Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T11:16:10.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Bang-bang controls of point processes

Published online by Cambridge University Press:  01 July 2016

P. Brémaud
P. Brémaud*
Affiliation:
CEREMADE, Université de Paris IX (Dauphine)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the problem of controlling the intensity of a point process in order to maximize the probability that the number of points in a fixed interval equals a given integer, under the constraint that the intensity belong to some closed interval of R+.

The problem is stated as a problem of optimization on the set of probabilities over the basic measurable space of point processes, and shown to be equivalent to a problem of deterministic control. Structural results concerning the set of optimal solutions are given. The existence of the latter is proven; the control is shown to be bang-bang and a complete solution can be obtained by application of Pontryagin's Maximum Principle.

Keywords

POINT PROCESSQUEUING PROCESSMARTINGALEPREDICTABLE PROCESSSTOCHASTIC INTEGRALOPTIMAL CONTROLBANG-BANG CONTROL
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 385 - 394
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Boel, R. (1974) Optimal Control of Jump Processes. Ph.D. Thesis, Memo ERL–M448, Department of EECS, University of California, Berkeley.Google Scholar
[2] Brémaud, P. (1972) A Martingale Approach to Point Processes. Ph.D. Thesis, Memo ERL–M 345, Department of EECS, University of California, Berkeley.Google Scholar
[3] Brémaud, P. (1974) The martingale theory of point processes with an intensity. In Proceedings of the IRIA Colloquium on Control Theory, Lect. Notes in Computer Science and Math. Systems 107, Springer-Verlag, 519–542.Google Scholar
[4] Doleans–Dade, C. and Meyer, P. A. (1970) Intégrales stochastiques par rapport aux martingales locales. In Seminaire de Probabilité Université de Strasbourg, IV, Lect. Notes in Mathematics 124, Springer–Verlag, 77–107.Google Scholar
[5] Jacod, J. (1973) On the stochastic intensity of a random point process. Technical Report, Department of Statistics, Princeton University.Google Scholar
[6] Jacod, J. (1975) Multivariate point process: Predictable projection Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
[7] Meyer, P. A. (1965) Probabilités et Potentiel. Hermann, Paris.Google Scholar
[8] Rishel, R. (1974) A minimum principle for controlled jump processes. In Proceedings of the IRIA Colloquium on Control Theory, Lect. Notes in Comp. Science and Math. Systems 107, Springer–Verlag, 499–518.Google Scholar
[9] Lee, E. B. and Markus, M. (1968) Foundations of Optimal Control Theory. Wiley, New York.Google Scholar