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:
