# 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 be the Cameron–Martin space, and denote classical Wiener space:

;

By the Sobolev embedding theorem, . Let

denote the inclusion map.

Suppose that is Fréchet differentiable. Then the Fréchet derivative is a map

i.e., for paths , is an element of , the dual space to . Denote by the continuous linear map defined by

sometimes known as the H-derivative. Now define to be the adjoint of in the sense that

Then the Malliavin derivative is defined by

The domain of is the set of all Fréchet differentiable real-valued functions on ; the codomain is .

The Skorokhod integral is defined to be the adjoint of the Malliavin derivative: