Stability of nonlinear regime-switching jump diffusion

Motivated by networked systems, stochastic control, optimization, and a wide variety of applications, this work is devoted to systems of switching jump diffusions. Treating such nonlinear systems, we focus on stability issues. One of the distinct features considered here is that the switching process depends on the jump diffusions. First asymptotic stability in the large is obtained. Then the study on exponential p$p$-stability is carried out. Connection between almost surely exponential stability and exponential p$p$-stability is exploited. Also presented are smooth-dependence on the initial data. Using the smooth-dependence, necessary conditions for exponential p$p$-stability are derived. Then criteria for asymptotic stability in distribution are provided. A couple of examples are given to illustrate our results.