Asymptotic stability of semi-Markov modulated jump diffusions

We consider the class of semi-Markov modulated jump diffusions (sMMJDs) whose operator turns out to be an integro-partial differential operator. We find conditions under which the solutions of this class of switching jump-diffusion processes are almost surely exponentially stable and moment exponentially stable. We also provide conditions that imply almost sure convergence of the trivial solution when the moment exponential stability of the trivial solution is guaranteed. We further investigate and determine the conditions under which the trivial solution of the sMMJD-perturbed nonlinear system of differential equations d X _t/dt = f (X _t) is almost surely exponentially stable. It is observed that for a one-dimensional state space, a linear unstable system of differential equations when stabilized just by the addition of the jump part of an sMMJD process does not get destabilized by any addition of a Brownian motion. However, in a state space of dimension at least two, we show that a corresponding nonlinear system of differential equations stabilized by jumps gets destabilized by addition of Brownian motion.