Integration by parts operator

Template:Short description Template:No footnotes 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

${\int }_E\mathrm{D}\phi (x)h(x)\mathrm{d}\mu (x)={\int }_E\phi (x)(Ah)(x)\mathrm{d}\mu (x)$

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.

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

$E^{\ast }\stackrel{i^{\ast }}{\to }{H}^{\ast }\cong H\stackrel{i}{\to }E.$

For h ∈ S, define Ah by

$(Ah)(x)=h(x)x-{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}}_H\mathrm{D}h(x).$

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

$S=\left\{h:C_0\to L_0^{2,1}|h\text{ is bounded and non-anticipating}\right\},$

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

${\int }_{C_0}\phi (x+\lambda h(x))\mathrm{d}\gamma (x)={\int }_{C_0}\phi (x)\exp (\lambda {\int }_0^1{\dot{h}}_s\cdot \mathrm{d}{x}_s-\frac{{\lambda }^2}{2}{\int }_0^1|{\dot{h}}_s{|}^2\mathrm{d}s)\mathrm{d}\gamma (x).$

Differentiating with respect to λ and setting λ = 0 gives

${\int }_{C_0}\mathrm{D}\phi (x)h(x)\mathrm{d}\gamma (x)={\int }_{C_0}\phi (x)(Ah)(x)\mathrm{d}\gamma (x),$

where (Ah)(x) is the Itō integral

${\int }_0^1{\dot{h}}_s\cdot \mathrm{d}{x}_s.$

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

• Template:Cite book Template:MathSciNet (See section 5.3)
• Template:Cite book Template:MathSciNet