Annals of Probability

On the expected diameter of an L2-bounded martingale

Lester E. Dubins, David Gilat, and Isaac Meilijson

Full-text: Open access


It is shown that the ratio between the expected diameter of an L2-bounded martingale and the standard deviation of its last term cannot exceed $\sqrt{3}$. Moreover, a one-parameter family of stopping times on standard Brownian motion is exhibited, for which the $\sqrt{3}$ upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost ct. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be bounded by $\sqrt{2}$ times the standard deviation of the last term of the martingale. These results complement the Dubins and Schwarz respective bounds 1 and $\sqrt{2}$ for the ratios between the expected maximum and maximal absolute value of the martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality.

Article information

Ann. Probab., Volume 37, Number 1 (2009), 393-402.

First available in Project Euclid: 17 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Brownian motion gambling theory martingale optimal stopping


E. Dubins, Lester; Gilat, David; Meilijson, Isaac. On the expected diameter of an L 2 -bounded martingale. Ann. Probab. 37 (2009), no. 1, 393--402. doi:10.1214/08-AOP406.

Export citation


  • [1] Dana, R.-A. and Jeanblanc, M. (2003). Financial Markets in Continuous Time. Springer, Berlin.
  • [2] Dubins, L. E. and Savage, L. J. (1965). How to Gamble if You Must. Inequalities for Stochastic Processes. McGraw-Hill, New York.
  • [3] Dubins, L. E. and Schwarz, G. (1988). A sharp inequality for sub-martingales and stopping-times. Soc. Math. de France Astérisque 157/8 129–145.
  • [4] Freedman, D. (1971). Brownian Motion and Diffusion. Holden-Day, San Francisco, CA.
  • [5] Gilat, D. (1977). Every nonnegative submartingale is the absolute value of a martingale. Ann. Probab. 5 475–481.
  • [6] Gilat, D. (1988). On the ratio of the expected maximum of a martingale and the $-norm of its last term. Israel J. Math. 63 270–280.
  • [7] Imhof, J.-P. (1985). On the range of Brownian motion and its inverse process. Ann. Probab. 13 1011–1017.
  • [8] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [9] Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars and Cie, Paris.
  • [10] Monroe, I. (1972). On embedding right-continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293–1311.
  • [11] Pitman, J. (1996). Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299–329.
  • [12] Skorokhod, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA.