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This work is concerned with numerical approximation of hybrid diffusions with regime switching modulated by continuous-time finite-state Markov chains. When using the Euler-Maruyama approximation algorithms, an important question is: Suppose the measure of the regime-switching diffusions converges to its invariant measure, will the sequence of measures of the approximation scheme converges to the same limit? This paper provides answers to this question. By appropriate interpolations and weak convergence methods, it shows that a suitably interpolated sequence resulted from the algorithm converges to the switching diffusion. The convergence to the invariant measure of the numerical algorithm is also studied.