Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

TL;DR

This paper introduces an exponential Euler numerical scheme for stiff stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, demonstrating strong convergence and stability.

Contribution

It presents a new exponential Euler method for such equations and proves its strong convergence rate and stability properties.

Findings

Converges with a rate close to the Hurst parameter H

Ensures existence of a unique stationary solution

Demonstrates pathwise asymptotic stability

Abstract

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H>1/2, which arise e. g., from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on (E. Buckwar, M.G. Riedler, and P.E. Kloeden 2011), we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.