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Intuitively speaking, the Malliavin derivative of a random variable F in Lp(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 variablesW(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 Lp(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 Lp(Ω); the domain of D is the closure of the smooth random variables in the seminorm :
For simplicity, consider now just the case p = 2. The Skorokhod integralδ is defined to be the L2-adjoint of the Malliavin derivative D. Just as D was not defined on the whole of L2(Ω), δ is not defined on the whole of L2(Ω; H): the domain of δ consists of those processes u in L2(Ω; H) for which there exists a constant C(u) such that, for all F in D1,2,
The Skorokhod integral of a process u in L2(Ω; H) is a real-valued random variable δu in L2(Ω); if u lies in the domain of δ, then δu is defined by the relation that, for all F ∈ D1,2,
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 Fj smooth and hj in H, then
The isometry property: for any process u in L2(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 DhX 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 D1,2 and a process u in dom(δ) is given by the formula
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