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Teor. Veroyatnost. i Primenen., 1993, Volume 38, Issue 2, Pages 288–330 (Mi tvp3941)  

This article is cited in 18 scientific papers (total in 18 papers)

Optimal stopping rules and maximal inequalities for Bessel processes

L. E. Dubinsa, L. A. Sheppb, A. N. Shiryaevc

a Department of Mathematics, University of California, Berkeley, CA, USA
b AT&T Bell Laboratories, Murray Hill, New Jersey, USA
c Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider, for Bessel processes $X\in\operatorname{Bes}^\alpha(x)$ with arbitrary order (dimension) $\alpha \in \mathbf{R}$, the problem of the optimal stopping (1.4) for which the gain is determined by the value of the maximum of the process $X$ and the cost which is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure (Theorem 1) and the price (Theorem 2). These results are used for the proof of maximal inequalities of the type
$$ \mathbf{E}\max\limits_{r\le\tau}X_r\le\gamma(\alpha)\sqrt {\mathbf{E}\tau}, $$
where $X \in\operatorname{Bes}^\alpha(0)$, $\tau$ is arbitrary stopping time, $\gamma(\alpha)$ is a constant depending on the dimension (order) $\alpha$. It is shown that $\gamma(\alpha)\sim\sqrt\alpha$ at $\alpha\to\infty$.

Keywords: Bessel processes, optimal stopping rules, maximal inequalities, moving boundary problem for parabolic equations (Stephan problem), local martingales, semimartingales, Dirichlet processes, local time, processes with reflection, Brownian motion with drift and reflection.

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English version:
Theory of Probability and its Applications, 1993, 38:2, 226–261

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Received: 02.10.1992

Citation: L. E. Dubins, L. A. Shepp, A. N. Shiryaev, “Optimal stopping rules and maximal inequalities for Bessel processes”, Teor. Veroyatnost. i Primenen., 38:2 (1993), 288–330; Theory Probab. Appl., 38:2 (1993), 226–261

Citation in format AMSBIB
\by L.~E.~Dubins, L.~A.~Shepp, A.~N.~Shiryaev
\paper Optimal stopping rules and maximal inequalities for Bessel processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1993
\vol 38
\issue 2
\pages 288--330
\jour Theory Probab. Appl.
\yr 1993
\vol 38
\issue 2
\pages 226--261

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    1. M. Jeanblanc-Picqué, A. N. Shiryaev, “Optimization of the flow of dividends”, Russian Math. Surveys, 50:2 (1995), 257–277  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. G. Peškir, A. N. Shiryaev, “The Khintchine inequalities and martingale expanding sphere of their action”, Russian Math. Surveys, 50:5 (1995), 849–904  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. Peskir G., “Optimal stopping of the maximum process: The maximality principle”, Annals of Probability, 26:4 (1998), 1614–1640  crossref  mathscinet  zmath  isi
    4. M. A. Urusov, A. S. Cherny, “Separating times for measures on filtered spaces”, Theory Probab. Appl., 48:2 (2004), 337–347  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Obloj J., Yor M., “On local martingale and its supremum: Harmonic functions and beyond”, From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 2006, 517–533  isi
    6. Cherny A., Urusov M., “On the absolute continuity and singularity of measures on filtered spaces: Separating times”, From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 2006, 125–168  isi
    7. M. V. Zhitlukhin, “A maximal inequality for skew Brownian motion”, Russian Math. Surveys, 64:5 (2009), 958–959  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. M. V. Zhitlukhin, A. A. Muravlev, “On Chernoffs hypotheses testing problem for the drift of a Brownian motion”, Theory Probab. Appl., 57:4 (2013), 708–717  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. Ya. A. Lyulko, “Exact inequalities for the maximum of a skew Brownian motion”, Moscow University Mathematics Bulletin, 67:4 (2012), 164–169  mathnet  crossref  mathscinet
    10. Glover K., Hulley H., Peskir G., “Three-Dimensional Brownian Motion and the Golden Ratio Rule”, Ann. Appl. Probab., 23:3 (2013), 895–922  crossref  isi
    11. Ya. A. Lyulko, A. N. Shiryaev, “Sharp maximal inequalities for stochastic processes”, Proc. Steklov Inst. Math., 287:1 (2014), 155–173  mathnet  crossref  crossref  isi  elib  elib
    12. Peskir G., “Quickest Detection of a Hidden Target and Extremal Surfaces”, Ann. Appl. Probab., 24:6 (2014), 2340–2370  crossref  isi
    13. Gapeev P.V. Rodosthenous N., “Optimal Stopping Problems in Diffusion-Type Models With Running Maxima and Drawdowns”, J. Appl. Probab., 51:3 (2014), 799–817  isi
    14. Gapeev P.V., Rodosthenous N., “on the Drawdowns and Drawups in Diffusion-Type Models With Running Maxima and Minima”, J. Math. Anal. Appl., 434:1 (2016), 413–431  crossref  isi
    15. Theory Probab. Appl., 61:1 (2017), 159–167  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. Gapeev P.V. Rodosthenous N., “Perpetual American options in diffusion-type models with running maxima and drawdowns”, Stoch. Process. Their Appl., 126:7 (2016), 2038–2061  crossref  mathscinet  zmath  isi  elib  scopus
    17. Gapeev V P. Rodosthenous N. Chinthalapati V.L.R., “On the Laplace Transforms of the First Hitting Times For Drawdowns and Drawups of Diffusion-Type Processes”, Risks, 7:3 (2019), 87  crossref  isi
    18. Gapeev V P., “Optimal Stopping Problems For Running Minima With Positive Discounting Rates”, Stat. Probab. Lett., 167 (2020), 108899  crossref  isi
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