Strong supermartingales and limits of nonnegative martingales

Given a sequence (Mn)∞n=1 of non-negative martingales starting at Mn0 = 1 we find a sequence of convex combinations (M̃n)∞n=1 and a limiting process X such that (M̃nτ) n=1 converges in probability to Xτ, for all finite stopping times τ. The limiting process X then is an optional strong supermartingale. A counter-example 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 ( ϕdXn)∞n=1 and explain the relation to the notion of the Fatou limit