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The uniform integrability of martingales. On a question by Alexander Cherny

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  • Ruf, Johannes

Abstract

Let X be a progressively measurable, almost surely right-continuous stochastic process such that Xτ∈L1 and E[Xτ]=E[X0] for each finite stopping time τ. In 2006, Cherny showed that X is then a uniformly integrable martingale provided that X is additionally nonnegative. Cherny then posed the question whether this implication also holds even if X is not necessarily nonnegative. We provide an example that illustrates that this implication is wrong, in general. If, however, an additional integrability assumption is made on the limit inferior of |X| then the implication holds. Finally, we argue that this integrability assumption holds if the stopping times are allowed to be randomized in a suitable sense.

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  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:10:p:3657-3662
    DOI: 10.1016/j.spa.2015.04.002
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    References listed on IDEAS

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    1. Yuri Kabanov & Robert Liptser, 2006. "From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift," Post-Print hal-00488295, HAL.
    2. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2009.
    3. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, July-Dece.
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    Cited by:

    1. 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.

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    Keywords

    Stopping time; Uniform integrability;

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