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Single jump processes and strict local martingales

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  • Herdegen, Martin
  • Herrmann, Sebastian

Abstract

Many results in stochastic analysis and mathematical finance involve local martingales. However, specific examples of strict local martingales are rare and analytically often rather unhandy. We study local martingales that follow a given deterministic function up to a random time γ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local martingales, uniformly integrable martingales that are not in H1, etc. As an application, we provide a construction of a (uniformly integrable) martingale M and a bounded (deterministic) integrand H such that the stochastic integral H•M is a strict local martingale.

Suggested Citation

  • Herdegen, Martin & Herrmann, Sebastian, 2016. "Single jump processes and strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 337-359.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:2:p:337-359
    DOI: 10.1016/j.spa.2015.09.003
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    1. Mark Loewenstein & Gregory A. Willard, 2000. "Local martingales, arbitrage, and viability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 16(1), pages 135-161.
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    Cited by:

    1. Martin HERDEGEN & Martin SCHWEIZER, 2015. "Economics-Based Financial Bubbles (and Why They Imply Strict Local Martingales)," Swiss Finance Institute Research Paper Series 15-05, Swiss Finance Institute.
    2. Michael Schatz & Didier Sornette, 2017. "Uniform Integrability of a Single Jump Local Martingale with State-Dependent Characteristics," Swiss Finance Institute Research Paper Series 17-21, Swiss Finance Institute.
    3. Martin Herdegen & Martin Schweizer, 2016. "Strong Bubbles And Strict Local Martingales," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-44, June.

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