Malliavin derivative
Encyclopedia
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.
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:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Malliavin derivative is a notion of derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
in the Malliavin calculus
Malliavin calculus
The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....
. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space
Classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain , taking values in a metric space . Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions...
, 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,
. Letdenote the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
.
Suppose that
is Fréchet differentiableFréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
. Then the Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
is a map
;i.e., for paths
, is an element of , the dual spaceDual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
to
. Denote by the continuousContinuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
linear map
defined bysometimes known as the H-derivative. Now define
to be the adjointAdjoint
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...
of
in the sense that.Then the Malliavin derivative
is defined byThe domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
of
is the set of all Fréchet differentiable real-valued functions on ; the codomainCodomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
is
.The Skorokhod integral
is defined to be the adjointAdjoint
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...
of the Malliavin derivative: