ON THE EXPECTED DIAMETER OF AN L2-BOUNDED MARTINGALE

• 成果类型: Article
• 署名作者: Dubins, Lester E.; Gilat, David; Meilijson, Isaac
• 署名单位: University of California System; University of California Berkeley; Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP406
• 发表日期: 2009
• 页码: 393-402
• 关键词: brownian-motion

摘要

It is shown that the ratio between the expected diameter of an L-2-bounded martingale and the standard deviation of its last term cannot exceed root 3. Moreover, I one-parameter family of stopping times on standard Brownian motion is exhibited, for which the root 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 root 2 times the standard deviation of the last term of the martingale. These results complement the Dubins and Schwarz respective bounds 1 and root 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.