On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

摘要

We study the joint laws of the maximum and minimum of a continuous, uniformly integrable martingale. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Azema & Yor, Perkins, Jacka and Cox & Obloj, this allows us to completely characterise the upper and lower bounds on all possible exito-exit probabilities, subject to a given terminal law of the martingale. In addition, we determine some further properties of these bounds, considered as functions of the maximum and minimum. (C) 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.