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:
Read more about this topic: Malliavin Derivative
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