The author considers a Markov process with state space and generator where if and , and . This defines a linear birth-death process modified to allow ’catastrophic’ decrements in the population size at a rate proportional to population size. This model, with specific catastrophe-size distributions , has previously been examined by the author, J. Gani and S. I. Resnick, ibid. 14, 709-731 (1982; Zbl 0496.92007).
Here the author is interested in the time T to extinction. He derives an expression for its probability generating function, criteria ensuring that and an asymptotic expression for , when this is not unity, as . In addition he derives a generating function for and obtains an asymptotic form of for large i when the process drifts to the origin. Finally, he considers the analogous problems for the Feller continuous- state branching process modified to allow downward jumps at a rate proportional to the level of the process.