Approximations of random periodic solutions for path-dependent stochastic differential equations with finite/infinite delay

The aim of this work is to show the numerical approximation of random periodic solutions for path-dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous-time Euler-Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path-dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean-square sense of a numerical approximation to path-dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path-dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure.