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  L2(E, μ; R) is said to be an integration by parts operator for μ if

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.

Examples

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).
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

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