Intro
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