Semi-implicit Taylor schemes for stiff rough differential equations

TL;DR

This paper introduces semi-implicit Taylor schemes tailored for solving multidimensional rough stochastic differential equations, especially effective for stiff equations driven by fractional Brownian motion, with proven convergence and practical advantages.

Contribution

The paper develops and analyzes new semi-implicit Taylor schemes for rough differential equations with unbounded drifts, extending numerical methods to a broader class of stochastic problems.

Findings

Schemes are well-posed and convergent.

Numerical experiments demonstrate effectiveness for stiff equations.

Methods outperform explicit schemes in stiff scenarios.

Abstract

We study a class of semi-implicit Taylor-type numerical methods that are easy to implement and designed to solve multidimensional stochastic differential equations driven by a general rough noise, e.g. a fractional Brownian motion. In the multiplicative noise case, the equation is understood as a rough differential equation in the sense of T.~Lyons. We focus on equations for which the drift coefficient may be unbounded and satisfies a one-sided Lipschitz condition only. We prove well-posedness of the methods, provide a full analysis, and deduce their convergence rate. Numerical experiments show that our schemes are particularly useful in the case of stiff rough stochastic differential equations driven by a fractional Brownian motion.