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On filtration enlargements and purely discontinuous martingales

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  • Ankirchner, Stefan

Abstract

Let M be a purely discontinuous martingale relative to a filtration . Given an arbitrary extension of the filtration , we will provide sufficient conditions for M to be a semimartingale relative to . Moreover we describe methods of how to find the Doob-Meyer decomposition with respect to the enlarged filtration. To this end we prove a new and more explicit version of the predictable representation property of Poisson random measures. Finally some concrete examples will show how the method developed may be applied.

Suggested Citation

  • Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:9:p:1662-1678
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    References listed on IDEAS

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    1. Nualart, David & Schoutens, Wim, 2000. "Chaotic and predictable representations for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 109-122, November.
    2. 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.
    3. Imkeller, Peter & Pontier, Monique & Weisz, Ferenc, 2001. "Free lunch and arbitrage possibilities in a financial market model with an insider," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 103-130, March.
    4. Josep Vives & Jorge A. León & Frederic Utzet & Josep L. Solé, 2002. "On Lévy processes, Malliavin calculus and market models with jumps," Finance and Stochastics, Springer, vol. 6(2), pages 197-225.
    5. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    6. 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.
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    Cited by:

    1. Dario Gasbarra & Jos'e Igor Morlanes & Esko Valkeila, 2011. "Initial Enlargement in a Markov chain market model," Papers 1108.2623, arXiv.org, revised Aug 2011.
    2. 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.
    3. Bernardo D'Auria & Jos'e A. Salmer'on, 2021. "Anticipative information in a Brownian-Poissonmarket: the binary information," Papers 2111.01529, arXiv.org.
    4. D'Auria, Bernardo & Salmerón Garrido, José Antonio, 2021. "Anticipative information in a Brownian-Poisson market: the binary information," DES - Working Papers. Statistics and Econometrics. WS 33624, Universidad Carlos III de Madrid. Departamento de Estadística.

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