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 Teor. Veroyatnost. i Primenen., 1996, Volume 41, Issue 4, Pages 810–826 (Mi tvp3203)

Martingales, Tauberian theorem, and strategies of gambling

A. A. Novikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Using the Tauberian theorem, we get an asymptotic relation between the tail of the distribution of the quadratic characteristic of a martingale and the expectation of its terminal value. In case of continuous martingales the following result is proven: if $\tau$ is a stopping time for a standard Wiener process Wt with integrable terminal value $W_\tau$, then
$$\liminf_{t\rightarrow \infty}(\mathbb P\{\tau >t\}\sqrt{t}) \ge \sqrt{\frac 2\pi }|\mathbb E W_\tau | .$$
Using a related result for discrete time martingales, we study asymptotic characteristics of some strategies of gambling and, in particular, Oscar's strategy.

Keywords: optimal stopping, local martingales, Wald equation, uniform integrability, sharp inequalities, gambling strategies, boundary crossing problem.

DOI: https://doi.org/10.4213/tvp3203

Full text: PDF file (766 kB)

English version:
Theory of Probability and its Applications, 1997, 41:4, 716–729

Bibliographic databases:

Citation: A. A. Novikov, “Martingales, Tauberian theorem, and strategies of gambling”, Teor. Veroyatnost. i Primenen., 41:4 (1996), 810–826; Theory Probab. Appl., 41:4 (1997), 716–729

Citation in format AMSBIB
\Bibitem{Nov96} \by A.~A.~Novikov \paper Martingales, Tauberian theorem, and strategies of gambling \jour Teor. Veroyatnost. i Primenen. \yr 1996 \vol 41 \issue 4 \pages 810--826 \mathnet{http://mi.mathnet.ru/tvp3203} \crossref{https://doi.org/10.4213/tvp3203} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1687109} \zmath{https://zbmath.org/?q=an:0895.60047} \transl \jour Theory Probab. Appl. \yr 1997 \vol 41 \issue 4 \pages 716--729 \crossref{https://doi.org/10.1137/S0040585X9797571X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000071926900008} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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