Preliminaries: the Malliavin derivative edit

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 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

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

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 edit

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

• 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
  $${\displaystyle \delta (Fu)=F\,\delta u-\langle \mathrm {D} F,u\rangle _{H}.}$$

   

An alternative to the Skorokhod integral is the Ogawa integral.

• "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.