Strong supermartingales and limits of nonnegative martingales

Abstract

Given a sequence $(M^n{)}_{n=1}^{\infty }$ of nonnegative martingales starting at $M_0^n=1$, we find a sequence of convex combinations $({\tilde{M}}^n{)}_{n=1}^{\infty }$ and a limiting process $X$ such that $({\tilde{M}}_{\tau }^n{)}_{n=1}^{\infty }$ 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 $(X^n{)}_{n=1}^{\infty }$, their left limits $(X_{-}^n{)}_{n=1}^{\infty }$ and their stochastic integrals $(\int \phi d{X}^n{)}_{n=1}^{\infty }$ and explain the relation to the notion of the Fatou limit.

Published at http://dx.doi.org/10.1214/14-AOP970 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Supermartingales as Radon-Nikodym densities and related measure extensions

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