Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative $α$-stable processes: Existence, uniqueness and numerical approximation

In this paper, we study the periodic measures within the context of time-inhomogeneous Markov processes. By extending the Hairer--Mattingly method to the time-inhomogeneous case, we first establish the existence, uniqueness and exponential ergodicity of periodic measures for general time-periodic Markov processes. These results are subsequently applied to time-inhomogeneous stochastic differential equations under the assumption that the drift is weakly dissipative and locally Lipschitz. We further employ the truncated Euler--Maruyama scheme to study numerical approximations of the periodic measures and derive the convergence rate between the exact and discrete periodic measures. These numerical results are new even in the time-homogeneous case under the assumptions considered in this paper.