A factorization of a Levy process over a phase-type horizon

This note provides a factorization of a Levy 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.