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

This article is cited in 15 scientific papers (total in 15 papers)

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
\begin{equation} \liminf_{t\rightarrow \infty}(\mathbb P\{\tau >t\}\sqrt{t}) \ge \sqrt{\frac 2\pi }|\mathbb E W_\tau | . \end{equation}
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.


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

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Received: 08.04.1996

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
\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
\jour Theory Probab. Appl.
\yr 1997
\vol 41
\issue 4
\pages 716--729

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    This publication is cited in the following articles:
    1. Vondracek Z., “Asymptotics of first–passage time over a one–sided stochastic boundary”, Journal of Theoretical Probability, 13:1 (2000), 279–309  crossref  mathscinet  zmath  isi  scopus
    2. Abundo M., “On the first–passage time of a diffusion process over a one–sided stochastic boundary”, Stochastic Analysis and Applications, 21:1 (2003), 1–23  crossref  mathscinet  zmath  isi  scopus
    3. Abundo M., “Limit at zero of the first–passage time density and the inverse problem for one–dimensional diffusions”, Stochastic Analysis and Applications, 24:6 (2006), 1119–1145  crossref  mathscinet  zmath  isi  scopus
    4. Liptser R., Novikov A., “Tail distributions of supremum and quadratic variation of local martingales”, From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 2006, 421–432  crossref  mathscinet  zmath  isi
    5. Kaji Sh., “The tail estimation of the quadratic variation of a quasi left continuous local martingale”, Osaka Journal of Mathematics, 44:4 (2007), 893–907  mathscinet  zmath  isi
    6. Doney R., Maller R., “Curve crossing for random walks reflected at their maximum”, Annals of Probability, 35:4 (2007), 1351–1373  crossref  mathscinet  zmath  isi  scopus
    7. Kaji Sh., “On the tail distributions of the supremum and the quadratic variation of a cadlag local martingale”, Seminaire de Probabilites XLI, Lecture Notes in Mathematics, 1934, 2008, 401–420  crossref  mathscinet  zmath  isi  scopus
    8. Kaji Sh., “The quadratic variations of local martingales and the first–passage times of stochastic integrals”, Journal of Mathematics of Kyoto University, 49:3 (2009), 491–502  crossref  mathscinet  zmath  isi  scopus
    9. Hulley H., Platen E., “A Visual Criterion for Identifying Ito Diffusions as Martingales or Strict Local Martingales”, Seminar on Stochastic Analysis, Random Fields and Applications, Progress in Probability, 63, 2011, 147–157  mathscinet  zmath  isi
    10. Aurzada F. Kramm T. Savov M., “First Passage Times of Levy Processes Over a One-Sided Moving Boundary”, Markov Process. Relat. Fields, 21:1 (2015), 1–38  mathscinet  zmath  isi  elib
    11. Theory Probab. Appl., 61:1 (2017), 140–151  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. Aurzada F. Kramm T., “The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes”, J. Theor. Probab., 29:3 (2016), 737–760  crossref  mathscinet  zmath  isi  elib  scopus
    13. Novikov A., Kaji Sh., “On Distibutions of First Passage Times of Martingales Arising in Some Gambling Problems”, Jpn. J. Ind. Appl. Math., 34:3, SI (2017), 859–871  crossref  mathscinet  zmath  isi  scopus
    14. Denisov D. Sakhanenko A. Wachtel V., “First-Passage Times For Random Walks With Nonidentically Distributed Increments”, Ann. Probab., 46:6 (2018), 3313–3350  crossref  mathscinet  zmath  isi  scopus
    15. Hulley H. Ruf J., “Weak Tail Conditions For Local Martingales”, Ann. Probab., 47:3 (2019), 1811–1825  crossref  mathscinet  isi  scopus
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