The Recurrence Classification of Risk and Storage Processes

Let X = {X(t), t ≥ 0} be a Markov storage process with compound Poisson input A = {A(t), t ≥ 0} and general release rule r(·). In a previous paper, a necessary and sufficient condition for the positive recurrence of X was obtained, and its stationary distribution was computed. Here we complete the recurrence classification of X, determining the conditions for null recurrence and transience.

Closely related to X is a Markov process X⁰ = {X⁰(t), t ≥ 0}. Its paths are absolutely continuous and increasing between downward jumps, the instantaneous rate of increase at time t being r(X⁰(t)). The jumps are generated by A but are truncated as necessary to keep X nonnegative. It is shown that X⁰ is positive recurrent iff X is transient, null recurrent iff X is null recurrent, and transient iff X is positive recurrent. Furthermore, in the case where X⁰ is transient, the probability that X⁰ ever hits zero (viewed as a function of the initial state) has the same density as the stationary distribution of X. A similar duality between the two processes is found in the case where X⁰ is positive recurrent.