A new class of exponential integrators for SDEs with multiplicative noise

In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take advantage of the exact solution of (generalized) geometric Brownian motion. We examine both Euler and Milstein versions of the scheme and prove strong convergence, taking care to deal with the dependence on the noise in the solution operator. 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 for low dimensional SDEs and high dimensional SDEs arising from the discretization of stochastic partial differential equations. We 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. Although our new schemes are derived and analysed under zero commutator conditions (1.2), our numerical investigations illustrate that the resulting methods rival traditional methods even when this does not hold.