Wong-Zakai approximations for quasilinear systems of It\^o's type stochastic differential equations driven by fBm with $H>1/2$

TL;DR

This paper extends Wong-Zakai approximations to systems of Itô stochastic differential equations driven by fractional Brownian motion with H>1/2, using a truncated Cameron-Martin expansion to handle the covariance structure.

Contribution

It introduces a novel approximation method for fBm-driven systems, employing a truncated Cameron-Martin expansion, and establishes convergence to the true solution.

Findings

The approximating sequence's law solves a Fokker-Planck equation.

The sequence converges to the solution of the Itô equation.

The method handles the covariance structure of fBm effectively.

Abstract

In a recent article Lanconelli and Scorolli (2021) extended to the multidimensional case a Wong-Zakai-type approximation for It\^o stochastic differential equations proposed by \Oksendal and Hu (1996). The aim of the current paper is to extend the latter result to system of stochastic differential equations of It\^o type driven by fractional Brownian motion (fBm) like those considered by Hu (2018). The covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli and instead we employ a truncated Cameron-Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables 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\^o equation, as the number of terms in the expansion goes to infinite.