Strong supermartingales and limits of non-negative martingales

Czichowsky, Christoph ORCID: 0000-0002-3513-6843 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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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
Date Deposited:	01 Sep 2015 13:33
Last Modified:	28 Nov 2024 18:03
Projects:	FA506041
Funders:	European Research Council
URI:	http://eprints.lse.ac.uk/id/eprint/63365

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