# Escape probabilities of compound renewal processes with drift

## TL;DR

This paper analyzes the escape probabilities of compound renewal processes with drift, providing integral equation solutions and explicit formulas for specific cases like Erlang and hypo-exponential arrivals, with applications in actuarial science.

## Contribution

It introduces a unified approach to compute escape probabilities for compound renewal processes with drift, including new solutions for Erlang, hypo-exponential, and rational Laplace transform cases.

## Key findings

- Explicit solutions for escape probabilities with Erlang and hypo-exponential arrivals
- Identification of solvable cases with two-sided jumps
- Connection to scale functions of diffusion processes

## Abstract

We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps arise we give a general solution for escape and survival probabilities under Erlang$(n)$ and hypo-exponential arrivals. These ideas are generalized to the class of arrival distributions having rational Laplace transforms. In a general situation with two-sided jumps we also identify important families of solvable cases. A parallelism with the "scale function" of diffusion processes is drawn.