Contractivity of stochastic θ-methods under non-global Lipschitz conditions

Abstract

The paper is devoted to address the numerical preservation of the exponential mean-square contractive character of the dynamics of stochastic differential equations (SDEs), whose drift and diffusion coefficients are subject to non-global Lipschitz assumptions. The conservative attitude of stochastic θ-methods is analyzed both for Itô and Stratonovich SDEs. The case of systems with linear drift is also analyzed in terms of spectral properties of the coefficient matrix of the drift. Numerical evidence on selected test problems confirms the effectiveness of the approach.