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Martingales with given maxima and terminal distributions

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Abstract

Let μ be any probability measure onR with λ |x|dμ(x)<∞, and let μ* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ<ν<μ* in the usual stochastic order, there is a martingale (X t)0≦t≦1 for which sup0≦t≦1 X t andX 1 have respective p.m. 'sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    Robert P. Kertz

  2. Institut für Mathematische Statistik und Wirtschaftsmathematik, Universität Göttingen, Lotzestrasse 13, 3400, Göttingen, FRG

    Uwe Rösler

Authors
  1. Robert P. Kertz
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  2. Uwe Rösler
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Supported in part by NSF grants DMS-86-01153 and DMS-88-01818.

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Kertz, R.P., Rösler, U. Martingales with given maxima and terminal distributions. Israel J. Math. 69, 173–192 (1990). https://doi.org/10.1007/BF02937303

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  • DOI: https://doi.org/10.1007/BF02937303

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