Abstract
Let (M t )0≤t≤1 be any martingale with initial law M 0 ~ μ0 and terminal law M 1 ~ μ1 and let S≡sup0≤t≤1 M t . Then there is an upper bound, with respect to stochastic ordering of probability measures, on the law of S.
An explicit description of the upper bound is given, along with a martingale whose maximum attains the upper bound.
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Hobson, D.G. (1998). The maximum maximum of a martingale. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101762
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DOI: https://doi.org/10.1007/BFb0101762
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