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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 763–767 (Mi tvp129)

Short Communications

On one extension of a martingale

B. D. Gnedenko

M. V. Lomonosov Moscow State University

Abstract: In this paper we introduce an $\varepsilon$-martingale and a strong $\varepsilon$-martingale. The first is defined by the inequality $|\mathbf{E}(X_t | \mathcal{F}_s)- X_s|\leq\varepsilon$, and the second one can be obtained from the $\varepsilon$-martingale by replacing in the definition fixed time moments with stopping times. The paper proves that a right-continuous $\varepsilon$-martingale is a strong $2\varepsilon$-martingale. At the same time we construct an example of a right-continuous $\varepsilon$-martingale which is not a strong $\varepsilon$-martingale for any $a<2$. We show that the dependence between $\varepsilon$-martingales and strong $\varepsilon$-martingales has no analogues for $\varepsilon$-submartingales. We also give the criterion for testing if a right-continuous with left limits process is a strong $\varepsilon$-martingale or not. The criterion is based on the possibility of uniform approximation of the process by a martingale with precision $\varepsilon/2$.

Keywords: $\varepsilon$-martingale, strong $\varepsilon$-martingale, Doob's stopping time theorem.

DOI: https://doi.org/10.4213/tvp129

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English version:
Theory of Probability and its Applications, 2006, 50:2, 659–662

Bibliographic databases:

Citation: B. D. Gnedenko, “On one extension of a martingale”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 763–767; Theory Probab. Appl., 50:2 (2006), 659–662

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp129
• https://doi.org/10.4213/tvp129
• http://mi.mathnet.ru/eng/tvp/v50/i4/p763

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This publication is cited in the following articles:
1. Bondarev B.V., Kozyr' S. M., “On the $\varepsilon$-sufficient control in a Merton problem with “physical” white noise”, Ukrainian Math. J., 61:8 (2009), 1215–1232
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