Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

Abstract

We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion are non-globally Lipschitz continuous. This stiffness may originate either from a linear operator in the drift, or from a perturbation of the nonlinear structures under discretisation, or both. Typical applications arise from the space discretisation of an SPDE, stochastic volatility models in finance, or certain ecological models. We prove that a timetepping strategy that adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is $(1-\varepsilon)/2$, for $\varepsilon\in(0,1)$, where $\varepsilon$ becomes arbitrarily small as the number of available finite moments for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift implicit method, three tamed type methods and a truncated method. Our numerical results show that the adaptive semi-implicit method is well suited as a general purpose solver, is more robust than the explicit time stepping methods and more efficient than the drift implicit method.