Exit Times, Overshoot and Undershoot for a Surplus Process in the Presence of an Upper Barrier

Abstract

We study the movement of a surplus process with initial capital u in the presence of two barriers: a lower barrier at zero and an upper barrier at b (b > u). More specifically, we consider the behaviour of the surplus: (a) in continuous time; and (b) only at claim arrival times. For each of these cases, we find the expected time until the process exits the interval [0,b]. We also obtain results related to the undershoot and overshoot of the surplus which, in particular for case (b) above, are derived under the assumption that the distribution of claim sizes and/or claim interarrival times belongs to the mixed Erlang class. In the final section we discuss the implementation of the methods in a number of examples using computer algebra software. These examples illustrate the efficiency of the methods even in fairly complicated cases.