A New Class of Exponential Integrators for Stochastic Differential Equations With Multiplicative Noise

TL;DR

This paper introduces novel exponential integrators for stochastic differential equations with multiplicative noise, leveraging exact solutions of geometric Brownian motion to improve convergence and efficiency over existing methods.

Contribution

The paper develops new exponential integrators for SDEs that utilize the exact solution of geometric Brownian motion, offering enhanced convergence rates and competitiveness for nonlinear noise.

Findings

Euler scheme achieves improved convergence rate for linear noise

Methods are competitive with existing integrators when using a homotopy parameter

Strong convergence is proven for the proposed schemes

Abstract

In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take the advantage of the exact solution of (generalised) geometric Brownian motion. We examine both Euler and Milstein versions of the scheme and prove strong convergence. For the special case of linear noise we obtain an improved rate of convergence for the Euler version over standard integration methods. We investigate the efficiency of the methods compared with other exponential integrators and show that by introducing a suitable homotopy parameter these schemes are competitive not only when the noise is linear but also in the presence of nonlinear noise terms.