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

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.