A Probabilistic Look at the Wiener-Hopf Equation

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.

[1]  Claudia Klüppelberg,et al.  Large deviations results for subexponential tails, with applications to insurance risk , 1996 .

[2]  Nimrod Bayer,et al.  On the identification of Wiener-Hopf factors , 1996, Queueing Syst. Theory Appl..

[3]  Richard L. Smith.  The extremal index for a Markov chain , 1992, Journal of Applied Probability.

[4]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[5]  Thomas Kailath,et al.  A view of three decades of linear filtering theory , 1974, IEEE Trans. Inf. Theory.

[6]  Frank Spitzer,et al.  The Wiener-Hopf equation whose kernel is a probability density. II , 1957 .

[7]  Sلأren Asmussen,et al.  Applied Probability and Queues , 1989 .

[8]  A. Pipkin,et al.  A Course on Integral Equations , 1991 .

[9]  Understanding the Wiener-Hopf factorization for the simple random walk , 1994 .

[10]  F. Spitzer.  Principles Of Random Walk , 1966 .

[11]  J.H.A. de Smit.  Explicit Wiener-Hopf Factorizations for the Analysis of Multi-Dimensional Queues , 1995 .

[12]  N. U. Prabhu.  Wiener-Hopf Techniques in Queueing Theory , 1974 .

[13]  George Weiss,et al.  Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations , 1958 .

[14]  A. Gut.  Stopped Random Walks , 1987 .

[15]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[16]  J. W. Cohen.  The Wiener-Hopf Technique in Applied Probability , 1975 .

[17]  H. W. Stolle,et al.  Theorie und Praxis der linearen Integralgleichungen , 1984 .

[18]  V. Malyshev.  Wiener-Hopf equations and their applications in probability theory , 1977 .

[19]  S. G. Mikhlin,et al.  Integral equations―a reference text , 1975 .

[20]  D. V. Lindley,et al.  The theory of queues with a single server , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[22]  T. M. Williams,et al.  Stochastic Storage Processes: Queues, Insurance Risk and Dams , 1981 .

[23]  P. Embrechts,et al.  Estimates for the probability of ruin with special emphasis on the possibility of large claims , 1982 .

[24]  F. Noether,et al.  Über eine Klasse singulärer Integralgleichungen , 1920 .

[25]  P. Embrechts,et al.  On closure and factorization properties of subexponential and related distributions , 1980, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[26]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[27]  C. Klüppelberg.  Subexponential distributions and integrated tails. , 1988 .