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

This article is cited in 1 scientific paper (total in 1 paper)

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:

Received: 31.03.2005

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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\paper On one extension of a martingale
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\pages 763--767
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\zmath{https://zbmath.org/?q=an:1128.60033}
\elib{http://elibrary.ru/item.asp?id=9157512}
\transl
\jour Theory Probab. Appl.
\yr 2006
\vol 50
\issue 2
\pages 659--662
\crossref{https://doi.org/10.1137/S0040585X97982037}
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  • https://doi.org/10.4213/tvp129
  • http://mi.mathnet.ru/eng/tvp/v50/i4/p763

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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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