Martingale (probability theory)
Generated by GPT-5-miniExpansion Funnel Raw 29 → Dedup 0 → NER 0 → Enqueued 0
| Martingale (probability theory) | |
|---|---|
.png) Thomas Steiner · CC BY-SA 3.0 · source | |
| Name | Martingale (probability theory) |
| Field | Probability theory |
| Introduced by | Paul Lévy |
| Introduced in | 1934 |
| Related | Stopping time, Filtration, Martingale convergence theorem |
Martingale (probability theory) A martingale is a class of stochastic process defined on a filtered probability space with a conditional expectation preserving property. Originating in work by Paul Lévy, formalized by Joseph L. Doob and developed in collaboration with results by Andrey Kolmogorov, Émile Borel, Albert Einstein and Norbert Wiener-era foundations, martingales underpin modern Kiyoshi Itō calculus, interactions with Ada Lovelace-era computation myths, and applications in the theories of Louis Bachelier, Robert Merton, Fischer Black, and Myron Scholes.
Definition and basic properties
A martingale is defined relative to a probability space together with a filtration and an integrable process such that conditional expectation given the past equals the present value, a property used by Joseph L. Doob in his work at University of Illinois Urbana–Champaign and in texts associated with Émile Borel and Andrey Kolmogorov. Key properties include preservation under taking conditional expectation, the tower property linked to Andrey Kolmogorov's extension theorem, and optional sampling relations studied by Paul Lévy and Joseph L. Doob. Martingales are often contrasted with submartingales and supermartingales in results developed by Per Martin-Löf and William Feller and are central to measure-theoretic probability approaches promoted by Andrey Kolmogorov and Norbert Wiener.
Examples and classes of martingales
Canonical examples include fair game models such as simple random walks studied by Andrey Kolmogorov and George Polya, discrete-time martingales arising in branching processes analyzed by John Kingman and Agnesi-era analogues, and continuous-time martingales like Brownian motion constructed by Norbert Wiener and further analyzed by Kiyoshi Itō and Paul Lévy. Martingales appear in Doob–Meyer decompositions associated with work by Joseph L. Doob and Claude Dellacherie, in local martingales central to Kiyoshi Itō theory and stochastic differential equations used by Anders Lindström-style researchers, and in square-integrable martingales connected to William Feller and Eugene Dynkin. Other classes include uniformly integrable martingales studied by Joseph L. Doob and bounded martingales featured in texts by William Feller and Persi Diaconis.
Convergence theorems and limit results
Fundamental convergence results include Doob's martingale convergence theorem proved by Joseph L. Doob, which builds on measure-theoretic foundations of Andrey Kolmogorov and uses techniques related to results by Paul Lévy and William Feller. Lp-convergence and almost-sure convergence for uniformly integrable martingales connect to work by Joseph L. Doob and Persi Diaconis, while the martingale central limit theorem has links to classical limit theorems of Andrey Kolmogorov and Paul Lévy. The optional sampling theorem and maximal inequalities of Joseph L. Doob yield uniform integrability criteria also explored by Claude Dellacherie and Paul-André Meyer.
Optional stopping and Doob's inequalities
The optional stopping theorem, formalized by Joseph L. Doob and influenced by stochastic stopping ideas traceable to Paul Lévy and Andrey Kolmogorov, gives conditions under which expectations are preserved at stopping times studied by Doob and Paul-André Meyer. Doob's maximal inequalities, central to martingale theory and developed by Joseph L. Doob, are applied in proofs by William Feller and in the theory of stochastic processes by Kiyoshi Itō and Paul-André Meyer. These results interact with predictable sigma-algebras and stopping time sigma-fields treated in the works of Claude Dellacherie and Paul-André Meyer.
Applications in stochastic processes and finance
Martingale methods are foundational in option pricing theories of Fischer Black, Myron Scholes, and Robert Merton, where equivalent martingale measures arise from the Girsanov theorem connected to Kiyoshi Itō calculus and work by Igor Girsanov. Martingales are used in the theory of Markov processes developed by Eugene Dynkin and Joseph L. Doob, in branching processes and coalescent theory influenced by John Kingman, and in sequential analysis and hypothesis testing influenced by Abraham Wald and Jerzy Neyman. Financial mathematics institutions such as Chicago Board Options Exchange and research by Fischer Black and Myron Scholes draw heavily on martingale pricing approaches.
Extensions and related concepts
Extensions encompass semimartingales central to the formulation of stochastic integration by Kiyoshi Itō and Paul-André Meyer, local martingales analyzed by Paul-André Meyer and Claude Dellacherie, and square-integrable martingales used in predictable representation theorems related to Eugene Dynkin and Kiyoshi Itō. Related concepts include the Doob–Meyer decomposition tied to Joseph L. Doob and Paul-André Meyer, the Girsanov theorem developed in the school of Kiyoshi Itō and Igor Girsanov, and applications in modern mathematical finance associated with Fischer Black, Myron Scholes, and Robert Merton.