Asmussen, Søren A probabilistic look at the Wiener-Hopf equation. (English) Zbl 0926.60040 SIAM Rev. 40, No. 2, 189-201 (1998). The Wiener-Hopf integral equation \[ Z(x)=z(x)+ \int^x_{-\infty} Z(x-y)F(dy), \quad x\geq 0, \] is analysed by purely probabilistic methods. The author investigates the nonnegative solutions \(Z\) of the equation assuming \(z\) to be nonnegative and bounded on finite intervals and \(F\) to be a probability measure with existing mean \(\mu\). He applies random walk techniques (ladder heights), supermartingales, coupling, \(0\)-1-laws, and exponential change of measure to characterize the set of solutions and to study their asymptotic properties, which strongly depend on the tails of the distribution \(F\). The cases \(\mu>0\) and \(\mu<0\) are intrinsically different. Reviewer: H.Wegmann (Darmstadt) Cited in 14 Documents MSC: 60G50 Sums of independent random variables; random walks 60K25 Queueing theory (aspects of probability theory) Keywords:coupling; exponential change of measure; first passage time; integral equation; ladder heights; Lindley equation; random walk; renewal equation; spread-out distribution; subexponential distribution; supermartingale PDF BibTeX XML Cite \textit{S. Asmussen}, SIAM Rev. 40, No. 2, 189--201 (1998; Zbl 0926.60040) Full Text: DOI OpenURL