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

Alexander M. G. Cox; Jan Obłój

doi:10.1016/j.spa.2015.03.005

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arXiv · 1406.0885

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

Alexander M. G. Cox·Jan Obłój

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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.

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Alexander M. G. Cox, Jan Obłój. 2015-03-28. On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale. https://doi.org/10.1016/j.spa.2015.03.005

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