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

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  • Cox, Alexander M.G.
  • Obłój, Jan

Abstract

We study the joint laws of the maximum and minimum of a continuous, uniformly integrable martingale. 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.

Suggested Citation

  • Cox, Alexander M.G. & Obłój, Jan, 2015. "On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3280-3300.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:3280-3300
    DOI: 10.1016/j.spa.2015.03.005
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