A projected Euler Method for Random Periodic Solutions of Semi-linear SDEs with non-globally Lipschitz coefficients

The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. Without relying on a priori high-order moment bounds of the numerical approximations, the mean square convergence rate of the approximation scheme is proved to be order $0.5$ for SDEs with multiplicative noise and order $1$ for SDEs with additive noise. Numerical examples are also provided to validate our theoretical findings.