A probabilistic look at the Wiener-Hopf equation. (English) Zbl 0926.60040

Asmussen, Søren

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)

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