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 $X_{\tau }\in L^1$ and $E[X_{\tau }]=E[X_0]$ for each finite stopping time $\tau$. 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 $\mathrm{\mid }X\mathrm{\mid }$ 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.