Enlargement of Filtration in Discrete Time Setting

We study, in a discrete time setting, the enlargement of filtration problem. We consider martingales in some filtration \(\mathbb{F}\), and we introduce a larger filtration \(\mathbb{G}\). We show that the martingale property in the \(\mathbb{G}\) filtration is, in general, lost, and we study the Doob decomposition of \(\mathbb{F}\)-martingales when they are considered as \(\mathbb{G}\)-adapted processes. This problem is closely linked with insider trading in a financial market.

We illustrate the study with examples. The goal is to show that results in continuous time are obtained in a very simple way in discrete time.

This is the joint work with Christophette Blanchet-Scalliet.