Exit, passage, and crossing times and overshoots for a Poisson compound process with an exponential component

Abstract

Integral transforms of the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time are found for a Poisson process with an exponentially distributed negative component. We obtain the distributions of the following functionals of the process on an exponentially distributed time interval: the supremum, infimum, and the value of the process, numbers of upcrossings and downcrossings, the number of passages into an interval and overshoots over a boundary of an interval. Introduction The main two-boundary functional for homogeneous stochastic processes with independent increments is the joint distribution of the first exit time from an interval and overshoot over the boundary at the exit time. Below we give a short survey of results related to this functional. We do not pretend to cover all the aspects of the theory; the survey reflects an author personal viewpoint and approach to the above functional in the case of stochastic processes with independent increments. Let ξ(t) ∈ R, t ≥ 0, ξ(0) = 0, be a homogeneous stochastic process with independent increments [1] whose cumulant is given by (Re p = 0) k(p) = 1 t lnE e−pξ(t) = 1 2 pσ − αp+ ∫ ∞ −∞ ( e−px − 1 + px 1 + x2 ) Π(dx). (1) Fix B > 0. Let y ∈ [0, B], x = B − y, and let χ = inf{t : ξ(t) / ∈ [−y, x]} be the first exit time of the process ξ(t) from the interval [−y, x]. We consider the random events A = {ξ(χ) > x} and Ay = {ξ(χ) < −y} meaning that the process exits the interval through the upper and lower boundary, respectively. Further let X = (ξ(χ)− x) IAx +(−ξ(χ)− y) IAy , P[A +Ay] = 1, be the overshoot of the process over the boundary at the first exit time from an interval where IA = IA(ω) is the indicator of a random event A. 2000 Mathematics Subject Classification. Primary 60J05, 60J10; Secondary 60J45.