On filtration enlargements and purely discontinuous martingales
type: Item
instance of: scholarly article
title: On filtration enlargements and purely discontinuous martingales
zbMATH Open document ID: 1157.60038
author: Stefan Ankirchner
published in: Stochastic Processes and their Applications
publication date: 2008-09-29T00:00:00Z
review text: The goal of the article under review is to know when a purely discontinuous martingale \(M_t\) under a filtration \({\mathcal F}_t\) remains a semimartingale after an enlargement of the filtration. If \({\mathcal G}_t\) is the enlarged filtration, a sufficient condition is given, and a new explicit formula for the Doob-Meyer decomposition of \(M_t\) under \({\mathcal G}_t\) is obtained. More precisely, the finite variation part of \(M_t\) is called the information drift of \(M_t\) with respect to \({\mathcal G}_t\), and the formula for this drift involves the predictable representation of the martingales \(E[1_A| {\mathcal F}_t]\). On the other hand, it is known that such a predictable representation can be obtained by means of a difference operator on the Poisson space, and this leads to another expression for the information drift. Finally, the method is applied to some concrete examples.
reviewed by: Jean Picard
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 5348271
zbMATH Keywords: enlargement of filtrations
zbMATH Keywords: semimartingale
zbMATH Keywords: Doob-Meyer decomposition
zbMATH Keywords: purely discontinuous martingale
zbMATH Keywords: Poisson random measure
zbMATH Keywords: predictable representation
MaRDI profile type: Publication
full work available at URL: j.spa.2007.10.007
OpenAlex ID: W2124206433
cites work: Q3562499
cites work: Metrics on the set of semimartingale filtrations
cites work: The Shannon information of filtrations and the additional logarithmic utility of insiders
cites work: Additional utility of insiders with imperfect dynamical information
cites work: Q3799426
cites work: White noise analysis for Lévy processes.
cites work: Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches
cites work: Free lunch and arbitrage possibilities in a financial market model with an insider.
cites work: Q3684922
cites work: Q4856610
cites work: Semi-martingales et grossissement d'une filtration
cites work: Grossissements de filtrations: exemples et applications. Séminaire de Calcul Stochastique 1982/83, Université Paris VI
cites work: On Lévy processes, Malliavin calculus and market models with jumps
cites work: Martingale Representation of Functionals of Lévy Processes
cites work: Chaotic and predictable representations for Lévy processes.
cites work: Q5533878
cites work: Duality formulas on the Poisson space
cites work: On the existence of smooth densities for jump processes
cites work: Q3996259
cites work: Q4226355
cites work: Canonical Lévy process and Malliavin calculus
cites work: Calcul stochastique d�pendant d'un param�tre
DOI: 10.1016/J.SPA.2007.10.007
description: scientific article; zbMATH DE number 5348271