On the expected diameter of an L2-bounded martingale

Abstract

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.