On higher order moments and recurrence of an SDE with switching

TL;DR

This paper investigates the second order recurrence properties of a multidimensional stochastic differential equation with switching regimes, using Markov chain embeddings and moment bounds to establish recurrence under certain conditions.

Contribution

It introduces a novel approach combining embedded Markov chains and martingale techniques to analyze recurrence of SDEs with switching regimes.

Findings

Established second order recurrence under specific conditions

Developed bounds for moments at jump times

Applied martingale properties to analyze switching SDEs

Abstract

Second order recurrence of a $d$-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities is established under suitable conditions. The approach is based on embedded Markov chains and a priori bounds for the moments of $X_t$ at moments of jumps of the discrete component, as well as on some simple martingale properties.