A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation

We consider a continuous-time Ehrenfest model defined over the integers from $-N$ to $N$, and subject to catastrophes occurring at constant rate. The effect of each catastrophe istantaneously resets the process to state 0. We investigate both the transient and steady-state probabilities of the above model. Further, the first passage time through state 0 is discussed. We perform a jump-diffusion approximation of the above model, which leads to the Ornstein-Uhlenbeck process with catastrophes. The underlying jump-diffusion process is finally studied, with special attention to the symmetric case arising when the Ehrenfest model has equal upward and downward transition rates.