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Some properties of the Itô–Wiener expansion of the solution of a stochastic differential equation and local times

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  • Rudenko, Alexey
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    Abstract

    In this paper, we use the formula for the Itô–Wiener expansion of the solution of the stochastic differential equation proven by Krylov and Veretennikov to obtain several results concerning some properties of this expansion. Our main goal is to study the Itô–Wiener expansion of the local time at the fixed point for the solution of the stochastic differential equation in the multidimensional case (when standard local time does not exist even for Brownian motion). We show that under some conditions the renormalized local time exists in the functional space defined by the L2-norm of the action of some smoothing operator.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 6 ()
    Pages: 2454-2479

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:6:p:2454-2479

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    Keywords: Itô–Wiener expansion; Stochastic differential equation; Local time; Renormalized local time; Second quantization operator;

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    1. Imkeller, Peter & Perez-Abreu, Victor & Vives, Josep, 1995. "Chaos expansions of double intersection local time of Brownian motion in and renormalization," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 1-34, March.
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