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Weak Tail Conditions for Local Martingales

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Abstract

The following conditions are necessary and jointly sufficient for an arbitrary càdlàg local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in L1 on the events that those times are finite; and (C) its almost sure limit is an integrable random variable.

Suggested Citation

  • Hardy Hulley & Johannes Ruf, 2019. "Weak Tail Conditions for Local Martingales," Published Paper Series 2019-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    • Handle: RePEc:uts:ppaper:2019-2
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    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.
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    3. 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.
    4. 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.
    5. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    6. 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.
    7. 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.
    8. Johannes Ruf, 2013. "Negative call prices," Annals of Finance, Springer, vol. 9(4), pages 787-794, November.
    9. Johannes Ruf, 2012. "Negative Call Prices," Papers 1204.1903, arXiv.org, revised Jan 2013.
    10. Hardy Hulley & Eckhard Platen, 2009. "A Visual Criterion for Identifying Ito Diffusions as Martingales or Strict Local Martingales," Research Paper Series 263, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    12. 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.
    13. Ruf, Johannes, 2015. "The uniform integrability of martingales. On a question by Alexander Cherny," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3657-3662.
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    Cited by:

    1. Alessandro Gnoatto & Martino Grasselli & Eckhard Platen, 2022. "Calibration to FX triangles of the 4/2 model under the benchmark approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 1-34, June.
    2. Alessandro Gnoatto & Martino Grasselli & Eckhard Platen, 2016. "A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds," Papers 1608.04683, arXiv.org, revised Mar 2018.
    3. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    4. Baldeaux, Jan & Ignatieva, Katja & Platen, Eckhard, 2018. "Detecting money market bubbles," Journal of Banking & Finance, Elsevier, vol. 87(C), pages 369-379.
    5. Martin Herdegen & Dorte Kreher, 2021. "Bubbles in discrete time models," Papers 2104.12740, arXiv.org, revised Jul 2022.
    6. Martin Herdegen & Dörte Kreher, 2022. "Bubbles in discrete-time models," Finance and Stochastics, Springer, vol. 26(4), pages 899-925, October.
    7. Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.

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