  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.
