The aim of this paper is to generalize two important results known for the Stratonovich and Itˆo 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.

Da Pelo Paolo, Lanconelli Alberto, Stan Aurel I. (2013). An Ito formula for a family of stochastic integrals and related Wong-Zakai type theorems. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 123(8), 3183-3200 [10.1016/j.spa.2013.03.005].

An Ito formula for a family of stochastic integrals and related Wong-Zakai type theorems

Lanconelli Alberto
Investigation
;
2013

Abstract

The aim of this paper is to generalize two important results known for the Stratonovich and Itˆo 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.
2013
Da Pelo Paolo, Lanconelli Alberto, Stan Aurel I. (2013). An Ito formula for a family of stochastic integrals and related Wong-Zakai type theorems. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 123(8), 3183-3200 [10.1016/j.spa.2013.03.005].
Da Pelo Paolo; Lanconelli Alberto; Stan Aurel I.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/662415
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