Stability of hybrid Lévy systems

Continuous-time stochastic systems have attracted a lot of attention recently, due to their wide-spread use in finance for modelling price-dynamics. More recently models taking into accounts shocks have been developed by assuming that the return process is an infinitesimal Lévy process. Lévy processes are also used to model the traffic in a telecommunication network. In this paper we focus on a particular technical problem: stability of time-varying stochastic systems driven or modulated by a Lévy process with discrete time interventions, such as parameter or state resetting. Such systems will be called hybrid Lévy systems. They are hybrid in the sense that jumps both in the dynamics may occur. The peculiarity of our systems is that the jump-times are defined by a more or less arbitrary point process, but there exists an asymmetry in the system dynamics. The novelty of our model relative to the theory of switching stochastic systems is two-fold. First, we allow slow time variation of the parameters, in a stochastic sense, without any statistical pattern, in the spirit of the classical stability result of Desoer, see [2]. Secondly, we allow certain jumps (resetting) in the system parameters almost without any a priori condition.