In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies several concepts:
 δ is an extension of the Itô integral to nonadapted processes;
 δ is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus);
 δ is an infinitedimensional generalization of the divergence operator from classical vector calculus.
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Contents
Definition
Preliminaries: the Malliavin derivative
Consider a fixed probability space (Ω, Σ, P) and a Hilbert space H; E denotes expectation with respect to P
Intuitively speaking, the Malliavin derivative of a random variable F in L^{p}(Ω) is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of H and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.
Consider a family of Rvalued random variables W(h), indexed by the elements h of the Hilbert space H. Assume further that each W(h) is a Gaussian (normal) random variable, that the map taking h to W(h) is a linear map, and that the mean and covariance structure is given by
for all g and h in H. It can be shown that, given H, there always exists a probability space (Ω, Σ, P) and a family of random variables with the above properties. The Malliavin derivative is essentially defined by formally setting the derivative of the random variable W(h) to be h, and then extending this definition to “smooth enough” random variables. For a random variable F of the form
where f : R^{n} → R is smooth, the Malliavin derivative is defined using the earlier “formal definition” and the chain rule:
In other words, whereas F was a realvalued random variable, its derivative DF is an Hvalued random variable, an element of the space L^{p}(Ω;H). Of course, this procedure only defines DF for “smooth” random variables, but an approximation procedure can be employed to define DF for F in a large subspace of L^{p}(Ω); the domain of D is the closure of the smooth random variables in the seminorm :
This space is denoted by D^{1,p} and is called the Watanabe–Sobolev space.
The Skorokhod integral
For simplicity, consider now just the case p = 2. The Skorokhod integral δ is defined to be the L^{2}adjoint of the Malliavin derivative D. Just as D was not defined on the whole of L^{2}(Ω), δ is not defined on the whole of L^{2}(Ω; H): the domain of δ consists of those processes u in L^{2}(Ω; H) for which there exists a constant C(u) such that, for all F in D^{1,2},
The Skorokhod integral of a process u in L^{2}(Ω; H) is a realvalued random variable δu in L^{2}(Ω); if u lies in the domain of δ, then δu is defined by the relation that, for all F ∈ D^{1,2},
Just as the Malliavin derivative D was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for “simple processes”: if u is given by
with F_{j} smooth and h_{j} in H, then
Properties
 The isometry property: for any process u in L^{2}(Ω; H) that lies in the domain of δ,
 If u is an adapted process, then for s > t, so the second term on the righthand side vanishes. The Skorokhod and Itô integrals coincide in that case, and the above equation becomes the Itô isometry.
 The derivative of a Skorokhod integral is given by the formula
 where D_{h}X stands for (DX)(h), the random variable that is the value of the process DX at “time” h in H.
 The Skorokhod integral of the product of a random variable F in D^{1,2} and a process u in dom(δ) is given by the formula
References
 Hazewinkel, Michiel, ed. (2001) [1994], "Skorokhod integral", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Ocone, Daniel L. (1988). "A guide to the stochastic calculus of variations". Stochastic analysis and related topics (Silivri, 1986). Lecture Notes in Math. 1316. Berlin: Springer. pp. 1&ndash, 79. MR953793
 SanzSolé, Marta (2008). "Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)" (PDF). Retrieved 20080709.