Wong-Zakai approximations for quasilinear systems of It\^o's type stochastic differential equations

TL;DR

This paper extends Wong-Zakai approximations to multidimensional quasilinear Itô SDE systems, demonstrating convergence of the approximations to the true solution via PDE analysis and distributional methods.

Contribution

It introduces a multidimensional Wong-Zakai approximation framework for quasilinear Itô SDE systems, utilizing hyperbolic PDEs and Wick products, extending prior scalar results.

Findings

The approximations converge to the Itô SDE solution as partition mesh tends to zero.

The law of the approximations solves a Fokker-Planck equation in distribution.

The approach generalizes scalar results to multidimensional systems.

Abstract

We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and {\O}ksendal in [7] for scalar quasi-linear It\^o stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of It\^o'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 lead 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 It\^o equation, as the mesh of the partition tends to zero.