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 H be the Cameron–Martin space, and C_{0} 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 C_{0}^{*}, the dual space to C_{0};. 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 \mathbf{F} of all Fréchet differentiable real-valued functions on C_{0};; the codomain is. The Skorokhod integral \delta; is defined to be the adjoint of the Malliavin derivative: