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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3 Citations (Scopus)
130 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)3280–3300
Number of pages21
JournalStochastic Processes and their Applications
Volume125
Issue number8
Early online date24 Mar 2015
DOIs
Publication statusPublished - Aug 2015

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