Illinois Journal of Mathematics

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

Ashkan Nikeghbali and Marc Yor

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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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258059492

Mathematical Reviews number (MathSciNet)
MR2247846

Zentralblatt MATH identifier
1101.60059

Subjects
Primary: 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G48: Generalizations of martingales

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


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