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Simplified calculus for semimartingales: Multiplicative compensators and changes of measure

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  • Alev{s} v{C}ern'y
  • Johannes Ruf

Abstract

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation when such compensation is meaningful. This generalization of the L\'evy--Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a signed stochastic exponential, which in turn has practical applications in mean--variance portfolio theory. Girsanov-type results based on multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the characteristic function of log returns for a popular class of minimax measures in a L\'evy setting.

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  • Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Papers 2006.12765, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2006.12765
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    References listed on IDEAS

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    1. Albert N. Shiryaev & Jan Kallsen, 2002. "The cumulant process and Esscher's change of measure," Finance and Stochastics, Springer, vol. 6(4), pages 397-428.
    2. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    3. Černý, Aleš & Ruf, Johannes, 2021. "Simplified stochastic calculus with applications in Economics and Finance," European Journal of Operational Research, Elsevier, vol. 293(2), pages 547-560.
    4. Ernst Eberlein & Antonis Papapantoleon & Albert N. Shiryaev, 2008. "Esscher transform and the duality principle for multidimensional semimartingales," Papers 0809.0301, arXiv.org, revised Nov 2009.
    5. Constantinos Kardaras & Johannes Ruf, 2020. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Finance and Stochastics, Springer, vol. 24(4), pages 871-901, October.
    6. Christian Bender & Christina Niethammer, 2008. "On q-optimal martingale measures in exponential Lévy models," Finance and Stochastics, Springer, vol. 12(3), pages 381-410, July.
    7. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
    8. Ruf, Johannes, 2013. "A new proof for the conditions of Novikov and Kazamaki," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 404-421.
    9. Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified stochastic calculus via semimartingale representations," Papers 2006.11914, arXiv.org, revised Jan 2022.
    10. Kallsen, Jan & Muhle-Karbe, Johannes, 2010. "Exponentially affine martingales, affine measure changes and exponential moments of affine processes," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 163-181, February.
    11. Kardaras, Constantinos, 2015. "On the stochastic behaviour of optional processes up to random times," LSE Research Online Documents on Economics 64965, London School of Economics and Political Science, LSE Library.
    12. Constantinos Kardaras & Johannes Ruf, 2019. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Papers 1912.04652, arXiv.org, revised Aug 2020.
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    Cited by:

    1. Aleš Černý & Christoph Czichowsky & Jan Kallsen, 2024. "Numeraire-Invariant Quadratic Hedging and Mean–Variance Portfolio Allocation," Mathematics of Operations Research, INFORMS, vol. 49(2), pages 752-781, May.
    2. Černý, Aleš & Ruf, Johannes, 2021. "Simplified stochastic calculus with applications in Economics and Finance," European Journal of Operational Research, Elsevier, vol. 293(2), pages 547-560.
    3. Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified stochastic calculus via semimartingale representations," Papers 2006.11914, arXiv.org, revised Jan 2022.

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