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 $(\tilde{M}^{n})^{\infty}_{n=1}$ and a limiting process $X$ such that $(\tilde{M}^{n}_{\tau})^{\infty}_{n=1}$ 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.