The Euler-Maruyama Approximation of State-Dependent Regime Switching Diffusions

In this paper, we consider the state-dependent regime switching diffusion process (X(t),R(t))t >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X(t), R(t))_{t \geqslant 0}$$\end{document}, where the drift term does not necessarily satisfy the dissipative condition for certain states of the switching component. We develop delicately the Lindeberg replacement trick and a change-of-measure technique to obtain the convergence rate between the law of (X(t),R(t))t >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X(t), R(t))_{t\geqslant 0}$$\end{document} and that of its Euler-Maruyama scheme with constant and decreasing step sizes. This convergence rate is quantified in terms of a function-class distance dG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\mathcal {G}}$$\end{document}. Moreover, we establish the ergodicity property of the Euler-Maruyama scheme. To illustrate our theoretical findings, we present in detail an example.