- Integration by parts operator
mathematics, an integration by parts operator is a linear operatorused to formulate integration by partsformulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysisand its applications.
Let "E" be a
Banach spacesuch that both "E" and its continuous dual space"E"∗ are separable spaces; let "μ" be a Borel measureon "E". Let "S" be any (fixed) subsetof 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
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 derivativeof "φ" at "x".
* 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
:For "h" ∈ "S", define "Ah" by
:This operator "A" is an integration by parts operator, also known as the
divergenceoperator; a proof can be found in Elworthy (1974).
:i.e., all bounded, adapted processes with
absolutely continuoussample 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 theoremimplies that
:Differentiating with respect to "λ" and setting "λ" = 0 gives
:where ("Ah")("x") is the
: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.
* cite book
last = Bell
first = Denis R.
title = The Malliavin calculus
publisher = Dover Publications Inc.
location = Mineola, NY
year = 2006
pages = pp. x+113
isbn = 0-486-44994-7 MathSciNet|id=2250060 (See section 5.3)
* cite book
last = Elworthy
first = K. David
chapter = Gaussian measures on Banach spaces and manifolds
title = Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol. II
pages = 151–166
publisher = Internat. Atomic Energy Agency
address = Vienna
year = 1974 MathSciNet|id=0464297
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