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Strong supermartingales and limits of non-negative martingales

Czichowsky, Christoph and Schachermayer, Walter (2016) Strong supermartingales and limits of non-negative martingales. Annals of Probability, 44 (1). pp. 171-205. ISSN 0091-1798

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Identification Number: 10.1214/14-AOP970

Abstract

Given a sequence (M n ) ∞ n=1 (Mn)n=1∞ of nonnegative martingales starting at M n 0 =1 M0n=1, we find a sequence of convex combinations (M ~ n ) ∞ n=1 (M~n)n=1∞ and a limiting process X X such that (M ~ n τ ) ∞ n=1 (M~τn)n=1∞ converges in probability to X τ Xτ, for all finite stopping times τ τ. The limiting process X 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 (Xn)n=1∞, their left limits (X n − ) ∞ n=1 (X−n)n=1∞ and their stochastic integrals (∫φdX n ) ∞ n=1 (∫φdXn)n=1∞ and explain the relation to the notion of the Fatou limit.

Item Type: Article
Official URL: http://www.imstat.org/aop/default.htm
Additional Information: © 2016 Institute of Mathematical Statistics
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 01 Sep 2015 13:33
Last Modified: 20 Oct 2019 02:17
Projects: FA506041
Funders: European Research Council
URI: http://eprints.lse.ac.uk/id/eprint/63365

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