Skorokhod integral

Consider a fixed probability space (Ω, Σ, P) and a Hilbert space H; E denotes expectation with respect to P

${\displaystyle \mathbf {E} [X]:=\int _{\Omega }X(\omega )\,\mathrm {d} \mathbf {P} (\omega ).}$

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

${\displaystyle \mathbf {E} [W(h)]=0,}$

${\displaystyle \mathbf {E} [W(g)W(h)]=\langle g,h\rangle _{H},}$

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

${\displaystyle F=f(W(h_{1}),\ldots ,W(h_{n})),}$

where f : Rⁿ → R is smooth, the Malliavin derivative is defined using the earlier “formal definition” and the chain rule:

${\displaystyle \mathrm {D} F:=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(W(h_{1}),\ldots ,W(h_{n}))h_{i}.}$

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 :

${\displaystyle \|F\|_{1,p}:={\big (}\mathbf {E} [|F|^{p}]+\mathbf {E} [\|\mathrm {D} F\|_{H}^{p}]{\big )}^{1/p}.}$

This space is denoted by D^1,p and is called the Watanabe–Sobolev space.

For simplicity, consider now just the case p = 2. The Skorokhod integral δ 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 D^1,2,

${\displaystyle {\big |}\mathbf {E} [\langle \mathrm {D} F,u\rangle _{H}]{\big |}\leq C(u)\|F\|_{L^{2}(\Omega )}.}$

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 ∈ D^1,2,

${\displaystyle \mathbf {E} [F\,\delta u]=\mathbf {E} [\langle \mathrm {D} F,u\rangle _{H}].}$

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

${\displaystyle u=\sum _{j=1}^{n}F_{j}h_{j}}$

with F_j smooth and h_j in H, then

${\displaystyle \delta u=\sum _{j=1}^{n}\left(F_{j}W(h_{j})-\langle \mathrm {D} F_{j},h_{j}\rangle _{H}\right).}$