Exponential ergodicity for SDEs driven by \(\alpha\)-stable processes with Markovian switching in Wasserstein distances
The object of study is an SDE with Markovian switching regime of the form \[dX_t=a(X_t,Y_t)dt+\sigma(X_t,Y_t)dZ_t,\] where \((Z_t)\) is a symmetric \(\alpha\)-stable Lévy process with \(\alpha\in(0,2)\) and \((Y_t)\) is a right-continuous Markov chain with finite state-space, independent of \((Z_t)\). The usual assumtions for existence and uniqueness of \((X_t)\) are supposed, i.e.,\ for fixed state the coefficient \(a\) is global Lipschitz continuous, \(\sigma\) is \(\gamma\)-Hölder continuous for \(\gamma\in\{\frac1{\alpha},1\}\) and the coefficients \(a,\sigma\) together fulfill a linear growth condition. The authors give sufficient conditions for exponential ergodicity in the Wasserstein distance of the Markovian semigroup \((P_t)\) of \((X_t,Y_t)\) to its stationary distribution using the theory of M-matrices, the coupling method and Lyapunov functionals.
scientific article; zbMATH DE number 6947312
scientific article