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The Splendors and Miseries of Martingales

Part of the Trends in the History of Science Book Series

Editors: Mazliak, Laurent; Shafer, Glenn

DOI: 10.1007/978-3-031-05988-9

ISBN: 978-3-031-05987-2

Publisher: Springer International Publishing - Springer Journals

Oct 18, 2022

1 - 10 of 19 Chapters

LitStream

The Origin and Multiple Meanings of Martingale

Mansuy, Roger

Pages: 3 –14

[A martingale can be a mathematical object, a gambling strategy in a casino, part of a horse’s harness, part of a sailboat’s rigging, a man’s coat, or even a courtesan. The wide variety of objects described by the word is both intriguing and challenging. Lexicographers have advanced various...
LitStream

Martingales at the Casino

Shafer, Glenn

Pages: 15 –50

[For more than two centuries, various betting strategies have been called martingales. What makes a betting strategy a martingale? Why have martingales been so popular among gamblers and casinos? The decade after the French revolution marks the beginning of a written record that casts some light...
LitStream

Émile Borel’s Denumerable Martingales, 1909–1949

Bru, Marie-France; Bru, Bernard

Pages: 51 –65

[At infinity, a fair game can become unfair. This paradox was first fully understood and mastered in the 1940s by Émile Borel, who resolved it using the theory of denumerable probability he had introduced in 1909 and the theory of martingales developed by his student Jean VilleVille, Jean in the...
LitStream

The Dawn of Martingale Convergence: Jessen’s Theorem and Lévy’s Lemma

Eid, Salah

Pages: 67 –104

[Jessen’s theorem and Lévy’s lemma, which both date from 1934, are the earliest known general formulations of the martingale convergence theorem. Børge Jessen worked within Lebesgue’s theory of integration; he saw his theorem as an extension of the Fubini-Lebesgue theorem of 1907–1920. Paul...
LitStream

Did Jean Ville Invent Martingales?

Shafer, Glenn

Pages: 107 –122

[Jean Ville, we are sometimes told, brought martingales into mathematical probability. In what way is this true? To respond thoughtfully, we must try to see probability theory as Ville and his contemporaries saw it in the 1930s and excavate the martingales already hidden in that theory. Then we...
LitStream

Paul Lévy’s Perspective on Jean Ville and Martingales

Mazliak, Laurent

Pages: 123 –145

[As the first part of this chapter explains, Paul Lévy’s theory of martingales was about extending the law of large numbers and other theorems about sequences of independent random variables to dependent random variables, Lévy showed that this extension is possible when each random variable has...
LitStream

Doob at Lyon: Bringing Martingales Back to France

Locker, Bernard; Mazliak, Laurent

Pages: 147 –166

[The evolution of Joseph Leo Doob’s work on probability is examined from the vantage point of his lecture on applications of the theory of martingales at the colloquium on probability at Lyon in 1948. During the 1940s, Doob had developed the theory of stochastic processes in Kolmogorov’s...
LitStream

Stochastic Processes in the Decades after 1950

Meyer, Paul-André; Shafer, Glenn

Pages: 169 –202

[Paul-André Meyer (1934–2003), founder and leader of the Strasbourg school of probability, worked from the 1960s into the 1990s on the theory of stochastic processes, emphasizing processes in continuous time. In this sweeping review, which first appeared in French in 2000, Meyer emphasizes the...
LitStream

Martingales in Japan

Watanabe, Shinzo

Pages: 203 –214

[Japanese mathematicians did not contribute directly to martingale theory before 1960, but many later contributions were based on the stochastic calculus that Kiyosi Itô first introduced in 1942. Itô’s collaboration with Henry McKean on the pathwise construction of diffusions attracted wide...
LitStream

My Encounters with Martingales

Krickeberg, Klaus

Pages: 215 –221

[The text below describes the mathematical activities of Klaus Krickeberg during the period when he was working on martingales, along with many other different mathematical subjects. This period extends from 1946 to 1964. His main results about martingales (semi-martingales always admitted) may...

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