On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale
arXiv:1406.0885 · doi:10.1016/j.spa.2015.03.005
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 Azéma & Yor, Perkins, Jacka and Cox & Obłój, 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.
19 pages, 4 figures. This is the authors' accepted version of the paper which will appear in Stochastic Processes and their Applications