Abstract:

This paper investigates the numerical approximation of random periodic solutions for stochastic differential equations (SDEs)

whose drift coefficient is only $\alpha$-H\"older continuous ($\alpha>0$) and diffusion coefficient only $(\frac12+\beta)$-H\"older continuous ($\beta>0$) in time. To overcome the limited convergence order of traditional methods caused by low temporal regularity, we propose two randomized schemes: the randomized Euler method (REM) and the randomized exponential integrator (REI).

These schemes offer complementary advantages: REM is simple to implement and broadly applicable, while REI achieves higher accuracy with a convergence order independent of $\beta$. By incorporating uniformly distributed random variables to sample the drift at randomized intermediate points, the random schemes attain higher convergence orders under low regularity conditions. Theoretical analysis shows that the mean-square convergence order of REM is $\min\left( \frac{1}{2}+\alpha, \frac{1}{2}+\beta, 1 \right)$, while REI achieves the order $\min\left( \frac{1}{2} + \alpha, 1 \right)$. Furthermore, we establish the existence and uniqueness of random periodic solutions for both numerical schemes and demonstrate their mean-square convergence to the exact random periodic solution of the SDE at the aforementioned orders. Numerical experiments are conducted to validate the theoretical findings.

Key words: Random periodic solution, Stochastic differential equation, Randomized Euler scheme, Randomized exponential integrator, Mean-square convergence