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This paper contains a review of the results of the martingale approach to point processes and its applications to the theory of dynamical systems (in the engineering sense), mainly estimation problems.

Some of the topics reviewed are: stochastic intensity, absolutely continuous changes of measure, martingale representations, changes of times, filtering, queues, etc.

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.

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Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage

Bang-bang controls of point processes

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2 results

Processus ponctuels et martingales: résultats récents sur la modélisation et le filtrage

Bang-bang controls of point processes