Euler-Maruyama's approximations of regime-switching jump diffusion processes

Let$(X_t, Z_t)_{t\ge0}$be the regime-switching jump diffusion process with invariant measure$\mu$, and we aim to approximate$\mu$using the well-known Euler-Maruyama (EM) scheme with constant step$\gamma$and decreasing step$\gamma_k$, respectively. Under some appropriate dissipative conditions and uniform ellipticity assumptions on the coefficients of the related stochastic differential equation (SDE), we show that the error between$\mu$and the invariant measure associated with the EM scheme is bounded by$O(\gamma^{1/2} )$(in the case of constant step) and $O(\gamma_k^{1/2} )$(in the case of variable step). In particular, we derive a faster convergence rate for the additive model and the continuous model. For the constant step approximation we use the Stein's method, while for the variable step we mainly rely on the method recently developed in Pag\`{e}s and Panloup (2020). This talk is based on a recent work together with Chen, Jin and Shen.