Euler–Maruyama approximation for stable stochastic differential equations with Markovian switching and related asymptotics

Abstract We investigate the EM approximation for double struck upper R Superscript d R d $\mathbb{R}^d$ -valued ergodic stochastic differential equations (SDEs) driven by rotationally invariant alpha α $\alpha$ -stable processes ( alpha element of left parenthesis 1 comma 2 right parenthesis α ∈ ( 1 , 2 ) $\alpha\in(1,2)$ ) with Markovian switching. The coefficient g violates the dissipative condition for certain states of the switching process. Using the Lindeberg principle, we establish quantitative error bounds between the original process left parenthesis upper X Subscript t Baseline comma upper R Subscript t Baseline right parenthesis Subscript t greater than or slanted equals 0 ( X t , R t ) t ⩾ 0 $(X_t,R_t)_{t\geqslant 0}$ and its Euler–Maruyama (EM) scheme under a specially designed metric. Furthermore, we derive both a central limit theorem and a moderate derivation principle for the empirical measures of both the SDE and its EM scheme. The theoretical results are subsequently validated through a concrete example.