## Illinois Journal of Mathematics

### Doob's maximal identity, multiplicative decompositions and enlargements of filtrations

#### Abstract

In the theory of progressive enlargements of filtrations, the supermartingale $Z_{t}=\mathbf{P}( g>t\mid \mathcal{F}_{t})$ associated with an honest time $g$, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale $Z_{t}$, based on Doob's maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales , using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.

#### Article information

Source
Illinois J. Math., Volume 50, Number 1-4 (2006), 791-814.

Dates
First available in Project Euclid: 12 November 2009

https://projecteuclid.org/euclid.ijm/1258059492

Digital Object Identifier
doi:10.1215/ijm/1258059492

Mathematical Reviews number (MathSciNet)
MR2247846

Zentralblatt MATH identifier
1101.60059

#### Citation

Nikeghbali, Ashkan; Yor, Marc. Doob's maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois J. Math. 50 (2006), no. 1-4, 791--814. doi:10.1215/ijm/1258059492. https://projecteuclid.org/euclid.ijm/1258059492