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

Let XX be a progressively measurable, almost surely right-continuous stochastic process such that Xτ∈L1Xτ∈L1 and E[Xτ]=E[X0]E[Xτ]=E[X0] for each finite stopping time ττ. In 2006, Cherny showed that XX is then a uniformly integrable martingale provided that XX is additionally nonnegative. Cherny then posed the question whether this implication also holds even if XX 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||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.