# Skorokhod integral

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 non-adapted processes;*δ*is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus);*δ*is an infinite-dimensional generalization of the divergence operator from classical vector calculus.

## 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 **R**-valued 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 real-valued random variable, its derivative D*F* is an *H*-valued random variable, an element of the space *L*^{p}(Ω;*H*). Of course, this procedure only defines D*F* for “smooth” random variables, but an approximation procedure can be employed to define D*F* 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 real-valued 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 right-hand 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 (D*X*)(*h*), the random variable that is the value of the process D*X*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 978-1-55608-010-4 - 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. 1–79. MR953793 - Sanz-Solé, Marta (2008). "Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)" (PDF). Retrieved 2008-07-09.