Cumulative Rewards for Birth–Death Processes with Catastrophes

This paper investigates cumulative reward functionals for birth–death processes with Poisson-driven total catastrophes. Using an auxiliary killed Markov process, we derive the integral equations for the distribution and expectation of the reward accumulated up to the first catastrophe, and establish the existence and uniqueness of bounded solutions. Explicit expressions are provided for constant birth–death rates. The asymptotic limits of the scaled expected reward are analyzed for vanishing and diverging catastrophe rates, with limiting behavior classified by the recurrence of the underlying catastrophe-free birth–death process.