Stability of jump diffusions with random switching

This paper summarizes what have been done in our recent paper [24], which is concerned with stability of a class of switching jump-diffusion processes. 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 system given by (X(t), α(t)) is the switching process α(t) depends on X(t). This paper focuses on the long-time behavior, namely, stability of the switching jump diffusions. First, the definitions of regularity and stability are recalled. It is then shown that under suitable conditions, the underlying systems are regular or 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 Liapunov 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.