An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

Paolo Da Pelo, Alberto Lanconelli and Aurel I. Stan

Stochastic Processes and their Applications, 2013, vol. 123, issue 8, 3183-3200

Abstract: The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.

Keywords: Stochastic differential equations; Second quantization operator; Heat equation (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (4)

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DOI: 10.1016/j.spa.2013.03.005

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