Exit problems for oscillating compound Poisson process

TL;DR

This paper derives Laplace transforms for key boundary functionals of oscillating compound Poisson processes, including first passage times and overshoot distributions, and analyzes their asymptotic behavior under specific conditions.

Contribution

It provides the first explicit Laplace transform formulas for boundary functionals of oscillating compound Poisson processes and studies their asymptotics.

Findings

Laplace transforms of first passage times are obtained.

Joint distribution of exit time and overshoot is characterized.

Asymptotic behavior of boundary functionals is established.

Abstract

In this article we determine the Laplace transforms of the main boundary functionals of the oscillating compound Poisson process. These are the first passage time of the level, the joint distribution of the first exit time from the interval and the value of the overshoot through the boundary. Under certain conditions we establish the asymptotic behaviour of the mentioned functionals.