Weak Error Estimates of Ergodic Approximations for Monotone Jump-diffusion SODEs
Provides theoretical guarantees for numerical approximation of invariant measures in monotone jump-diffusion SODEs, addressing a specific open problem for the backward Euler method.
The paper proves exponential ergodicity for the stochastic theta method (θ∈(1/2,1]) for monotone jump-diffusion SODEs and establishes weak error estimates for the backward Euler method, achieving a one-order convergence rate between exact and numerical invariant measures in the jump-free case, resolving an open question.
We first derive the exponential ergodicity of the stochastic theta method (STM) with $θ\in (1/2,1]$ for monotone jump-diffusion stochastic ordinary differential equations (SODEs) under a dissipative condition. Then we establish the weak error estimates of the backward Euler method (BEM), corresponding to the STM with $θ=1$. In particular, the time-independent estimate for the BEM in the jump-free case yields a one-order convergence rate between the exact and numerical invariant measures, answering a question left in {\it Z. Liu and Z. Liu, J. Sci. Comput. (2025) 103:87}.