Malliavin Derivative - Definition

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:

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