# A factorization of a L\'evy process over a phase-type horizon

## TL;DR

This paper extends the Wiener-Hopf factorization to Lévy processes over phase-type horizons, providing new representations and explicit formulas for joint distributions, thereby advancing the theoretical understanding of such stochastic processes.

## Contribution

It introduces a novel factorization of Lévy processes over phase-type horizons, including time-reversal-based representations and explicit joint law formulas.

## Key findings

- Multiple time-reversal representations yield the same factor.
- Explicit formulas for joint law of supremum and terminal value.
- Extension of Wiener-Hopf factorization to phase-type horizons.

## Abstract

This note provides a factorization of a L\'evy pocess over a phase-type horizon $\tau$ given the phase at the supremum, thereby extending the Wiener-Hopf factorization for $\tau$ exponential. One of the factors is defined using time reversal of the phase process. It is shown that there are a variety of time-reversed representations, all yielding the same factor. Consequences of this are discussed and examples provided. Additionally, some explicit formulas for the joint law of the supremum and the terminal value of the process at $\tau$ are given.