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 formula_1 be the Cameron–Martin space, and formula_2 denote classical Wiener space:
formula_3;
formula_4
By the Sobolev embedding theorem, formula_5. Let
formula_6
denote the inclusion map.
Suppose that formula_7 is Fréchet differentiable. Then the Fréchet derivative is a map
formula_8
i.e., for paths formula_9, formula_10 is an element of formula_11, the dual space to formula_12. Denote by formula_13 the continuous linear map formula_14 defined by
formula_15
sometimes known as the "H"-derivative. Now define formula_16 to be the adjoint of formula_17 in the sense that
formula_18
Then the Malliavin derivative formula_19 is defined by
formula_20
The domain of formula_19 is the set formula_22 of all Fréchet differentiable real-valued functions on formula_12; the codomain is formula_24.
The Skorokhod integral formula_25 is defined to be the adjoint of the Malliavin derivative:
formula_26
References.
 ;templatestyles src="Reflist/styles.css" / ;