A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes

We consider a continuous-time Markov chain {Ñ(t), t ⩾ 0} with state space S. A point process {X(t),t ⩾ 0}, using a function A:[0,∞)→S in the following way, influences the evolution of {Ñ(t), t ⩾ 0}: whenever an event of {X(t)} occurs at time t, the Markov chain of interest goes immediately at state A(t) and evolves according to the dynamics of {Ñ(t)} until the next event of {X(t)} and so on. We study the transient and the limiting distribution of the resulting process and prove some stochastic comparison facts. Several examples that demonstrate the applicability of the model are also included.