Study of stochastic systems with catastrophe and its jump-diffusion approximation

Abstract

Many discrete stochastic systems are often investigated under suitable limiting conditions leading to diffusion processes. Consider a stochastic model defined over the integers from -N to N, and subject to catastrophes occurring at constant rate. The effect of each catastrophe instantaneously resets the process to state 0. Both the transient and steady-state probabilities of the above model are investigated. Further, the first passage time through state 0 is discussed. In this work, we perform a suitable scaling limit on the continuous time stochastic model that leads to a suitable jump-diffusion process of the Ornstein-Uhlenbeck type. Finally, the jump-diffusion approximation of the above model is studied.