Skorokhod integral

In mathematics, the Skorokhod integral, also named Hitsuda–Skorokhod integral, often denoted ${\displaystyle \delta }$, is an

. Part of its importance is that it unifies several concepts:

• ${\displaystyle \delta }$ is an extension of the
  Itô integral to non-adapted processes
  ;
• ${\displaystyle \delta }$ is the
  adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus
  );
• ${\displaystyle \delta }$ is an infinite-dimensional generalization of the divergence operator from classical vector calculus.

The integral was introduced by Hitsuda in 1972^[1] and by Skorokhod in 1975.^[2]

Definition

Preliminaries: the Malliavin derivative

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

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

${\displaystyle W(h)}$, 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

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 "

" random variables. For a random variable ${\displaystyle F}$ of the form

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:

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 :

This space is denoted by ${\displaystyle \mathbf {D} ^{1,p}}$ and is called the Watanabe–Sobolev space.

The Skorokhod integral

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 ${\displaystyle \mathbf {D} ^{1,2}}$,

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 ${\displaystyle F\in \mathbf {D} ^{1,2}}$,

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

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

Properties

• The isometry property: for any process ${\displaystyle u}$ in ${\displaystyle \mathbf {D} ^{1,p}}$ that lies in the domain of ${\displaystyle \delta }$,
  $${\displaystyle \mathbf {E} {\big [}(\delta u)^{2}{\big ]}=\mathbf {E} \int |u_{t}|^{2}dt+\mathbf {E} \int D_{s}u_{t}\,D_{t}u_{s}\,ds\,dt.}$$

  
  If ${\displaystyle u}$ is an adapted process, then ${\displaystyle D_{s}u_{t}=0}$ for ${\displaystyle s>t}$, 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
  $${\displaystyle \mathrm {D} _{h}(\delta u)=\langle u,h\rangle _{H}+\delta (\mathrm {D} _{h}u),}$$

  
  where ${\displaystyle \mathrm {D} _{h}X}$ stands for ${\displaystyle (\mathrm {D} X)(h)}$, the random variable that is the value of the process ${\displaystyle \mathrm {D} X}$ at "time" ${\displaystyle h}$ in ${\displaystyle H}$.
• The Skorokhod integral of the product of a random variable ${\displaystyle F}$ in ${\displaystyle \mathbf {D} ^{1,2}}$ and a process ${\displaystyle u}$ in ${\displaystyle \operatorname {dom} (\delta )}$ is given by the formula
  $${\displaystyle \delta (Fu)=F\,\delta u-\langle \mathrm {D} F,u\rangle _{H}.}$$

  

Alternatives

An alternative to the Skorokhod integral is the Ogawa integral.

References

• "Skorokhod integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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.