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

Alexander M. G. Cox, Jan Obłój

OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/fac1bdf9-f581-4606-882e-a6eacfd6bb0f • DOI: https://doi.org/10.1016/j.spa.2015.03.005

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.