Proceedings of the Steklov Institute of Mathematics

, Volume 287, Issue 1, pp 155–173 | Cite as

Sharp maximal inequalities for stochastic processes

  • Ya. A. LyulkoEmail author
  • A. N. Shiryaev


This work is a survey of existing methods and results in the problem of estimating the mathematical expectation of the maximum of a random process up to an arbitrary Markov time. Both continuous-time (standard Brownian motion, skew Brownian motion, Bessel processes) and discrete-time (symmetric Bernoulli random walk and its modulus) processes are considered.


Brownian Motion Random Walk Function Versus STEKLOV Institute Mathematical Expectation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland, Amsterdam, 1981; Nauka, Moscow, 1986).zbMATHGoogle Scholar
  2. 2.
    L. E. Dubins, L. A. Shepp, and A. N. Shiryaev, “Optimal stopping rules and maximal inequalities for Bessel processes,” Teor. Veroyatn. Primen. 38(2), 288–330 (1993) [Theory Probab. Appl. 38, 226–261 (1994)].zbMATHMathSciNetGoogle Scholar
  3. 3.
    K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths (Springer, Berlin, 1965; Mir, Moscow, 1965).CrossRefzbMATHGoogle Scholar
  4. 4.
    Ya. A. Lyul’ko, “Exact inequalities for the maximum of a skew Brownian motion,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 4, 26–31 (2012) [Moscow Univ. Math. Bull. 67, 164–169 (2012)].Google Scholar
  5. 5.
    A. S. Mishchenko, “A discrete Bessel process and its properties,” Teor. Veroyatn. Primen. 50(4), 797–806 (2005) [Theory Probab. Appl. 50, 700–709 (2006)].CrossRefMathSciNetGoogle Scholar
  6. 6.
    D. L. Burkholder, B. J. Davis, and R. F. Gundy, “Integral inequalities for convex functions of operators on martingales,” in Proc. 6th Berkeley Symp. Math. Stat. Probab., Univ. California, 1970 (Univ. Calif. Press, Berkeley, CA, 1972), Vol. 2, pp. 223–240.Google Scholar
  7. 7.
    D. L. Burkholder and R. F. Gundy, “Extrapolation and interpolation of quasi-linear operators on martingales,” Acta Math. 124, 249–304 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    L. E. Dubins and L. J. Savage, Inequalities for Stochastic Processes: How to Gamble If You Must (Dover, New York, 1976).zbMATHGoogle Scholar
  9. 9.
    L. E. Dubins and G. Schwarz, “A sharp inequality for sub-martingales and stopping times,” in Colloque Paul Lévy sur les processus stochastiques 1987 (Soc. Math. France, Paris, 1988), Astérisque 157–158, pp. 129–145.Google Scholar
  10. 10.
    T. Fujita, “A random walk analogue of Lévy’s theorem,” Stud. Sci. Math. Hung. 45(2), 223–233 (2008).zbMATHGoogle Scholar
  11. 11.
    S. E. Graversen and G. Peškir, “On Wald-type optimal stopping for Brownian motion,” J. Appl. Probab. 34, 66–73 (1997).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    S. E. Graversen and G. Peškir, “On Doob’s maximal inequality for Brownian motion,” Stoch. Processes Appl. 69, 111–125 (1997).CrossRefzbMATHGoogle Scholar
  13. 13.
    S. E. Graversen and G. Peškir, “Maximal inequalities for Bessel processes,” J. Inequal. Appl. 2, 99–119 (1998).zbMATHMathSciNetGoogle Scholar
  14. 14.
    S. E. Graversen and G. Peškir, “Optimal stopping and maximal inequalities for geometric Brownian motion,” J. Appl. Probab. 35, 856–872 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    S. E. Graversen and G. Peškir, “Maximal inequalities for the Ornstein-Uhlenbeck process,” Proc. Am. Math. Soc. 128(10), 3035–3041 (2000).CrossRefzbMATHGoogle Scholar
  16. 16.
    J. M. Harrison and L. A. Shepp, “On skew Brownian motion,” Ann. Probab. 9(2), 309–313 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    S. D. Jacka, “Optimal stopping and best constants for Doob-like inequalities. I: The case p = 1,” Ann. Probab. 19(4), 1798–1821 (1991).CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    A. Lejay, “On the constructions of the skew Brownian motion,” Probab. Surv. 3, 413–466 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    H. P. McKean, Jr., “The Bessel motion and a singular integral equation,” Mem. Coll. Sci., Univ. Kyoto., Ser. A: Math. 33(2), 317–322 (1960).zbMATHMathSciNetGoogle Scholar
  20. 20.
    G. Peškir, “A change-of-variable formula with local time on curves,” J. Theor. Probab. 18(3), 499–535 (2005).CrossRefzbMATHGoogle Scholar
  21. 21.
    G. Peškir and A. N. Shiryaev, “Maximal inequalities for reflected Brownian motion with drift,” Theory Probab. Math. Stat. 63, 137–143 (2001).Google Scholar
  22. 22.
    G. Peškir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems (Birkhäuser, Basel, 2006).zbMATHGoogle Scholar
  23. 23.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. (Springer, Berlin, 1999).CrossRefzbMATHGoogle Scholar
  24. 24.
    J. B. Walsh, “A diffusion with a discontinuous local time,” in Temps locaux. Exposés du séminaire J. Azema-M. Yor (1976–1977) (Soc. Math. France, Paris, 1978), Astérisque 52–53, pp. 37–45.Google Scholar
  25. 25.
    G. Wang, “Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion,” Proc. Am. Math. Soc. 112(2), 579–586 (1991).CrossRefzbMATHGoogle Scholar
  26. 26.
    M. V. Zhitlukhin, “A maximal inequality for skew Brownian motion,” Stat. Decis. 27, 261–280 (2009).zbMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.International Laboratory of Quantitative FinanceNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations