Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations
Provides theoretical error bounds for numerical approximation of random periodic solutions, relevant for researchers in stochastic dynamics.
The paper proves existence of random periodic solutions for SDEs with multiplicative noise and shows that Euler-Maruyama and Milstein schemes approximate them with mean-square errors of √Δt and Δt, respectively.
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 $\sqrt {Δt}$ in the mean-square sense in Euler-Maruyama method and $Δt$ in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.