Initial enlargement of filtrations and entropy of Poisson compensators
type: Item
instance of: scholarly article
title: Initial enlargement of filtrations and entropy of Poisson compensators
zbMATH Open document ID: 1266.60078
author: Stefan Ankirchner
author: Jakub Zwierz
published in: Journal of Theoretical Probability
publication date: 2011-03-31T00:00:00Z
review text: Consider a Poisson random measure \(\mu\), denote by \(\mathbb{F}\) the smallest initial filtration satisfying the usual conditions and containing the one generated by \(\mu\), and let \(\mathbb{G}\) be the initial enlargement of \(\mathbb{F}\) with the \(\sigma\)-field generated by a random variable \(G\). The authors show that the mutual information between the enlarging random variable \(G\) and the \(\sigma\)-algebra generated by the Poisson random measure \(\mu\) equals the expected relative entropy of the \(\mathbb{G}\)-compensator relative to the \(\mathbb{F}\)-compensator of the random measure \(\mu\). This result provides some insight into how the Doob-Meyer decomposition of a stochastic process changes when the filtration is enlarged. In particular, the authors prove that, if \(\mathbb{G}\) is an initial enlargement of \(\mathbb{F}\) by some random variable \(G\) with finite entropy, then every square integrable \((\mathbb{F},\mathbb{P})\)-martingale remains a \((\mathbb{G},\mathbb{P})\)-semimartingale and has an information drift that is integrable.
reviewed by: Antonis Papapantoleon
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 5872576
zbMATH Keywords: initial enlargement of filtration
zbMATH Keywords: Poisson random measure
zbMATH Keywords: mutual information
zbMATH Keywords: semimartingale
zbMATH Keywords: embedding
MaRDI profile type: Publication
full work available at URL: s10959-010-0292-9
OpenAlex ID: W2047297517
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DOI: 10.1007/S10959-010-0292-9
description: scientific article; zbMATH DE number 5872576