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]

Definition

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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:

 

See also

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References

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