Malliavin derivative

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In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense.

[edit] Definition

Let $i : H \to C_{0}$ denote classical Wiener space:

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

$C_{0} := C_{0} ([0, T]; \mathbb{R}^{n}) := \{ \mathrm{continuous\,paths\,starting\,at\,0} \}$ ;

$i : H \to C_{0}$ is the inclusion map.

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

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

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

$\mathrm{D}_{H} F (\sigma) := \mathrm{D} F (\sigma) \circ i,$

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

$\langle \nabla_{H} F (\sigma), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (\sigma) (h) = \lim_{t \to 0} \frac{F (\sigma + t i(h)) - F(x)}{t}$ .

Then the Malliavin derivative $D t$ is defined by

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

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

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

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