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January 2016 Strong supermartingales and limits of nonnegative martingales
Christoph Czichowsky, Walter Schachermayer
Ann. Probab. 44(1): 171-205 (January 2016). DOI: 10.1214/14-AOP970

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.

Citation

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Christoph Czichowsky. Walter Schachermayer. "Strong supermartingales and limits of nonnegative martingales." Ann. Probab. 44 (1) 171 - 205, January 2016. https://doi.org/10.1214/14-AOP970

Information

Received: 1 December 2013; Revised: 1 July 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1339.60045
MathSciNet: MR3456335
Digital Object Identifier: 10.1214/14-AOP970

Subjects:
Primary: 60G48, 60H05

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • January 2016
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