Variable-Step Euler–Maruyama Approximations of Regime-Switching Jump Diffusion Processes

Let $$\left( X_{t},Z_{t}\right) _{t\ge 0}$$ X t , Z t t ≥ 0 be the regime-switching jump diffusion process with invariant measure $$\mu $$ μ . We aim to approximate $$\mu $$ μ using the Euler–Maruyama (EM) scheme with decreasing step sequence $$\Gamma =(\gamma _n)_{n\in {\mathbb {N}}}$$ Γ = ( γ n ) n ∈ N . 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 $$\mu $$ μ and the invariant measure associated with the EM scheme is bounded by $$O(\sqrt{\gamma _n})$$ O ( γ n ) . In particular, we derive a better convergence rate $$O(\gamma _n)$$ O ( γ n ) for the additive case and the continuous case.