s | ≤ ǫ ǫ s − Moreover, we clearly observe that +,2 2 1/2 1 1/2 + 2 + E[ Js ] sup E[ F ′′(Xt) ] (E[(∆ Xs) ]+ V (s+) V (s)). (5.8) | | ≤ 2 t [0,T ] | | − ∈ ! + We can now deduce from (5.7), (5.8), and (5.1) that the sum s D [0,T ) Js converges absolutely in ∈ X ∩ (L2 ). In particular, X P ∞ + + S J (h)= S(J )(h), h W ∗ , s s ∈ A 2 s D [0,T ) s D [0,T ) ∈ X∩ ∈ X∩ Let us summarize the foregoing: The sum + + ψ (V (s+) V (s),X ) F (X ) F ′(X )∆ X (5.9) F − s+ − s − s s s D [0,T ) ∈ X∩ 1 + 2 + F ′′(X )(E[(∆ X ) ]+ V (s+) V (s)) −2 s s − 2 converges absolutely in (LX ) and, due to (5.6), its S-transform is given by S( ψ (V (s+) V +(s),X ))(h) (S F (X ))(h) (5.10) F − s+ − s s D [0,T ) ∈ X∩ + 1 + (S F ′(X ))∆ h(s) (S F ′′(X ))∆ V (s) − s − 2 s C. Bender /Itˆo’s Formula for Gaussian Processes 17 ∗ for h W2 . The jumps from the left can be treated in the same way. The only difference is that we apply∈ A 2 2E[X ∆−X ] E[(∆−X ) ] (V (s ) V −(s)) = V (s) V (s ) = ∆−V (s) s s − s − − − − − instead of (5.5), which explains the change of sign in front of the second derivative term compared to the corresponding term resulting in the case of the jumps from the right. We finally obtain that F (Xs) ψF (V (s ) V −(s),Xs ) F ′(Xs)∆−Xs (5.11) − − − − − s D (0,T ] ∈ X∩ 1 2 + F ′′(X )(E[(∆−X ) ]+ V (s ) V −(s)) 2 s s − − 2 converges absolutely in (LX ) and its S-transform is given by (S F (Xs))(h) S( ψF (V (s ) V −(s),Xs ))(h) (5.12) − − − − s D (0,T ] ∈ X∩ 1 (S F ′(X ))∆−h(s) (S F ′′(X ))∆−V (s) − s − 2 s ∗ for h W2 . We∈ next A discuss the integral with respect to the variance V . The subexponential growth condition (2.3) again ensures that T 2 E[ F ′′(X ) ]d V (s) < , | s | | | ∞ Z0 where V denotes the total variation of V . Thus, by Fubini’s theorem and H¨older’s inequality, | | T 2 F ′′(X )dV (s) (L ). s ∈ X Z0 Another application of Fubini’s theorem then yields T T S F ′′(Xs)dV (s) (h)= (S F ′′(Xs))(h) dV (s), (5.13) Z0 ! Z0 ∗ for every h W2 . We can finally∈ A combine (5.9)–(5.13) with Proposition 5.2 in order to show that the S-transform of 1 T F (X ) F (X ) F ′′(X )dV (s) T − 0 − 2 s Z0 F (Xs) ψF (V (s ) V −(s),Xs ) F ′(Xs)∆−Xs − − − − − − s D (0,T ] ∈ X∩ 1 2 + F ′′(X )(E[(∆−X ) ]+ V (s ) V −(s)) 2 s s − − + + ψ (V (s+) V (s),X ) F (X ) F ′(X )∆ X − F − s+ − s − s s s D [0,T ) ∈ X∩ 1 + 2 + F ′′(X )(E[(∆ X ) ]+ V (s+) V (s)) −2 s s − at every h W ∗ is given by the Henstock-Kurzweil integral ∈ A 2 T (S F ′(Xs))(h) dh(s). Z0 T Hence, the Wick-Skorokhod integral 0 F ′(Xs)d⋄Xs exists and the asserted Itˆoformula is valid. We close this section with a simplifiedR version of the Itˆoformula in Theorem 5.1 as announced in (1.2). To this end, we assume that X is stochastically RCLL, i.e. for every t [0,T ) and every sequence (t ) ∈ n converging to t from the right, Xtn converges to Xt in probability, and, moreover: For every t (0,T ] 2 ∈ there is a random variable Xt (LX ) such that for every sequence (tn) converging to t from the left, − ∈ 2 Xtn converges to Xt in probability. By Gaussianity, both limits also hold strongly in (LX ). In particular, − X is weakly regulated with Xt+ = Xt. C. Bender /Itˆo’s Formula for Gaussian Processes 18 Corollary 5.3. Suppose the centered Gaussian process X satisfies the following assumptions: X is stochastically RCLL, W ∗ is dense in HX , the variance function V of X is of bounded variation, and A 2 2 E[(∆−X ) ] < . s ∞ s D ∈XX 2 Assume F (R) and F, F ′, F ′′ satisfy the growth condition (2.3) with λ = supt [0,T ] V (t). Then, ∈ C ∈ T 2 F ′(Xs )d⋄Xs exists and the following Itˆoformula holds in (LX ): 0+ − R T T 1 c F (XT ) = F (X0)+ F ′(Xs )d⋄Xs + F ′′(Xs )dV (s) − 2 − Z0+ Z0+ + F (Xs) F (Xs ) F ′(Xs )∆−Xs − − − − s D (0,T ] ∈ X∩ +F ′′(Xs )E[Xs (∆−Xs)] . − − c Here, V denotes the continuous part of V , the set DX of stochastic discontinuities is at most countable 2 and the sum converges absolutely in (LX ). Note that E[Xs (∆−Xs)] = 0, if X is martingale. Hence, in view of Theorem 4.7, Corollary 5.3 contains the classical− Itˆoformula (1.3) for Gaussian martingales as special case. (Recall that the pathwise jumps of a Gaussian martingale occur only at the deterministic times of stochastic discontinuities). Proof of Corollary 5.3. As X is stochastically right-continuous, the rational span of X ; t ([0,T ) Q) T { t ∈ ∩ ∪{ }} is dense in HX . Hence, condition (H) holds for X. Moreover, V ±(t) = V (t ), because X is stochas- + ± tically RCLL, as already observed at the end of Example 2.4. Finally, ∆ Xt = 0 for every t [0,T ]. Consequently, Theorem 5.1 is applicable and simplifies in the following way: ∈ F (X ) F (X ) T − 0 T 1 T = F ′(X )d⋄X + F ′′(X )dV (s) s s 2 s Z0 Z0 1 2 + F (Xs) F (Xs ) F ′(Xs)∆−Xs + F ′′(Xs)E[(∆−Xs) ]. − − − 2 s D (0,T ] ∈ X∩ As V is right-continuous, µV has no atom at 0. Hence, 1 T 1 T F ′′(X )dV (s)= F ′′(X )dV (s) 2 s 2 s Z0 Z0+ T 1 c 1 = F ′′(Xs )dV (s)+ F ′′(Xs)∆−V (s). 2 0+ − 2 Z s DX (0,T ] ∈ X∩ 1 2 Noting that 2 (E[(∆−Xs) ] + ∆−V (s)) = E[Xs∆−Xs], we obtain T 1 c F (XT ) F (X0) F ′′(Xs )dV (s) − − 2 − Z0+ F (Xs) F (Xs ) F ′(Xs )∆−Xs − − − − − s D (0,T ] ∈ X∩ +F ′′(Xs )E[Xs (∆−Xs)] − − T = F ′(Xs)d⋄Xs (F ′(Xs) F ′(Xs ))∆−Xs 0 − − − Z s DX (0,T ] ∈ X∩ (F ′′(Xs)E[Xs∆−Xs] F ′′(Xs )E[Xs ∆−Xs]) , − − − − C. Bender /Itˆo’s Formula for Gaussian Processes 19 where, by the local Lipschitz continuity of F ′ and the growth condition on F ′′, the sum on the right-hand 2 side can be seen to converge absolutely in (LX ). We now compute the S-transform of the right-hand side at h W ∗ . Applying the right-continuity of h and the analogue of (5.4), we observe that it is given by ∈ A 2 T (S F ′(Xs))(h) dh(s) S(F ′(Xs) F ′(Xs ))(h)∆−h(s) 0+ − − − Z s DX (0,T ] ∈ X∩ T = (S F ′(Xs ))(h) dh(s). − Z0+ T Again, we may conclude that the Wick-Skorokhod integral 0+ F ′(Xs )d⋄Xs exists and that the asserted Itˆoformula holds. − R 6. Subordinated fractional Brownian motion H H In this section, we consider processes of the form Xt = BAt where B is a fractional Brownian motion with Hurst parameter H (0, 1) and A is a L´evy subordinator independent of BH , see [34], pp. 137–142, for background information∈ on L´evy subordinators. The resulting process X has stationary increments and features jumps (unless the subordinator is deterministic, which we rule out for the remainder of the H section). If the subordinator A is a Gamma process, then BAt is known as fractional Laplace motion and has been applied, e.g., as a model for hydraulic conductivity [25] and sediment transport [12]. Subordi- nated fractional Brownian motion has also been proposed as a risk process in actuarial mathematics, see [8]. H Apparently, subordinated fractional Brownian motion Xt = BAt itself is not a Gaussian process, but X becomes Gaussian after conditioning on the path of the subordinator. This fact can be used to extend our integration theory to subordinated fractional Brownian motion and to derive an Itˆoformula for this class of processes. Of course, this reasoning applies to a much wider class of conditionally Gaussian processes. Exploiting the independence of BH and A, the obvious generalization of the Wick-Skorokhod integral is the following: We first condition on the paths of the subordinator A, then integrate in the sense of Gaus- sian Wick-Skorokhod integration, and finally evaluate the integral along the paths of the subordinator. This formally leads to the definition T T a a z(s,X) d⋄Xs = z(s,X ) d⋄Xs 0 0 Z Z ! a=A | a H where Xs := Ba(s) for a deterministic nondecreasing function a with right-continuous paths. In order to define the integral rigorously, we denote by D([0,T ]) the space of RCLL real-valued functions on [0,T ] and equip it with the Borel-σ-field (generated by the Skorokhod topology). For every a H RCLL nondecreasing function a : [0,T ] R, we consider the Gaussian process Xt = Ba(t), t [0,T ]. Since, → ∈ 1 Cov(Xa,Xa)= (a(t)2H + a(s)2H a(t) a(s) 2H ) (6.1) t s 2 − | − | (cp. Example 3.1), we observe that the covariance function is, for fixed s, a bounded variation function in t. In particular, each of the processes Xa satisfies condition (H). Definition 6.1. Suppose that Zt = z(t,X) for a measurable function z : [0,T ] D([0,T ]) R and A is T ×a a → a set of RCLL nondecreasing functions such that P ( A A )=1. If z(s,X ) d⋄X exists for every { ∈ } 0 s A D D R T a a a and there is a measurable function Iz : ([0,T ]) ([0,T ]) R such that 0 z(s,X ) d⋄Xs = I ∈(Xa,a) for every a A, then we define × → z ∈ R T Zs d⋄Xs := Iz(X, A). Z0 A routine application of Fubini’s theorem shows that the integral is (in case of existence) uniquely defined up to modification. C. Bender /Itˆo’s Formula for Gaussian Processes 20 We can now state an Itˆoformula for subordinated fractional Brownian motion in our framework. In the statement of the formula, we apply that any L´evy subordinator can be represented as At = σt + (∆As) 0 1 2H 2 Theorem 6.2. Suppose H 1/2 or 0+ x ν(dx) < . If F (R) and F , F ′, and F ′′ are bounded, T ≥ ∞ ∈C then F ′(Xs )d⋄Xs exists and the following Itˆoformula holds P -almost surely: 0+ − R R T T 2H 1 F (XT ) = F (0) + F ′(Xs )d⋄Xs + σH F ′′(Xs )As − ds − − Z0+ Z0+ + F (Xs) F (Xs ) F ′(Xs )∆Xs − − − − s (0,T ] ∈X 1 2H 2H 2H + F ′′(Xs )(As As (∆As) ) . 2 − − − − Here, the sum runs pathwise over the set of jump times of A (which includes the jump times of X) and is absolutely convergent P -almost surely. Morever, σ is the constant slope of the continuous part of A, T 2H 1 and the term σH F ′′(Xs )As − ds is to be read as zero, if σ =0. 0+ − In the BrownianR motion case H =1/2, the processes of the form X = B1/2 are L´evy processes and, t At in particular, include the class of CGMY-processes which were introduced in financial modeling in [6] and are among the most popular stock price models with jumps. As expected, our Itˆoformula takes the same form as the classical one for L´evy processes, if H = 1/2. We again emphasize that the covariance function of fractional Brownian motion is visible in the additional second derivative term of the jump component. Proof. We first define A as the set of nondecreasing RCLL functions on [0,T ] of the form a(t) = σt + 2H 0 1 2H 2H E (∆As) 1 0<∆A 1 = T x ν(dx) { s≤ } 0 F (x(s)) F (x(s )) F ′(x(s ))∆x(s) − − − − − s (0,T ] ∈X 1 2H 2H 2H + F ′′(x(s ))(a(s) a(s ) (∆a(s)) ) . 2 − − − − ! C. Bender /Itˆo’s Formula for Gaussian Processes 21 Note that the assumptions on F ensure that the sum over the jumps and the integral (if σ> 0) converge. a We emphasize that P ( X A′ ) = 1 for every a A, and P ( X A′ ) = 1. The first identity holds, because { ∈ } ∈ { ∈ } a 2 H H 2 2H E (∆Xs ) = E[ Ba(s) Ba(s ) ]= (∆a(s)) < . | − − | ∞ s (0,T ] s; ∆a(s)=0 s (0,T ] ∈X X 6 ∈X The second identity is a consequence of Fubini’s theorem and the first one, since a P ( X A′ )= P ( X A′ )PA(da)=1. { ∈ } A { ∈ } Z It, thus, remains to show that, for every a A, ∈ T T a a a a 2H 1 F ′(Xs ) d⋄Xs = F (XT ) F (0) σH F ′′(Xs )a(s) − ds − − − − Z0 Z0+ a a a a F (Xs ) F (Xs ) F ′(Xs )∆Xs − − − − − s (0,T ] ∈X 1 a 2H 2H 2H + F ′′(Xs )(a(s) a(s ) (∆a(s)) ) . 2 − − − − This equation can easily be reduced to Corollary 5.3. To this end we note, that by continuity of BH , a D a = s; ∆a(s) =0 s; ∆X =0 . Hence, the (pathwise) sum over s (0,T ] can be replaced by X { 6 }⊃{ s 6 } ∈ the sum over s D a (0,T ]. Moreover, by (6.1), ∈ X ∩ a a a a 1 2H 2H 2H E[Xs (∆−Xs )] = E[Xs (∆Xs )] = (a(s) a(s ) (∆a(s)) ), − − 2 − − − and so the sum over the jumps coincides with the corresponding sum in Corollary 5.3. We finally note a 2H that the variance of X is given by Va(t) = a(t) . Hence, by the chain rule in Theorem A.1, the continuous part of Va is zero, if σ = 0, and otherwise equals t c 2H 1 Va (t)=2Hσ a(s) − ds. Z0 Hence, T T a 2H 1 1 a c σH F ′′(Xs )a(s) − ds = F ′′(Xs )dVa (s), − 2 − Z0+ Z0+ which finishes the reduction to Corollary 5.3 and the proof. Example 6.3. (i) One of the best studied subordinated fractional Brownian motion is fractional Laplace motion, where the subordinator is a Gamma process A, i.e. a pure jump L´evy process whose L´evy measure has the density 1 λx ρ(x)= 1(0, ](x)cx− e− ∞ with respect to the Lebesgue measure, for positive constants c and λ. The integrability condition on the L´evy measure imposed in Theorem 6.2 holds for every H (0, 1), and so our Itˆoformula applies to the full range of Hurst parameters. Fractional Laplace motion∈ has heavier tails than a normal distribution and features a stochastic self-similarity property. It has long-range dependence if and only if H > 1/2, and its distribution is infinite divisible, if and only if H 1/2. For a proof of these properties we refer to [16]. ≥ (ii) If A is an inverse Gaussian subordinator, i.e. a pure jump L´evy process whose L´evy measure has the density 3/2 λx ρ(x)= 1(0, ](x)cx− e− ∞ H with respect to the Lebesgue measure (c,λ > 0), then Xt = BAt is known as fractional inverse Gaussian process. It can be obtained as scaling limit of continuous time random walks with correlated jumps [19], has long-range dependence for H > 1/2 [20], and fails to have an infinite divisible distribution for H < 1/2 [35]. The integrability condition on the L´evy measure imposed in Theorem 6.2 holds for H > 1/4, and so our Itˆoformula applies to this range of Hurst parameters. C. Bender /Itˆo’s Formula for Gaussian Processes 22 7. Regularity in the Cameron-Martin space In this section, we study the regularity of the elements of the Cameron-Martin space of X as required in condition (H). We provide sufficient conditions in terms of the mixed planar variation of the co- variance function, the quadratic variation of X, and finally consider the case, when X has an integral representation with respect to a Gaussian martingale. The first criterion is formulated in terms of the mixed planar variation and is taken from Friz et al. [11]. Theorem 7.1. Suppose that the covariance function of X is of finite (1,p)-planar variation for some p 1 in the following sense: There is a constant K > 0 such that for every partition π = 0= t 2 2 (∆Xs) = (∆Xs) s (0,t] s (0,t] D ∈X ∈ X∩ X 2 and that this sum converges absolutely in (LX ), because E[ (∆X )2 2]1/2 = √3 E[(∆X )2] < . | s | s ∞ s (0,t] D s (0,t] D ∈ X∩ X ∈ X∩ X 2 2 As ∆h(s) = E[(exp⋄(h) 1)(∆X ) ] due to (2.2), we, thus, obtain, for every h CM , | | − s ∈ X σ (h)= ∆h(s) 2 = ∆h(s) 2 < . 2 | | | | ∞ s (0,T ] s (0,T ] D ∈X ∈ X∩ X Let (πk) be a sequence of partitions of [0,T ], whose mesh size tends to zero. Then, by Theorem 3.50 in [15] (X X )2 (∆X )2 v(T ) ti − ti−1 − s → t π 0 s D (0,T ] i∈Xk\{ } ∈ X∩ C. Bender /Itˆo’s Formula for Gaussian Processes 23 2 in (LX ) as k goes to infinity, which in turn implies, 2 2 E[(exp⋄(h) 1)(X X ) ] E[(exp⋄(h) 1)(∆X ) ] 0. − ti − ti−1 − − s → t π 0 s D (0,T ] i∈Xk\{ } ∈ X∩ By (2.2), the left-hand side equals, 2 2 (h(ti) h(ti 1)) ∆h(s) − − − | | t π 0 s D (0,T ] i∈Xk\{ } ∈ X∩ Thus, 2 2 lim (h(ti) h(ti 1)) = ∆h(s) = σ2(h). k − − | | →∞ t π 0 s (0,T ] i∈Xk\{ } ∈X This convergence along the mesh size clearly implies convergence along the direction of refinement of partitions, as required in the definition of W2∗. We finally assume that (Mt)t R is a a two-sided RCLL Gaussian martingale with variance function ∈ VM and X has an integral representation of the form + ∞ Xt = gt(s)dMs, (7.1) Z−∞ 2 where gt L (R, dVM ) for every t [0,T ] and the integral is understood in the Itˆosense. In particular, H is then∈ a closed subspace of ∈H = f(s)dM ; f L2(R, dV ) and we denote by π the X M { s ∈ M } HX orthogonal projection on HX . We also extend gt to t R by setting gt := gT for t>T and gt := g0 for t< 0. R ∈ 2 Proposition 7.3. Suppose A is a dense subset of L (R, dVM ). Then, the functions of the form [0,T ] R, t E Xt f(s)dMs , f A → 7→ R ∈ Z constitute a dense subset of the Cameron-Martin space CMX of X. Proof. For f A let f := πH ( f(s)dMs). Then, E[X f(s)dMs]= E[X f] is indeed an element of the ∈ X · · Cameron-Martin space of X. However, f(s)dMs; f A is dense in HM by the Itˆoisometry, which implies that π ( f(s)dM );Rf A is{ dense in H .∈ ThisR } completes the proof. { HX s ∈ } R X The following theoremR provides a simple sufficient condition for X to be W2∗-dense in terms of the integral kernel gt in the representation (7.1) and the adjoint of the densely defined linear operator K : L2(R, dV ) L2(R, dV ), 1 g g . M → M (a,b] 7→ b − a Theorem 7.4. X is W2∗-dense, if the adjoint operator K∗ of K is densely defined, i.e.: The set D(K∗), 2 2 consisting of those f L (R, dV ) such that there is an K∗f L (R, dV ) satisfying ∈ M ∈ M t (gt(s) g0(s))f(s)dVM (s)= K∗f(s)dVM (s) R − Z Z0 for every t [0,T ], is dense in L2(R, dV ). ∈ M Note that by Theorem VII.2.3 in [36] the sufficient condition in Theorem 7.4 is equivalent to the closability of the operator K. Proof. By the previous proposition, the functions of the form t t K∗f(s)dVM (s)+ g0(s)f(s)dVM (s) 7→ R Z0 Z are then dense in CMX , because by Itˆo’s isometry, E Xt f(s)dMs = gt(s)f(s)dVM (s). R R Z Z These functions are clearly of bounded variation, and, hence, members of W2∗. C. Bender /Itˆo’s Formula for Gaussian Processes 24 8. Comparison to the literature We finally explain how condition (H) can be verified in the literature on Itˆo’s formula for Gaussian processes in the Skorokhod sense. In all the cases discussed below, the authors assume or show that the variance function V of X is of bounded variation and continuous: 1. Al`os et al. [1]: The authors assume an integral representation of the form (7.1) with respect to a Brownian motion M = W (on the interval [0,T ]). In the more general ‘singular’ case [1, Theorem 1], condition (K3) entails that t E[X T f(s)dW ] is continuous and of bounded variation for 7→ t 0 s every step function f. Hence, by Proposition 7.3, X is W2∗-dense and, moreover, X is stochastically continuous [15, Theorem 8.21]. In particular,R condition (H) is satisfied. 2. Mocioalca and Viens [26]: Again an integral representation of the form (7.1) with respect to a Brownian motion M = W on an interval is supposed. Proposition 15 in [26] shows that the assumptions of Theorem 7.4 are satisfied. Hence, a dense subset of the Cameron-Martin space of X is absolutely continuous with respect to the Lebesgue measure with square integrable density. This again implies that (H) holds and X is stochastically continuous. 3. Nualart and Taqqu [29]: In this reference X is supposed to have continuous paths. The key as- sumption is that X has zero planar quadratic variation along sufficiently uniform partitions. Itˆo’s formula can be recovered by our techniques under their assumptions, if the notions such as W2∗- regularity are also formulated in terms of these sufficiently uniform partitions only. However, in our general framework of Gaussian processes with stochastic discontinuities it does not seem to be possible to restrict to sufficiently uniform partitions as suggested in [29] in the continuous case. 4. Kruk et al. [17]: The authors assume that X is stochastically continuous and has a covariance measure. By their Remark 3.1 the latter property is equivalent to the property that the covariance function of X has finite planar (1,1)-variation. By Theorem 7.1, X is W2∗-regular and, in particular, satisfies (H). 5. Nualart and Taqqu [30]: Here the authors assume that X has continuous paths and satisfies the quadratic variation property in our Theorem 7.2. Hence, X is stochastically continuous, W2∗-regular, and satisfies (H). 6. Lei and Nualart [24] and Hu et al. [13]: The assumptions in these two references include that, for every s [0,T ], the covariance function t R(t,s) of X is absolutely continuous with respect to the Lebesgue∈ measure. Thus, X is stochastically7→ continuous and satisfies (H) by Remark 4.9 (ii). 7. Alpay and Kipnis [2]: The authors consider a class of Gaussian stationary increment processes and show in their equation (5.1) that the S-transform t (SX )(h) of X is absolutely continuous with 7→ t respect to the Lebesgue measure for a dense subspace of HX . Hence, by (2.1), X is W2∗-dense and satisfies (H), because it also is stochastically continuous. 8. Lebovits [22]: X is defined in terms of an integral representation of the form (7.1) with respect to a two-sided Brownian motion M = W . The conditions on the kernel g are such that t t 7→ E[Xt R fdW ] is absolutely continuous with respect to the Lebesgue measure for every Schwartz function f, see Remark 1 in [22]. As the Schwartz functions are dense in L2(R, dt), Proposition 7.3 againR implies that X is stochastically continuous and that (H) is satisfied. We emphasize again that in all these references, the Gaussian process X is stochastically continuous and, hence, Itˆo’s formula has the form (1.1) without jump terms, while the main contribution of the present paper is to understand the influence of the stochastic discontinuities on the Gaussian Itˆoformula. Appendix A: A chain rule for the Henstock-Kurzweil integral In this appendix, we prove a chain rule for the Henstock-Kurzweil integral. The general lines of proof closely follow the arguments in Norvaiˇsa [27]. We have, however, to deal with a case of ‘mixed’ regularity which is not covered there. Here is what we are going to show. Theorem A.1. Suppose u W ∗([0,T ]), u W ([0,T ]) (i.e, of bounded variation), and let u := 1 ∈ 2 2 ∈ 1 (u1,u2). Denote Si := inf ui(t), sup ui(t) , i =1, 2. t [0,T ] t [0,T ] " ∈ ∈ # Suppose G 1(S S ; R) and such that there is a constant K 0 and a continuous function ∈ C 1 × 2 1 ≥ C. Bender /Itˆo’s Formula for Gaussian Processes 25 K : S1 S2 S2 R 0 satisfying K(x1, x2, x2)=0 and × × → ≥ ∂G ∂G (x , x ) (y , x ) K x y (A.1) ∂x 1 2 − ∂x 1 2 ≤ 1| 1 − 1| 1 1 ∂G ∂G (x , x ) (x ,y ) K(x , x ,y ) x y 1/2 ∂x 1 2 − ∂x 1 2 ≤ 1 2 2 | 2 − 2| 1 1 T for every x ,y S and x ,y S . Then, ∂G (u(s)) du (s) exists as Henstock-Kurzweil integral 1 1 1 2 2 2 0 ∂x1 1 and ∈ ∈ R G(u(T )) G(u(0)) − T ∂G T ∂G = (u(s)) du (s)+ (u(s)) du (s) ∂x 1 ∂x 2 Z0 1 Z0 2 ∂G ∂G + G(u(s)) G(u(s )) (u(s))∆−u1(s) (u(s))∆−u2(s) − − − ∂x1 − ∂x2 s (0,T ] ∈X ∂G + ∂G + + G(u(s+)) G(u(s)) (u(s))∆ u1(s) (u(s))∆ u2(s). − − ∂x1 − ∂x2 s [0,T ) ∈X Here, both sums converge absoulutely. Note first that the integral with respect to u2 exists as Lebesgue-Stieltjes integral, because u2 is of bounded variation and the integrand is regulated. The chain rule above, thus, holds as a consequence of Theorem 4.2 (with α = 1) in Norvaiˇsa [27] under the stronger Lipschitz condition ∂G ∂G ∂G ∂G (x , x ) (y ,y ) + (x , x ) (y ,y ) ∂x 1 2 − ∂x 1 2 ∂x 1 2 − ∂x 1 2 1 1 2 2 K 1( x1 y1 + x2 y2 ), ≤ | − | | − | which is not satisfied in our application of this chain rule in Section 5. As in Norvaiˇsa [27], we shall show that the chain rule is valid in the sense of Young-Stieltjes integration and then exploit the relationship between the Young-Stieltjes integral and the Henstock-Kurzweil integral as stated in Dudley and Norvaiˇsa [10, Part I, Theorem F.2]. We, thus, first recall the construction of the Young-Stieltjes integral. A Young-tagged partition τ := ((si 1,si),yi); i = 1,...,n of the interval [0,T ], by definition, { − } satisfies 0 = s0 < s1 < ...sn = T and yi (si 1,si), whereas in a tagged partition the tag point ∈ − yi lies in the closed interval [si 1,si]. Given such a Young-tagged partition τ and regulated functions u, r : [0,T ] R, one considers the− Young-Stieltjes sums → n + SYS(u, r, τ) := u(si 1)∆ r(si 1)+ u(yi)(r(si ) r(si 1+)) + u(si)∆−r(si) − − − − − i=1 X The Young-Stieltjes integral of u with respect to r is said to exist, if there is a real number I such that, for every ǫ> 0, there is a Young-tagged partition χ such that for every refinement τ of χ S (u, r, τ) I < ǫ. | YS − | In this case, I is defined to be the value of this integral. In the above, a Young-tagged partition τ = ((si 1,si),yi); i = 1,...,n is said to be a refinement of a Young-tagged partition χ, if the partition {κ(τ)− := s ,...,s is a refinement} of the partition κ(χ), i.e. κ(τ) κ(χ). { 0 n} ⊃ Proof of Theorem A.1. Define S := S S , S′ := S S S . We consider for every Young-tagged 1 × 2 1 × 2 × 2 partition τ := ((si 1,si),yi); i =1,...,n { − } n 1 − ∂G V +(τ) = G(u(s +)) G(u(s )) (u(s ))∆+u (s ) i − i − ∂x i 1 i i=0 1 X ∂G + (u(si))∆ u2(si) −∂x2 C. Bender /Itˆo’s Formula for Gaussian Processes 26 n ∂G V −(τ) = G(u(s )) G(u(s )) (u(s ))∆−u (s ) i − i− − ∂x i 1 i i=1 1 X ∂G (u(si))∆−u2(si) −∂x2 n ∂G R(τ) = G(u(si )) G(u(si 1+)) (u(yi)) − − − − ∂x i=1 1 X ∂G (u1(si ) u1(si 1+)) (u(yi))(u2(si ) u2(si 1+)). × − − − − ∂x2 − − − Then, ∂G ∂G S (u),u , τ = G(u(T )) G(u(0)) S (u),u , τ (A.2) YS ∂x 1 − − YS ∂x 2 1 2 + V (τ) V −(τ) R(τ). − − − ∂G We fix an arbitrary ǫ > 0. As (u) is regulated and u2 is of bounded variation, the Young-Stieltjes ∂x2 integral of ∂G u with respect to u exists and coincides with the Henstock-Kurzweil integral and the ∂x2 2 Lebesgue-Stieltjes◦ integral by Theorems I.4.2, I.F.2 and Corollary II.3.20 in Dudley and Norvaiˇsa [10]. Hence, there is a Young-tagged partition χ of [0,T ] such that for all refinements τ of χ ∂G T ∂G SYS (u),u2, τ (u(s)) du2(s) < ǫ/4. (A.3) ∂x2 − 0 ∂x2 Z We next treat the jumps from the right. Suppose s [0,T ) such that ∆+u (s) = 0 for l = 1 or l = 2. ∈ l 6 Note that the set of such time points s is at most countable, since u1 and u2 are regulated. By the mean value theorem there are θ [u (s) u (s+),u (s) u (s+)], l =1, 2 such that for θ = (θ ,θ ) l ∈ l ∧ l l ∨ l 1 2 ∂G + ∂G + G(u(s+)) G(u(s)) = (θ)∆ u1(s)+ (θ)∆ u2(s). − ∂x1 ∂x2 Define + ∂G + ∂G + Vs := G(u(s+)) G(u(s)) (u(s))∆ u1(s) (u(s))∆ u2(s). − − ∂x1 − ∂x2 ∂G Let K2 := max ′ K(x1, x2, x˜2) and K3 := 2 max (x1, x2) . Then, by (A.1) and (x1,x2,x˜2) S (x1,x2) S ∂x2 Young’s inequality, ∈ ∈ | | V + ∆+u (s) (K u (s) θ + K u (s) θ 1/2)+ K ∆+u (s) | s | ≤ | 1 | 1| 1 − 1| 2| 2 − 2| 3| 2 | (K + K /2) ∆+u (s) 2 + (K + K /2) ∆+u (s) . ≤ 1 2 | 1 | 3 2 | 2 | Hence, for some constant K˜ > 0 V + K˜ ∆+u (s) 2 + ∆+u (s) < , | s |≤ | 1 | | 2 | ∞ s [0,T ) s [0,T ) s [0,T ) ∈X ∈X ∈X because u1 W2∗ and u2 is of bounded variation. Thus, the sum over the jumps from the right in the asserted chain∈ rule converges absolutely. We can, then, find a finite subset µ [0,T ) such that ⊂ V + ǫ/4. | s |≤ s [0,T ) µ ∈ X \ By passing to a refinement, if necessary, we can assume without loss of generality, that µ κ(χ). Then, for every refinement τ of χ ⊂ + ∂G + V (τ) G(u(s+)) G(u(s)) (u(s))∆ u1(s) − − − ∂x1 s [0,T ) ∈X ∂G + (u(s))∆ u2(s) ǫ/4. (A.4) −∂x2 ≤ C. Bender /Itˆo’s Formula for Gaussian Processes 27 By the same argument, the sum over the jumps from the left converges absolutely and for every refinement τ of χ ∂G V −(τ) G(u(s)) G(u(s )) (u(s))∆−u1(s) − − − − ∂x1 s (0,T ] ∈X ∂G (u(s))∆−u2(s) ǫ/4. (A.5) −∂x2 ≤ It remains to treat the remainder term R(τ). Again, by the mean value theorem, there are θl,i [ul(si ) ∈ − ∧ ul(si 1+),ul(si ) ul(si 1+)], l =1, 2, i =1,...,n, such that for θi := (θ1,i,θ2,i), i =1,...,n, − − ∨ − n ∂G ∂G R(τ) = (θi) (u(yi)) (u1(si ) u1(si 1+)) ∂x − ∂x − − − i=1 1 1 Xn ∂G ∂G + (θi) (u(yi)) (u2(si ) u2(si 1+)) ∂x − ∂x − − − i=1 2 2 X =: (I) + (II). In order to estimate these two terms separately, we need some extra notation. We write Osc(u, E) := max sup ul(s) ul(t) ; s,t E) , E [0,T ], l=1,2 {| − | ∈ } ⊂ for the oscillation of u over the set E. We denote the total variation of u2 over [0,T ] by v1(u2) and the 2-variation of u1 over the open interval (si 1,si) by v2(u1, (si 1,si)), i.e. − − v2(u1, (si 1,si)) − m 2 := sup u1(tj ) u1(tj 1) ; m N, si 1 n (I) K1 θ1,i u1(yi) u1(si ) u1(si 1+) | | ≤ | − || − − − | i=1 Xn 1/2 + K(θ1,i,θ2,i,u2(yi)) θ2,i u2(yi) u1(si ) u1(si 1+) | − | | − − − | i=1 nX K1 + K2 2 K1 2 u1(si ) u1(si 1+) + θ1,i u1(yi) ≤ 2 | − − − | 2 | − | i=1 Xn 1 + K(θ ,θ ,u (y )) θ u (y ) 2 1,i 2,i 2 i | 2,i − 2 i | i=1 X n 2K1 + K2 v2(u1, (si 1,si)) ≤ 2 − i=1 X v (u ) + 1 2 sup K(x , x , x˜ ); 2 { 1 2 2 (x1, x2, x˜2) S′, x2 x˜2 max Osc(u, (si 1,si))) . ∈ | − |≤ j=1,...,n − } Moreover, ∂G ∂G (II) v1(u2) sup (z1) (z2) ; | | ≤ {|∂x2 − ∂x2 | z1,z2 S, z1 z2 max Osc(u, (si 1,si))) , ∈ | − |∞ ≤ j=1,...,n − } C. Bender /Itˆo’s Formula for Gaussian Processes 28 2 ∂G where denotes the maximum norm in R . By uniform continuity of on S and of K on S′, there ∂x2 is a δ >| ·0 |∞ such that ∂G ∂G ǫ max K(x , x , x˜ ), (x , x ) (˜x , x˜ ) 1 2 2 ∂x 1 2 − ∂x 1 2 ≤ 16v (u ) 2 2 1 2 for every (x1, x2), (˜x1, x˜2) S with (x1, x2) (˜x1, x˜2) δ. As u 1 and u2 are regulated, there is a partition λ = t ,...,t of∈ [0,T ] such| that − |∞ ≤ { 0 m} max Osc(u, (tj 1,tj )) δ. j=1,...,m − ≤ ˜ Finally, by the equivalent characterization of W2∗ in Lemma 4.1 of Norvaiˇsa [27], there is a partition λ of [0,T ] such that for every refinement x ,...,x of λ˜ { 1 k} k ǫ v2(u1, (xj 1, xj )) . − ≤ 8K +4K j=1 1 2 X We may and shall again assume that λ λ˜ κ(χ), by passing to a refinement of χ, if necessary. Then, for every refinement τ of χ, ∪ ⊂ ǫ R(τ) . (A.6) | |≤ 4 Gathering (A.2)–(A.6), we observe that T ∂G (u(s)) du (s) exists as Young-Stieltjes integral and satisfies 0 ∂x1 1 the asserted chain rule. 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