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Optimization Applications of Compensators of Poisson Random Measures

Published online by Cambridge University Press:  27 July 2009

Shengwu He
Shengwu He
Affiliation:
Department of Mathematical Statistics East China, Normal University Shanghai, People's Republic of China
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Abstract

Compensator or dual predictable projection is one of the main concepts of the general theory of stochastic processes, established by a French probabilistic school. In general, it is a considerably abstract concept and difficult to grasp. But in the Poisson case, it affords very powerful tools to calculate the expectation of certain stochastic integrals with respect to Poisson processes and random measures. Naturally, compensators can be used to solve certain problems of applied probability, involved in Poisson processes and random measures. In this direction, Brémaud and Jacod [1] had used the compensators of Poisson processes to solve the optimal dispatching problem, investigated by Ross [4] originally. In this note, we use the compensators of Poisson random measures to solve two optimal stopping problems: house selling and the burglary problem. The solutions are simple and completely rigorous as well. Our objective is to attract more attention to this method of using compensators. To facilitate reading, we introduce some fundamental results on marked point processes and Poisson random measures and their compensators.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 1 , January 1989 , pp. 149 - 155
Copyright
Copyright © Cambridge University Press 1989

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References

Brémaud, P. & Jacod, J. (1977). Processus ponctuels et martingales: résultats récents sur la modelisation et le filtrage. Advances in Applied Probability 9: 362–369.CrossRefGoogle Scholar
Chow, Y.S., Robbins, H. & Siegmund, D. (1974). Great expectation: the theory of optimal stopping. New York: Houghton Mifflin Company.Google Scholar
Jacod, J. & Shiryaev, A.N. (1987). Limit theorems for stochastic processes. Springer-Verlag.Google Scholar
Ross, S. (1970). Applied probability models with optimization applications. San Francisco: Holden-Day.Google Scholar