Integration by parts operator explained

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} D \varphi(x) h(x) d \mu(x) = \int _{E} \varphi(x) (A h)(x) 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 ^{*} \xrightarrow{i ^{*} } H^ \cong H \xrightarrow E.

For h ∈ S, define Ah by

(A h)(x) = h(x) x - trace _{H} 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\{ \left. h \colon C _{0} \to L ^{2 , 1} _{0} \right| h isboundedandnon-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}} \varphi (x + λ h(x)) d \gamma(x) = \int _{C _{0}} \varphi(x) \exp \left( λ \int ^{1} _{0} • h _{s} ⋅ d x _{s} - (λ ^{2})/(2) \int ^{1} _{0} | • h _{s} | ^{2} d s \right) d \gamma(x).

Differentiating with respect to λ and setting λ = 0 gives

\int _{C _{0}} D \varphi(x) h(x) d \gamma(x) = \int _{C _{0}} \varphi(x) (A h) (x) d \gamma(x),

where (Ah)(x) is the Itō integral

\int ^{1} _{0} • h _{s} ⋅ 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

• Book: Bell , Denis R. . The Malliavin calculus. Dover Publications Inc.. Mineola, NY. 2006. x+113. 0-486-44994-7. (See section 5.3)
• Book: Elworthy , K. David . Gaussian measures on Banach spaces and manifolds. Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II. 151 - 166. Internat. Atomic Energy Agency. Vienna. 1974.