FENG, C., LIU, Y. and ZHAO, H., 2017. Numerical approximation of random periodic solutions of stochastic differential equations. Zeitschrift für Angewandte Mathematik und Physik, 68 (5), 119.
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 −∞ 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 moldi
ied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pullback 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 √∆t in the mean-square sense in Euler-
Maruyama method and ∆t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.
This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 International Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/
CF and HZ would like to acknowledge the financial support of Royal Society Newton Advanced Fellowship grant NA150344.