The Annals of Probability

Strong supermartingales and limits of nonnegative martingales

Christoph Czichowsky and Walter Schachermayer

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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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 171-205.

Received: December 2013
Revised: July 2014
First available in Project Euclid: 2 February 2016

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Zentralblatt MATH identifier

Primary: 60G48: Generalizations of martingales 60H05: Stochastic integrals

Komlós’s lemma limits of nonnegative martingales Fatou limit optional strong supermartingales predictable strong supermartingales limits of stochastic integrals convergence in probability at all finite stopping times substitute for compactness


Czichowsky, Christoph; Schachermayer, Walter. Strong supermartingales and limits of nonnegative martingales. Ann. Probab. 44 (2016), no. 1, 171--205. doi:10.1214/14-AOP970.

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