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.
Read more about this topic: Integration By Parts Operator
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