Skorokhod integral

Consider a fixed probability space ${\displaystyle (\Omega ,\Sigma ,\mathbf {P} )}$ and a Hilbert space ${\displaystyle H}$; ${\displaystyle \mathbf {E} }$ denotes expectation with respect to ${\displaystyle \mathbf {P} }$

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

Intuitively speaking, the Malliavin derivative of a random variable ${\displaystyle F}$ in ${\displaystyle L^{p}(\Omega )}$ is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of ${\displaystyle H}$ and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.

Consider a family of ${\displaystyle \mathbb {R} }$-valued random variables ${\displaystyle W(h)}$, indexed by the elements ${\displaystyle h}$ of the Hilbert space ${\displaystyle H}$. Assume further that each ${\displaystyle W(h)}$ is a Gaussian (normal) random variable, that the map taking ${\displaystyle h}$ to ${\displaystyle W(h)}$ 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 ${\displaystyle g}$ and ${\displaystyle h}$ in ${\displaystyle H}$. It can be shown that, given ${\displaystyle H}$, there always exists a probability space ${\displaystyle (\Omega ,\Sigma ,\mathbf {P} )}$ 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 ${\displaystyle W(h)}$ to be ${\displaystyle h}$, and then extending this definition to "smooth enough" random variables. For a random variable ${\displaystyle F}$ of the form

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

where ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ 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 ${\displaystyle F}$ was a real-valued random variable, its derivative ${\displaystyle \mathrm {D} F}$ is an ${\displaystyle H}$-valued random variable, an element of the space ${\displaystyle L^{p}(\Omega ;H)}$. Of course, this procedure only defines ${\displaystyle \mathrm {D} F}$ for "smooth" random variables, but an approximation procedure can be employed to define ${\displaystyle \mathrm {D} F}$ for ${\displaystyle F}$ in a large subspace of ${\displaystyle L^{p}(\Omega )}$; the domain of ${\displaystyle \mathrm {D} }$ 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 ${\displaystyle \mathbf {D} ^{1,p}}$ and is called the Watanabe–Sobolev space.

For simplicity, consider now just the case ${\displaystyle p=2}$. The Skorokhod integral ${\displaystyle \delta }$ is defined to be the ${\displaystyle L^{2}}$-adjoint of the Malliavin derivative ${\displaystyle \mathrm {D} }$. Just as ${\displaystyle \mathrm {D} }$ was not defined on the whole of ${\displaystyle L^{2}(\Omega )}$, ${\displaystyle \delta }$ is not defined on the whole of ${\displaystyle L^{2}(\Omega ;H)}$: the domain of ${\displaystyle \delta }$ consists of those processes ${\displaystyle u}$ in ${\displaystyle L^{2}(\Omega ;H)}$ for which there exists a constant ${\displaystyle C(u)}$ such that, for all ${\displaystyle F}$ in ${\displaystyle \mathbf {D} ^{1,2}}$,

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

The Skorokhod integral of a process ${\displaystyle u}$ in ${\displaystyle L^{2}(\Omega ;H)}$ is a real-valued random variable ${\displaystyle \delta u}$ in ${\displaystyle L^{2}(\Omega )}$; if ${\displaystyle u}$ lies in the domain of ${\displaystyle \delta }$, then ${\displaystyle \delta u}$ is defined by the relation that, for all ${\displaystyle F\in \mathbf {D} ^{1,2}}$,

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

Just as the Malliavin derivative ${\displaystyle \mathrm {D} }$ was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if ${\displaystyle u}$ is given by

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

with ${\displaystyle F_{j}}$ smooth and ${\displaystyle h_{j}}$ in ${\displaystyle H}$, then

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