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The cumulant process and Esscher's change of measure

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Author Info

  • Albert N. Shiryaev

    () (Steklov Mathematical Institute, Gubkina St. 8, 117966 Moscow, Russia Manuscript)

  • Jan Kallsen

    () (Institut für Mathematische Stochastik, Universität Freiburg, Eckerstraße 1, 79104 Freiburg i. Br., Germany)

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    Abstract

    In this paper two kinds of cumulant processes are studied in a general setting. These processes generalize the cumulant of an infinitely divisible random variable and they appear as the exponential compensator of a semimartingale. In a financial context cumulant processes lead to a generalized Esscher transform. We also provide some new criteria for uniform integrability of exponential martingales.

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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 6 (2002)
    Issue (Month): 4 ()
    Pages: 397-428

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    Handle: RePEc:spr:finsto:v:6:y:2002:i:4:p:397-428

    Note: received: January 2001; final version received: November 2001
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    Web page: http://www.springerlink.com/content/101164/

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    Related research

    Keywords: Cumulant process; stochastic logarithm; exponential transform; exponential compensator; exponentially special semimartingale; Esscher transform; uniform integrability;

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    Cited by:
    1. Blei, Stefan & Engelbert, Hans-Jürgen, 2009. "On exponential local martingales associated with strong Markov continuous local martingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2859-2880, September.
    2. Mayerhofer, Eberhard & Muhle-Karbe, Johannes & Smirnov, Alexander G., 2011. "A characterization of the martingale property of exponentially affine processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 568-582, March.
    3. Anastasia Ellanskaya & Lioudmila Vostrikova, 2013. "Utility maximisation and utility indifference price for exponential semi-martingale models with random factor," Papers 1303.1134, arXiv.org.
    4. Grigelionis, Bronius & Mackevicius, Vigirdas, 2003. "The finiteness of moments of a stochastic exponential," Statistics & Probability Letters, Elsevier, vol. 64(3), pages 243-248, September.
    5. Hubalek, Friedrich & Sgarra, Carlo, 2009. "On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2137-2157, July.
    6. Matthias Fengler & Helmut Herwartz & Christian Werner, 2010. "A dynamic copula approach to recovering the index implied volatility skew," University of St. Gallen Department of Economics working paper series 2010 1132, Department of Economics, University of St. Gallen, revised Nov 2011.
    7. Liuren Wu, 2006. "Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns," The Journal of Business, University of Chicago Press, vol. 79(3), pages 1445-1474, May.

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