Using Stein’s Method to Analyze Euler–Maruyama Approximations of Regime-Switching Jump Diffusion Processes

For a kind of regime-switching jump diffusion process $$(X_t,Z_t)_{t\ge 0}$$ ( X t , Z t ) t ≥ 0 , under some conditions, it is exponentially ergodic under the weighted total variation distance with ergodic measure $$\mu $$ μ . We use the Euler–Maruyama scheme of the process $$(X_t,Z_t)_{t\ge 0}$$ ( X t , Z t ) t ≥ 0 which has an ergodic measure $$\mu _{\eta }$$ μ η ( $$\eta $$ η is the step size of the Euler–Maruyama scheme) to approximate the ergodic measure $$\mu $$ μ . Furthermore, we use Stein’s method to prove that the convergence rate of $$\mu _{\eta }$$ μ η to $$\mu $$ μ is $$\eta ^{\frac{1}{2}}$$ η 1 2 in terms of some function-class distance $$d_{{\mathcal {G}}}(\mu ,\mu _{\eta })$$ d G ( μ , μ η ) .