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

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