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 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}, DF(σ) is an element of C_{0}^{*}, the dual space to C0. Denote by DHF 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 DHF 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 Dt 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 Dt is the set \mathbf{F} of all Fréchet differentiable real-valued functions on C0; the codomain is L^{2} ([0, T]; \mathbb{R}^{n}).

The Skorokhod integral δ 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}).
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