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Israel Journal of Mathematics

, Volume 69, Issue 2, pp 173–192 | Cite as

Martingales with given maxima and terminal distributions

  • Robert P. Kertz
  • Uwe Rösler
Article

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.

Keywords

Brownian Motion Probability Measure Induction Hypothesis Probability Space Weak Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 1990

Authors and Affiliations

  • Robert P. Kertz
    • 1
  • Uwe Rösler
    • 2
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Institut für Mathematische Statistik und WirtschaftsmathematikUniversität GöttingenGöttingenFRG

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