Integration by parts operator

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.

Definition

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 Ã¢Â†Â’ L²(E, μ; R) is said to be an integration by parts operator for μ if

for every C¹ function φ : E Ã¢Â†Â’ R and all h Ã¢ÂˆÂˆ S for which either side of the above equality makes sense. In the above, Dφ(x) denotes the Fréchet derivative of φ at x.

Examples

• Consider an abstract Wiener space i : H Ã¢Â†Â’ 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 Dφ 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.

References

• (See section 5.3)