Exponential Euler method for stiff SDEs driven by fractional Brownian motion

TL;DR

This paper proves that the exponential Euler method for stiff stochastic differential equations driven by fractional Brownian motion achieves an optimal convergence order of one, improving upon previous results and utilizing Malliavin calculus techniques.

Contribution

It introduces a new proof of the exponential Euler scheme's convergence order of one for fractional Brownian motion-driven SDEs using Malliavin derivatives.

Findings

Convergence order of one achieved for the exponential Euler scheme.

Improved theoretical understanding of numerical methods for fractional SDEs.

Utilization of Malliavin calculus to establish optimal convergence rates.

Abstract

In a recent paper by Kamrani et al. (2024), exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise was discussed, and the convergence order close to the Hurst parameter H was proved. Utilizing the technique of Malliavin derivative, we prove the exponential Euler scheme and obtain a convergence order of one, which is the optimal rate in numerical simulation.