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Analyticity of Poisson-driven stochastic systems

Published online by Cambridge University Press:  01 July 2016

Michael A. Zazanis*
Affiliation:
Northwestern University
*
Postal address: Northwestern University, Department of Industrial Engineering and Management Sciences, 2145 Sheridan Rd, Evanston, IL 60208, USA.

Abstract

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ,Eλ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.

MSC classification

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

Research supported in part by NSF Grants ECS-8811003 and DDM-8905638.

References

Bernstein, S. (1914) Sur la définition et les propriétés des fonctions analytiques d'une variable réelle. Math. Ann. 75, 449468.CrossRefGoogle Scholar
Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Glynn, P. W. (1987) Likelihood ratio gradient estimation: an overview. Proc. Winter Simulation Conf. , ed. Thesen, A., Grant, H. and Kelton, W.D., pp. 366374.CrossRefGoogle Scholar
Lipster, E. and Shiryaev, B. (1977) Statistics of Stochastic Processes , Vol. I. Springer-Verlag, New York.Google Scholar
Reiman, M. and Simon, B. (1988) Light traffic limits of sojourn time distributions in Markovian queueing networks. Commun. Statist.Stoch. Models 4, 191233.CrossRefGoogle Scholar
Reiman, M. and Simon, B. (1989) Open queueing systems in light traffic. Math. Operat. Res. 14, 2659.Google Scholar
Reiman, M. and Weiss, A. (1989) Simulation via likelihood ratios. Operat. Res. 37, 830846.Google Scholar
Widder, D. V. (1946) The Laplace Transform. Princeton University Press.Google Scholar
Wong, E. and Hajek, B. (1986) Stochastic Processes in Engineering Systems. Springer-Verlag, New York.Google Scholar