Wong-Zakai approximations for quasilinear systems of Ito's type stochastic differential equations

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:141

Article

Lanconelli, Alberto¹  Scorolli, Ramiro¹ 

[1] Univ Bologna, Dipartimento Sci Stat Paolo Fortunati, Bologna, Italy

关键词: Stochastic differential equations;    Wong-Zakai approximation;    Wick product;    Fokker-Planck equation;   

DOI  :  10.1016/j.spa.2021.07.007

来源: Elsevier

【 摘 要 】

We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Oksendal (1996) for scalar quasi-linear Ito stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Ito's SDEs, we consider for any finite partition of the time interval [0, T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong-Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker-Planck equation and that the sequence converges to the solution of the Ito equation, as the mesh of the partition tends to zero. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

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