- 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.[citation needed]
Definition
Let H be the Cameron-Martin space, and C0 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 Dt is defined by
The domain of Dt 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
References
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