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'' → ''L''²(''E'', ''μ''; R) is said to be an integration by parts operator for ''μ'' if

:$\int_ \mathrm \varphi(x) h(x) \, \mathrm \mu(x) = \int_ \varphi(x) (A h)(x) \, \mathrm \mu(x)$

for every ''C''¹ function ''φ'' : ''E'' → R and all ''h'' ∈ ''S'' for which either side of the above equality makes sense. In the above, D''φ''(''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 ''C''¹ functions from ''E'' into ''E''^∗; ''E''^∗ can be thought of as a subspace of ''E'' in view of the inclusions

::$E^ \xrightarrow H^ \cong H \xrightarrow E.$

:For ''h'' ∈ ''S'', define ''Ah'' by

::$(A h)(x) = h(x) x - \mathrm_ \mathrm h(x).$

: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 ''C''₀ of continuous paths in R^''n'' starting at zero and defined on the unit interval , 1has another integration by parts operator. Let ''S'' be the collection

::$S = \left\,$

:i.e., all bounded, adapted processes with absolutely continuous sample paths. Let ''φ'' : ''C''₀ → R be any ''C''¹ function such that both ''φ'' and D''φ'' are bounded. For ''h'' ∈ ''S'' and ''λ'' ∈ R, the Girsanov theorem implies that

::$\int_ \varphi (x + \lambda h(x)) \, \mathrm \gamma(x) = \int_ \varphi(x) \exp \left( \lambda \int_^ \dot_ \cdot \mathrm x_ - \frac \int_^ , \dot_ , ^ \, \mathrm s \right) \, \mathrm \gamma(x).$

:Differentiating with respect to ''λ'' and setting ''λ'' = 0 gives

::$\int_ \mathrm \varphi(x) h(x) \, \mathrm \gamma(x) = \int_ \varphi(x) (A h) (x) \, \mathrm \gamma(x),$

:where (''Ah'')(''x'') is the Itō integral

::$\int_^ \dot_ \cdot \mathrm x_.$

: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

* (See section 5.3)
* {{MathSciNet, id=0464297

Integral calculus
Measure theory
Operator theory
Stochastic calculus