Malliavin derivative

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.^{[citation needed]}

Definition

Let H be the Cameron-Martin space, and C₀ denote classical Wiener space:

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

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

By the Sobolev embedding theorem, $H \subset C_0$. Let

$i : H \to C_{0}$

denote 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

$\int_0^T \left(\partial_t \nabla_H F(\sigma)\right) \cdot \partial_t h := \langle \nabla_{H} F (\sigma), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (\sigma) (h) = \lim_{t \to 0} \frac{F (\sigma + t i(h)) - F(\sigma)}{t}$.

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:

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

See also

• H-derivative

References