Analytic Proof of Pecherskii–Rogozin Identity and Wiener–Hopf Factorization

Abstract

We present an analytic proof of the Pecherskii–Rogozin identity and the Wiener–Hopf factorization. The proof is rather general and requires only one mild restriction on the tail of the Lévy measure. The starting point of the proof of the Pecherskii–Rogozin identity is a two-dimensional integral equation satisfied by the joint distribution of the first passage time and the overshoot. This equation is reduced to a one-dimensional Wiener–Hopf integral equation, which is then solved using classical techniques from the theory of the Riemann boundary value problems. The Wiener–Hopf factorization is then derived as a corollary of the Pecherskii–Rogozin identity.