On higher order moments and rates of convergence for SDEs with switching

Second order recurrence are established for a $d$-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities, under suitable conditions. As a corollary, the rate of convergence towards the invariant regime of order $t^{-2}$ is claimed. The approach is based on embedded Markov chains and a priori bounds for the moments of the diffusion component at times of jumps of the discrete component, as well as on certain simple martingale properties.