Integration By Parts Operator - Examples

Examples

• 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 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.