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arXiv · 1501.05922

The Uniform Integrability of Martingales. On a Question by Alexander Cherny

Johannes Ruf
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Abstract

Let $X$ be a progressively measurable, almost surely right-continuous stochastic process such that $X_τ\in L^1$ and $E[X_τ] = E[X_0]$ 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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Johannes Ruf. 2015-05-01. The Uniform Integrability of Martingales. On a Question by Alexander Cherny. https://doi.org/10.1016/j.spa.2015.04.002

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