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Initial Enlargement of Filtrations and Entropy of Poisson Compensators

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  • Stefan Ankirchner

    (Humboldt-Universität zu Berlin)

  • Jakub Zwierz

    (Uniwersytet Wroclawski)

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 $\mathbb{G}$ .

Suggested Citation

  • Stefan Ankirchner & Jakub Zwierz, 2011. "Initial Enlargement of Filtrations and Entropy of Poisson Compensators," Journal of Theoretical Probability, Springer, vol. 24(1), pages 93-117, March.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-010-0292-9
    DOI: 10.1007/s10959-010-0292-9
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    References listed on IDEAS

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    1. Peter Imkeller, 2003. "Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 153-169, January.
    2. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    3. Stefan Ankirchner & Steffen Dereich & Peter Imkeller, 2005. "The Shannon information of filtrations and the additional logarithmic utility of insiders," Papers math/0503013, arXiv.org, revised May 2006.
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