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.

Definition

Let [math]\displaystyle{ H }[/math] be the Cameron–Martin space, and [math]\displaystyle{ C_{0} }[/math] denote classical Wiener space:

[math]\displaystyle{ 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} \} }[/math];

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

By the Sobolev embedding theorem, [math]\displaystyle{ H \subset C_0 }[/math]. Let

[math]\displaystyle{ i : H \to C_{0} }[/math]

denote the inclusion map.

Suppose that [math]\displaystyle{ F : C_{0} \to \mathbb{R} }[/math] is Fréchet differentiable. Then the Fréchet derivative is a map

[math]\displaystyle{ \mathrm{D} F : C_{0} \to \mathrm{Lin} (C_{0}; \mathbb{R}); }[/math]

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

[math]\displaystyle{ \mathrm{D}_{H} F (\sigma) := \mathrm{D} F (\sigma) \circ i : H \to \mathbb{R}, }[/math]

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

[math]\displaystyle{ \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}. }[/math]

Then the Malliavin derivative [math]\displaystyle{ \mathrm{D}_{t} }[/math] is defined by

[math]\displaystyle{ \left( \mathrm{D}_{t} F \right) (\sigma) := \frac{\partial}{\partial t} \left( \left( \nabla_{H} F \right) (\sigma) \right). }[/math]

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

The Skorokhod integral [math]\displaystyle{ \delta\; }[/math] is defined to be the adjoint of the Malliavin derivative:

[math]\displaystyle{ \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}). }[/math]

See also

• H-derivative