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

Johannes Ruf

Stochastic Processes and their Applications, 2015, vol. 125, issue 10, 3657-3662

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.

Keywords: Stopping time; Uniform integrability (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2015.04.002

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