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] Let

denote classical Wiener space: By the Sobolev embedding theorem,

Let denote the inclusion map.

is Fréchet differentiable.

Then the Fréchet derivative is a map i.e., for paths

, the dual space to

the continuous linear map

defined by sometimes known as the H-derivative.

in the sense that Then the Malliavin derivative

of all Fréchet differentiable real-valued functions on

The Skorokhod integral

is defined to be the adjoint of the Malliavin derivative: