On the expected diameter of an L2-bounded martingale

TL;DR

This paper establishes upper bounds on the expected diameter of L2-bounded martingales relative to the standard deviation of their last term, introduces optimal stopping times, and compares these bounds with related martingale measures.

Contribution

It provides new bounds for the expected diameter of martingales, constructs optimal stopping times achieving these bounds, and relates diameter to maximal drawdown, extending classical martingale inequalities.

Findings

Expected diameter ratio cannot exceed sqrt(3).

Constructed stopping times attain the sqrt(3) bound.

Expected maximal drawdown is bounded by sqrt(2) times the last term's standard deviation.

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 c t. 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.