Numerical Schemes for Rough Parabolic Equations

TL;DR

This paper develops numerical approximation schemes for nonlinear rough parabolic equations driven by rough signals, extending previous work to cover cases including multidimensional fractional Brownian motion with Hurst index greater than 1/3.

Contribution

It introduces a novel numerical scheme combining rough paths theory with stochastic PDE discretization techniques for rough parabolic equations.

Findings

Scheme effectively approximates solutions driven by rough signals.

Applicable to equations with multidimensional fractional Brownian motion.

Extends previous theoretical results to practical numerical methods.

Abstract

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H \textgreater{} 1/3.