Strong supermartingales and limits of nonnegative martingales

Abstract

Given a sequence $(M^n)^{\infty}_{n=1}$ of nonnegative martingales starting at $M^n_0=1$, we find a sequence of convex combinations $(\widetilde{M}^n)^{\infty}_{n=1}$ and a limiting process $X$ such that $(\widetilde{M}^n_{\tau})^{\infty}_{n=1}$ converges in probability to $X_{\tau}$, for all finite stopping times $\tau$. 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 $(X^n)^{\infty}_{n=1}$, their left limits $(X^n_-)^{\infty}_{n=1}$ and their stochastic integrals $(\int\varphi \,dX^n)^{\infty}_{n=1}$ and explain the relation to the notion of the Fatou limit.