Doob's martingale inequality

In mathematics, Doob's martingale inequality, also known as Kolmogorov's submartingale inequality, is a fundamental result in the study of stochastic processes.

Key aspects of the inequality include:

  • It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
  • By bounding the running maximum of a stochastic process using only its terminal expectation, it provides a powerful tool for analyzing the extreme behaviors of sample paths.
  • As the name suggests, the result is usually given in the case that the process is a martingale, but the core mathematics inherently apply to submartingales.
  • The inequality is due to the American mathematician Joseph L. Doob.
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