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

TL;DR

This paper characterizes the joint distribution of a continuous martingale's maximum, minimum, and terminal value, providing explicit bounds and solutions that are tight, thereby advancing the understanding of martingale exit probabilities.

Contribution

It introduces explicit, tight bounds on joint exit probabilities of martingales with given terminal laws and constructs novel solutions to the Skorokhod embedding problem.

Findings

Derived explicit bounds on exit probabilities of martingales.

Constructed novel solutions to the Skorokhod embedding problem.

Fully characterized bounds based on terminal law and properties of maximum and minimum.

Abstract

We study the joint laws of a continuous, uniformly integrable martingale, its maximum, and its minimum. 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 Az\'ema & Yor, Perkins, Jacka and Cox & Ob{\l}\'oj, this allows us to completely characterise the upper and lower bounds on all possible exit/no-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.