Long-term adaptive symplectic numerical integration of linear stochastic oscillators driven by additive white noise

Abstract

In this paper, we present an adaptive variable step size numerical scheme for the integration of linear stochastic oscillator equations driven by additive Brownian white noise. We first show that traditional adaptive schemes based on local error estimation destroy the long-time behavior of the underlying method. As a remedy, we extend the idea presented in Hairer and Soderlind (SIAM J. Sci. Comput. 26(6), 1838–1851 2005) to the stochastic setting and show that using step density control mechanisms based on time regularization and local error tracking, we are able to obtain numerical schemes which preserve the important qualitative features of the solution process such as symmetry, time reversibility, symplecticity, linear growth rate of the second moment, and infinite oscillation. Numerical experiments confirm the theoretical findings of the paper.