## The Annals of Probability

### 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.

#### Article information

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

Dates
Revised: July 2014
First available in Project Euclid: 2 February 2016

https://projecteuclid.org/euclid.aop/1454423038

Digital Object Identifier
doi:10.1214/14-AOP970

Mathematical Reviews number (MathSciNet)
MR3456335

Zentralblatt MATH identifier
1339.60045

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

#### Citation

Czichowsky, Christoph; Schachermayer, Walter. Strong supermartingales and limits of nonnegative martingales. Ann. Probab. 44 (2016), no. 1, 171--205. doi:10.1214/14-AOP970. https://projecteuclid.org/euclid.aop/1454423038

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