Strong supermartingales and limits of nonnegative martingales
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Czichowsky, Christoph [2] ; Schachermayer, Walter [1]
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[1] University of Vienna
University of Vienna
Innere Stadt, Austria
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[2] london School of Economics and Political Sciences
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 1, 2016, págs. 171-205
- Idioma: inglés
- DOI: 10.1214/14-AOP970
- Enlaces
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Resumen
Given a sequence (Mn)∞n=1 of nonnegative martingales starting at Mn0=1, we find a sequence of convex combinations (M~n)∞n=1 and a limiting process X such that (M~nτ)∞n=1 converges in probability to Xτ, for all finite stopping times τ. 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 (Xn)∞n=1, their left limits (Xn−)∞n=1 and their stochastic integrals (∫φdXn)∞n=1 and explain the relation to the notion of the Fatou limit.
