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Strict local martingales with jumps

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  • Protter, Philip

Abstract

A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, control, and especially in mathematical finance. We give a method for constructing such examples and analyze one particular method in detail.

Suggested Citation

  • Protter, Philip, 2015. "Strict local martingales with jumps," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1352-1367.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1352-1367
    DOI: 10.1016/j.spa.2014.10.018
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    References listed on IDEAS

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    1. Hardy Hulley, 2010. "The Economic Plausibility of Strict Local Martingales in Financial Modelling," Research Paper Series 279, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    3. Xin Guo & Yan Zeng, 2008. "Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem," Papers 0801.3191, arXiv.org.
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    5. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    6. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    7. Martin Keller-Ressel, 2014. "Simple examples of pure-jump strict local martingales," Papers 1405.2669, arXiv.org, revised Jun 2015.
    8. Robert Jarrow & Philip Protter & A. Sezer, 2007. "Information reduction via level crossings in a credit risk model," Finance and Stochastics, Springer, vol. 11(2), pages 195-212, April.
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    Cited by:

    1. Robert A. Jarrow, 2015. "Asset Price Bubbles," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 201-218, December.
    2. Keller-Ressel, Martin, 2015. "Simple examples of pure-jump strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4142-4153.
    3. Li, Xue-Mei, 2017. "Strict local martingales: Examples," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 65-68.
    4. Aditi Dandapani & Philip Protter, 2019. "Strict Local Martingales Via Filtration Enlargement," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-28, December.

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