Foundations of Modern Probability pp 119-139 | Cite as

# Martingales and Optional Times

Chapter

## Abstract

Filtrations and optional times; random time-change; martingale property; optional stopping and sampling; maximum and upcrossing inequalities; martingale convergence, regularity, and closure; limits of conditional expectations; regularization of submartingales

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## References

- Martingales were first introduced by Bernstein (1927, 1937) in his efforts to relax the independence assumption in the classical limit theorems. Both Bernstein and Levy (1935a-b, 1954) extended Kolmogorov’s maximum inequality and the central limit theorem to a general martingale context. The
*term*martingale (originally denoting part of a horse’s harness and later used for a special gambling system) was introduced in the probabilistic context by Ville (1939).Google Scholar - The first martingale convergence theorem was obtained by Jessen (1934) and Lévy (1935b), both of whom proved Theorem 7.23 for filtrations generated by sequences of independent random variables. A submartin-gale version of the same result appears in Sparre- Andersen and Jessen (1948). The independence assumption was removed by Levy (1954), who also noted the simple martingale proof of Kolmogorov’s zero-one law and obtained his conditional version of the Borel-Cantelli lemma.Google Scholar
- The general convergence theorem for discrete-time martingales was proved by Doob (1940), and the basic regularity theorems for continuous-time martingales first appeared in Doob (1951). The theory was extended to submartingales by Snell (1952) and Doob (1953). The latter book is also the original source of such fundamental results as the martingale closure theorem, the optional sampling theorem, and the
*L*^{p}-inequality.Google Scholar - Though hitting times have long been used informally, general optional times seem to appear for the first time in Doob (1936). Abstract filtrations were not introduced until Doob (1953). Progressive processes were introduced by Dynkin (1961), and the modern definition of the σ-fields
*F*_{τ}is due to Yushkevich.Google Scholar - Elementary introductions to martingale theory are given by many authors, including Williams (1991). More information about the discrete-time case is given by Neveu (1975) and Chow and Teicher (1997). For a detailed account of the continuous-time theory and its relations to Markov processes and stochastic calculus, see Dellacherie and Meyer (1975–87).Google Scholar

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© Springer Science+Business Media New York 2002