In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications.
Let E be a Banach space such that both E and its continuous dual space E∗ are separable spaces; let μ be a Borel measure on E. Let S be any (fixed) subset of the class of functions defined on E. A linear operator A : S → L2(E, μ; R) is said to be an integration by parts operator for μ if
∫
E
D
φ
(
x
)
h
(
x
)
d
μ
(
x
)
=
∫
E
φ
(
x
)
(
A
h
)
(
x
)
d
μ
(
x
)
for every C1 function φ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ(x) denotes the Fréchet derivative of φ at x.
Consider an abstract Wiener space i : H → E with abstract Wiener measure γ. Take S to be the set of all C1 functions from E into E∗; E∗ can be thought of as a subspace of E in view of the inclusions
For
h ∈
S, define
Ah by
This operator
A is an integration by parts operator, also known as the divergence operator; a proof can be found in Elworthy (1974).
The classical Wiener space C0 of continuous paths in Rn starting at zero and defined on the unit interval [0, 1] has another integration by parts operator. Let S be the collection
i.e., all bounded, adapted processes with absolutely continuous sample paths. Let
φ :
C0 →
R be any
C1 function such that both
φ and D
φ are bounded. For
h ∈
S and
λ ∈
R, the Girsanov theorem implies that
Differentiating with respect to
λ and setting
λ = 0 gives
where (
Ah)(
x) is the Itō integral
The same relation holds for more general
φ by an approximation argument; thus, the Itō integral is an integration by parts operator and can be seen as an infinite-dimensional divergence operator. This is the same result as the integration by parts formula derived from the Clark-Ocone theorem.
