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

Abstract

Let X be a progressively measurable, almost surely right-continuous stochastic process such that X τ ∈ 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.

Piecewise constant local martingales with bounded numbers of jumps

“…The next example illustrates that the assumptions of Corollary 2 are not sufficient to guarantee that M is a uniformy integrable martingale, even if there is only one jump possible, that is, even if N = 1. The example is adapted from Ruf (2015), where it is used as a counterexample for a different conjecture.…”