Strong supermartingales and limits of nonnegative martingales Christoph Czichowsky, Walter Schachermayer

Given a sequence (Mn).n=1 of nonnegative martingales starting at Mn0=1, we find a sequence of convex combinations (~Mn).n=1 and a limiting process X such that (~Mn.).n=1 converges in probability to X., for all finite stopping times .. The limiting process X then is an optional strong supermartingale. A counterexample reveals that the convergence in probability cannot be replaced by almost sure convergence in this statement. We also give similar convergence results for sequences of optional strong supermartingales (Xn).n=1, their left limits (Xn.).n=1 and their stochastic integrals (..dXn).n=1 and explain the relation to the notion of the Fatou limit.