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January 2009 On the expected diameter of an L2-bounded martingale
Lester E. Dubins, David Gilat, Isaac Meilijson
Ann. Probab. 37(1): 393-402 (January 2009). DOI: 10.1214/08-AOP406


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.


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Lester E. Dubins. David Gilat. Isaac Meilijson. "On the expected diameter of an L2-bounded martingale." Ann. Probab. 37 (1) 393 - 402, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1159.60020
MathSciNet: MR2489169
Digital Object Identifier: 10.1214/08-AOP406

Primary: 60G44
Secondary: 60G40

Keywords: Brownian motion , gambling theory , martingale , Optimal stopping

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 1 • January 2009
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