Skorokhod integral

In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of . It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies several concepts:

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 : Rⁿ → 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 DF is an H-valued 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 :

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