Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations

Abstract

In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and moldiflied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt{\Delta t}$ in the mean-square sense in Euler-Maruyama method and $\Delta t$ in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.

31 pages, 4 figures

References in corpus (6)

• Random Periodic Solutions of Random Dynamical Systems
• Pathwise Random Periodic Solutions of Stochastic Differential Equations
• Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding
• Random Periodic Processes, Periodic Measures and Ergodicity
• Taylor expansions of solutions of stochastic partial differential equations with additive noise
• A random dynamical systems perspective on stochastic resonance

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• Anticipating Random Periodic Solutions--II. SPDEs with Multiplicative Linear Noise
• Random periodic solutions of non-autonomous stochastic differential equations
• Weak Random Periodic Solutions of Random Dynamical Systems
• The random periodic solution of a stochastic differential equation with a monotone drift and its numerical approximation
• The Galerkin analysis for the random periodic solution of semilinear stochastic evolution equations