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This work is concerned with the theory of initial and progressive enlargements of areference filtration \hbox{$\mathbb{F}$} $F$ with a random time τ. We provide, under anequivalence assumption, slightly stronger than the absolute continuity assumption ofJacod, alternative proofs to results concerning canonical decomposition of an \hbox{$\mathbb{F}$} $F$ -martingalein the enlarged filtrations. Also, we address martingales’ characterization in theenlarged filtrations in terms of martingales in the reference filtration, as well aspredictable representation theorems in the enlarged filtrations.

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.

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Carthaginian enlargement of filtrations ∗

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

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2 results

Carthaginian enlargement of filtrations∗

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

Carthaginian enlargement of filtrations ∗