A maximal inequality for supermartingales

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A tight upper bound is given on the distribution of the maximum of a supermartin-gale. Specifically, it is shown that if Y is a semimartingale with initial value zero and quadratic variation process [Y, Y] such that Y + [Y,Y] is a supermartingale, then the probability the maximum of Y is greater than or equal to a positive constant a is less than or equal to 1/(1 + a). The proof makes use of the semimartingale calculus and is inspired by dynamic programming.

Original languageEnglish (US)
Article number55
JournalElectronic Communications in Probability
StatePublished - Aug 14 2014


  • Drift
  • Martingale
  • Maximal inequality
  • Semimartingale calculus

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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