Malliavin derivative

Let $H$ be the Cameron–Martin space, and $C\_\{0\}$ denote classical Wiener space:

:$H\; :=\; \backslash \{\; f\; \backslash in\; W^\{1,2\}\; ([0,\; T];\; \backslash mathbb\{R\}^\{n\})\; \backslash ;|\backslash ;\; f(0)\; =\; 0\; \backslash \}\; :=\; \backslash \{\; \backslash text\{paths\; starting\; at\; 0\; with\; first\; derivative\; in\; \}\; L^\{2\}\; \backslash \}$;

:$C\_\{0\}\; :=\; C\_\{0\}\; ([0,\; T];\; \backslash mathbb\{R\}^\{n\})\; :=\; \backslash \{\; \backslash text\{continuous\; paths\; starting\; at\; 0\}\; \backslash \};$

By the Sobolev embedding theorem, $H\; \backslash subset\; C\_0$. Let

:$i\; :\; H\; \backslash to\; C\_\{0\}$

denote the inclusion map.

Suppose that $F\; :\; C\_\{0\}\; \backslash to\; \backslash mathbb\{R\}$ is Fréchet differentiable. Then the Fréchet derivative is a map

:$\backslash mathrm\{D\}\; F\; :\; C\_\{0\}\; \backslash to\; \backslash mathrm\{Lin\}\; (C\_\{0\};\; \backslash mathbb\{R\});$

i.e., for paths $\backslash sigma\; \backslash in\; C\_\{0\}$, $\backslash mathrm\{D\}\; F\; (\backslash sigma)\backslash ;$ is an element of $C\_\{0\}^\{*\}$, the dual space to $C\_\{0\}\backslash ;$. Denote by $\backslash mathrm\{D\}\_\{H\}\; F(\backslash sigma)\backslash ;$ the continuous linear map $H\; \backslash to\; \backslash mathbb\{R\}$ defined by

:$\backslash mathrm\{D\}\_\{H\}\; F\; (\backslash sigma)\; :=\; \backslash mathrm\{D\}\; F\; (\backslash sigma)\; \backslash circ\; i\; :\; H\; \backslash to\; \backslash mathbb\{R\},$

sometimes known as the H-derivative. Now define $\backslash nabla\_\{H\}\; F\; :\; C\_\{0\}\; \backslash to\; H$ to be the adjoint of $\backslash mathrm\{D\}\_\{H\}\; F\backslash ;$ in the sense that

:$\backslash int\_0^T\; \backslash left(\backslash partial\_t\; \backslash nabla\_H\; F(\backslash sigma)\backslash right)\; \backslash cdot\; \backslash partial\_t\; h\; :=\; \backslash langle\; \backslash nabla\_\{H\}\; F\; (\backslash sigma),\; h\; \backslash rangle\_\{H\}\; =\; \backslash left(\; \backslash mathrm\{D\}\_\{H\}\; F\; \backslash right)\; (\backslash sigma)\; (h)\; =\; \backslash lim\_\{t\; \backslash to\; 0\}\; \backslash frac\{F\; (\backslash sigma\; +\; t\; i(h))\; -\; F(\backslash sigma)\}\{t\}.$

Then the Malliavin derivative $\backslash mathrm\{D\}\_\{t\}$ is defined by

:$\backslash left(\; \backslash mathrm\{D\}\_\{t\}\; F\; \backslash right)\; (\backslash sigma)\; :=\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash left(\; \backslash left(\; \backslash nabla\_\{H\}\; F\; \backslash right)\; (\backslash sigma)\; \backslash right).$

The domain of $\backslash mathrm\{D\}\_\{t\}$ is the set $\backslash mathbf\{F\}$ of all Fréchet differentiable real-valued functions on $C\_\{0\}\backslash ;$; the codomain is $L^\{2\}\; ([0,\; T];\; \backslash mathbb\{R\}^\{n\})$.

The Skorokhod integral $\backslash delta\backslash ;$ is defined to be the adjoint of the Malliavin derivative:

:$\backslash delta\; :=\; \backslash left(\; \backslash mathrm\{D\}\_\{t\}\; \backslash right)^\{*\}\; :\; \backslash operatorname\{image\}\; \backslash left(\; \backslash mathrm\{D\}\_\{t\}\; \backslash right)\; \backslash subseteq\; L^\{2\}\; ([0,\; T];\; \backslash mathbb\{R\}^\{n\})\; \backslash to\; \backslash mathbf\{F\}^\{*\}\; =\; \backslash mathrm\{Lin\}\; (\backslash mathbf\{F\};\; \backslash mathbb\{R\}).$

• H-derivative

{{reflist}}

Category:Generalizations of the derivative

Category:Stochastic calculus

Category:Malliavin calculus