The Splendors and Miseries of Martingales: Their History from the Casino to Mathematics

Introduction
Contents
Part I In the Beginning
1 The Origin and Multiple Meanings of Martingale
	1 Introduction
	2 From Probability Back to Gambling
	3 Are Martingales Foolish?
	4 An Excursion Around Martigues
	5 Back to Harnesses
	6 The Ultimate Treachery of Martingales
2 Martingales at the Casino
	1 Prelude
	2 Introduction
	3 The Casino
		3.1 Trente et Quarante
		3.2 The Business Model
		3.3 The Paris Casinos
	4 Gamblers\' Fallacies
		4.1 Two Moralists
		4.2 The Blatant Rogue
		4.3 The Failed Mathematician
		4.4 The Many-Talented Gambler
	5 Betting Systems and Game Theory
3 Émile Borel\'s Denumerable Martingales, 1909–1949
	1 Introduction
	2 Martingales of Fathers of Families
	3 Borel\'s Martingales
4 The Dawn of Martingale Convergence: Jessen\'s Theorem and Lévy\'s Lemma
	1 Introduction
	2 Jessen\'s Theorem
		2.1 Magister Thesis 1929
		2.2 Doctoral Thesis 1930
		2.3 The Acta Article 1934
		2.4 A Probabilistic Interlude 1934–1935
		2.5 After 1934
	3 Lévy\'s Lemma
		3.1 Before 1930
		3.2 Lévy\'s Denumerable Probabilities
*-20pt Part II Ville, Lévy and Doob
5 Did Jean Ville Invent Martingales?
	1 Introduction
	2 A Glimpse of Jean Ville
	3 Probability as Ville Encountered It in the Early 1930s
	4 Martingales in Probability Before Ville
	5 Combining Game Theory with Denumerable Probability
	6 Legacy
	7 A Final Question
6 Paul Lévy\'s Perspective on Jean Ville and Martingales
	1 Introduction
	2 Lévy and His Martingale Condition
		2.1 Lévy\'s Growing Interest in Probability
		2.2 Genesis of Lévy\'s Martingale Condition
		2.3 Chapter VIII of the Book Théorie de l\'addition des variables aléatoires
	3 Lévy Versus Ville
	4 Conclusion
7 Doob at Lyon: Bringing Martingales Back to France
	1 The Colloquium
	2 Paul Lévy
	3 Jean Ville
	4 Joseph Doob
	5 At the Colloquium
	6 Doob\'s Lecture
		6.1 Strong Law of Large Numbers
		6.2 Inverse Probability
*-20pt Part III Modern Probability
8 Stochastic Processes in the Decades after 1950
	1 Introduction
	2 Probability Around 1950
		2.1 Early Developments
		2.2 ``Stochastic Processes\'\'
	3 The Great Topics of the Years 1950–1965
		3.1 Markov Processes
		3.2 Development of Soviet Probability
		3.3 Classical Potential Theory and Probability
		3.4 Theory of Martingales
		3.5 Markov Processes and Potential
		3.6 Special Markov Processes
		3.7 Connections Between Markov Processes and Martingales
	4 The Period 1965–1980
		4.1 The Stochastic Integral
		4.2 Markov Processes
		4.3 General Theory of Processes
		4.4 Inequalities of Martingales and Analysis
		4.5 Martingale Problems
		4.6 ``Stochastic Mechanics\'\'
		4.7 Relations to Physics
	5 After 1980
		5.1 The ``Malliavin Calculus\'\'
		5.2 Stochastic Differential Geometry
		5.3 Distributions and White Noise
		5.4 Large Deviations
		5.5 Noncommutative Probability
		5.6 Omissions
9 Martingales in Japan
	1 Before 1960: Itô\'s Stochastic Analysis
	2 Japanese Contributions to Martingales from 1961 to 1970
		2.1 The Doob–Meyer Decomposition Theorem for Supermartingales
		2.2 Stochastic Integrals for Square-Integrable Martingales and Semimartingales
		2.3 Martingale Representation Theorems
	3 Japanese Contributions to Martingales After 1971
		3.1 Fisk–Stratonovich Symmetric Stochastic Integrals. Itô\'s Circle Operation
		3.2 Itô–Tanaka\'s Formula and Local Times
		3.3 Problems Concerning Filtrations
10 My Encounters with Martingales
	1 Studying at the University of Berlin Right After the War
	2 Collecting Building Blocks for Martingale Theory
	3 A Year in Illinois
	4 Final Work Till 1964
*-20pt Part IV Modern Applications
11 Martingales in the Study  of Randomness
	1 Introduction
	2 Richard von Mises\'s Collectives
	3 Abraham Wald\'s Clarification
	4 Jean Ville\'s Martingales
	5 The Status Quo of the 1950s
	6 The Invention of the Algorithmic Definition  of Randomness in the 1960s
		6.1 Kolmogorov
		6.2 Solomonoff
		6.3 Chaitin
	7 Martin-Löf\'s Definition of Randomness
	8 Claus-Peter Schnorr\'s Computable Martingales
	9 Leonid Levin\'s Semimeasures
	10 Characterizing Martin-Löf Randomness Using Complexity
		10.1 Leonid Levin in the Soviet Union
		10.2 Monotone Complexity: Levin and Schnorr
		10.3 Prefix Complexity
	11 After the 1970s
12 Encounters with Martingales in Statistics and Stochastic Optimization
	1 Introduction
	2 Setting the Stage
		2.1 Harold Hotelling
		2.2 Abraham Wald
		2.3 Herbert Robbins
	3 Sequential Testing and Confidence Intervals
		3.1 Wald\'s Seminal Work During the Second World War
		3.2 Sequential Tests with Power 1 and Confidence Sequences
		3.3 BHAT and Time-Sequential Survival Analysis
	4 Martingales in Sequential Design of Experiments  and Bandit Problems
	5 Stochastic Approximation (SA) and Adaptive SA
	6 Martingales and Biorhythms in Time Series
	7 Martingales in Stochastic Optimization, 1987–2021
		7.1 Contextual Bandits in Reinforcement Learning and Personalization, Modified Gradient Boosting and SA in AI
		7.2 Joint State and Parameter Estimation in Hidden Markov Models, with Uncertainty Quantification
	8 Concluding Remarks
13 Martingales in Survival Analysis
	1 Introduction
	2 The Hazard Rate and a Martingale Estimator
	3 Stochastic Integration and Statistical Estimation
	4 Stopping Times, Unbiasedness and Independent Censoring
	5 Martingale Central Limit Theorems
	6 Two-Sample Tests for Counting Processes
	7 The Copenhagen Environment
	8 From Kaplan–Meier to the Empirical Transition Matrix
	9 Pustulosis Palmo-Plantaris and ps: [/EMC pdfmark [/Subtype /Span /ActualText (k) /StPNE pdfmark [/StBMC pdfmarkkps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Sample Tests
	10 The Cox Model
	11 The Monograph Statistical Models Based on Counting Processes
	12 Limitations of Martingales
14 Encounters with Martingales in Stochastic Control
	1 Introduction
	2 Frequency Domain Methods for Control and Estimation
	3 Time Domain Methods for Control and Estimation
	4 Nonlinear Stochastic Control
	5 Some Other Related Stochastic Optimization Problems
	6 Appendix (by Laurent Mazliak): Martingale Problems and Stochastic Control of General Processes
		6.1 Strong and Weak Solutions of Stochastic Differential Equations. Martingale Problems
		6.2 General Formulation of a Control Problem
*-20pt Part V Documents
15 Analysis or Probability? Eight Letters Between Børge Jessen and Paul Lévy
	1 Introduction
	2 Lévy to Jessen. Paris, 27 September 1934
	3 Lévy to Jessen. Paris, 4 April 1935
	4 Jessen to Lévy. Undated Draft, About 8 April 1935
	5 Lévy to Jessen. Hennequeville, 24 April 1935
	6 Lévy to Jessen. Paris, 3 May 1935
	7 Jessen to Lévy. Copenhagen, 11 August 1935
	8 Lévy to Jessen. S. Cristina, 23 August 1935
	9 Bohr and Jessen to Lévy. Copenhagen, 14 July 1947
16 Counterexamples to Abstract Probability: Ten Letters by Jessen, Doob and Dieudonné
	1 Introduction
	2 Jessen to Doob, 11 May 1948
	3 Doob to Jessen, 17 May 1948
	4 Jessen to Doob, 29 May 1948
	5 Doob to Jessen, 4 June 1948
	6 Jessen to Dieudonné, 17 June 1948
	7 Dieudonné to Jessen, Nancy, 28 June 1948
	8 Jessen to Dieudonné, 13 September 1948
	9 Jessen to Doob, 13 September 1948
	10 Jessen to Doob, 17 May 1949
	11 Jessen to Doob, 23 June 1949
17 Jean Ville Remembers Martingales
	1 Introduction
	2 Letter from Crépel to Ville, 22 August 1984
	3 Crépel\'s Interview of Ville, 27 August 1984
		3.1 Mathematics in France in the 1930s
		3.2 Vienna and Karl Menger
		3.3 Random Sequences and Martingales
		3.4 Probability Back in France
		3.5 Other Aspects of Probability
		3.6 Economics
		3.7 Computing at the University of Paris
	4 Letter from Crépel to Ville, 21 January 1985
	5 Letter from Ville to Crépel, 2 February 1985
		5.1 First Note
		5.2 Second Note
		5.3 Third Note
18 Seven Letters from Paul Lévy to Maurice Fréchet
	1 Introduction
19 Andrei Kolmogorov and Leonid Levin on Randomness
	1 Introduction
	2 Letter from Kolmogorov to Fréchet, 1939
	3 Abstracts of Three Talks by Kolmogorov, 1967–1974
		3.1 31 October 1967
		3.2 23 November 1971
		3.3 16 April 1974
	4 Three Letters from Levin to Kolmogorov 1970–1971
		4.1 Letter I
		4.2 Letter II
		4.3 Letter III
Index