Adaptive time-stepping for the strong numerical solution of stochastic differential equations

Models based on stochastic differential equations are of high interest today due to their many important practical applications. Thus the need for efficient and accurate numerical methods to approximate their solution. In this paper, we propose several adaptive time-stepping strategies for the strong numerical solution of stochastic differential equations in Itô form, driven by multiple Wiener processes satisfying the commutativity condition. The adaptive schemes are based on I and PI control, and allow arbitrary values of the stepsize. The explicit Milstein method is applied to approximate the solution of the problem and the adaptive implementations are based on estimates of the local error obtained using Richardson extrapolation. Numerical tests on several models arising in applications show that our adaptive time-stepping schemes perform better than the fixed stepsize alternative and an adaptive Brownian tree time-stepping strategy.