Initial Enlargement of Filtrations and Entropy of Poisson Compensators

Stefan Ankirchner () and Jakub Zwierz ()
Additional contact information
Stefan Ankirchner: Humboldt-Universität zu Berlin
Jakub Zwierz: Uniwersytet Wroclawski

Journal of Theoretical Probability, 2011, vol. 24, issue 1, 93-117

Abstract: Abstract Let μ be a Poisson random measure, let $\mathbb{F}$ be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let $\mathbb{G}$ be the initial enlargement of $\mathbb{F}$ with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ-algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the $\mathbb{G}$ -compensator relative to the $\mathbb{F}$ -compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob–Meyer decompositions of stochastic processes when the filtration is enlarged from $\mathbb{F}$ to $\mathbb{G}$ . In particular, we show that if the mutual information between G and the σ-algebra generated by the Poisson random measure μ is finite, then every square-integrable $\mathbb{F}$ -martingale is a $\mathbb{G}$ -semimartingale that belongs to the normed space $\mathcal{S}^{1}$ relative to $\mathbb{G}$ .

Keywords: Initial enlargement of filtration; Poisson random measure; Entropy; Mutual information; Semimartingale; Embedding; 60G44; 60H30; 94A17 (search for similar items in EconPapers)
Date: 2011
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10959-010-0292-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-010-0292-9

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-010-0292-9

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().