First-exit times for increasing compound processes

Abstract

For a process that increases linearly, with unit slope, between jumps of i.i.d. positive sizes occurring at renewal times, we present methods to compute the distribution of the first time a prespecified level is reached or exceeded, and of the position at this time. In the exponential case the Laplace transform of this first-exit time is derived in closed form. A general formula for the distribution of the stopping time is given, and shown to yield explicit results in certain cases. An effective method of successive approximation is also derived. The problem is equivalent to that of determining the distribution of the total ON time in [0t:] of a system changing between the states ON and OFF at the points of an alternating renewal process