Geodesic
On one extension of a martingale
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 763-767 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {On one extension of a martingale},
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     url = {http://geodesic.mathdoc.fr/item/TVP_2005_50_4_a6/}
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B. D. Gnedenko. On one extension of a martingale. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 4, pp. 763-767. http://geodesic.mathdoc.fr/item/TVP_2005_50_4_a6/

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