A Probabilistic Look at the Wiener--Hopf Equation

Abstract

Existence, uniqueness, and asymptotic properties of solutions Z to the Wiener--Hopf integral equation $Z(x)$ $ =$ $z(x)+\int_{-\infty}^xZ(x-y)F(dy)$, $x\ge 0$, are discussed by purely probabilistic methods, involving random walks, supermartingales, coupling, the Hewitt--Savage 0--1 law, ladder heights, and exponential change of measure.