The Martingale Theory of Point Processes Over the Real Half Line Admitting an Intensity

Abstract

In [2] we have shown how the frame of martingale theory and stochastic integration could be systematically used in the theory of point processes, and, more generally, of jump processes. This new point of view has proven to be particularly adapted to problems of a dynamical nature such as filtering and optimal stopping for instance. It established a formal bridge between the theory of Wiener processes with a drift and point processes with an intensity through the martingale point of view. The methods used in the first case in the problems of detection and filtering can be applied word for word to solve the corresponding problems in the theory of jump processes: a striking example of this situation is the extension of the Fujisaki, Kallianpur and Kunita [11] martingale method of filtering (see [7] [18] and [19]). The martingale representation theorem plays a central role in the martingale point of view for jump processes. In this article we obtain it through the theorem of transformation of probabilities of [2] and recent results of J. Jacod [14]. The method seems to be a “natural” one and will find other applications.