Stability of Regime-Switching Jump Diffusions

This work is concerned with the stability of a class of switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a random switching device. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. A distinct feature of the two-component process $(X(t),\alpha(t))$ considered in this paper is that the switching process $\alpha(t)$ depends on the $X(t)$ process. This paper focuses on the long-time behavior, namely, stability of the switching jump diffusions. First, the definitions of regularity and stability are recalled. Next it is shown that under suitable conditions, the underlying systems are regular or have no finite explosion time. To study stability of the trivial solution (or the equilibrium point 0), systems that are linearizable (in the x variable) in a neighborhood of 0 are considered. Sufficient conditions for stability and instability are obtained. Then, almost sure stability is examined by treating a Lyapunov exponent. The stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients. To close the gap, a transformation technique is used to obtain a necessary and sufficient condition for stability.