Informationen zum Autor Glenn Shafer is University Professor at Rutgers University. Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London. Shafer and Vovk are the authors of Probability and Finance: It's Only a Game , published by Wiley and co-authors of Algorithmic Learning in a Random World . Shafer's other previous books include A Mathematical Theory of Evidence and The Art of Causal Conjecture. Klappentext Game-theoretic probability and finance come of ageGlenn Shafer and Vladimir Vovk's Probability and Finance, published in 2001, showed that perfect-information games can be used to define mathematical probability. Based on fifteen years of further research, Game-Theoretic Foundations for Probability and Finance presents a mature view of the foundational role game theory can play. Its account of probability theory opens the way to new methods of prediction and testing and makes many statistical methods more transparent and widely usable. Its contributions to finance theory include purely game-theoretic accounts of Ito's stochastic calculus, the capital asset pricing model, the equity premium, and portfolio theory.Game-Theoretic Foundations for Probability and Finance is a book of research. It is also a teaching resource. Each chapter is supplemented with carefully designed exercises and notes relating the new theory to its historical context.Praise from early readers"Ever since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure-theoretic. In this ground-breaking work, Shafer and Vovk give a game-theoretic foundation instead. While being just as rigorous, the game-theoretic approach allows for vast and useful generalizations of classical measure-theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades." - Peter Grünwald, CWI and University of Leiden"Shafer and Vovk have thoroughly re-written their 2001 book on the game-theoretic foundations for probability and for finance. They have included an account of the tremendous growth that has occurred since, in the game-theoretic and pathwise approaches to stochastic analysis and in their applications to continuous-time finance. This new book will undoubtedly spur a better understanding of the foundations of these very important fields, and we should all be grateful to its authors." - Ioannis Karatzas, Columbia University Zusammenfassung Earlier edition published in 2001 as: Probability and finance: it's only a game! Inhaltsverzeichnis Preface xi Acknowledgments xv Part I Examples in Discrete Time 1 1 Borel's Law of Large Numbers 5 1.1 A Protocol for Testing Forecasts 6 1.2 A Game-Theoretic Generalization of Borel's Theorem 8 1.3 Binary Outcomes 16 1.4 Slackenings and Supermartingales 18 1.5 Calibration 19 1.6 The Computation of Strategies 21 1.7 Exercises 21 1.8 Context 24 2 Bernoulli's and De Moivre's Theorems 31 2.1 Game-Theoretic Expected Value and Probability 33 2.2 Bernoulli's Theorem for Bounded Forecasting 37 2.3 A Central Limit Theorem 39 2.4 Global Upper Expected Values for Bounded Forecasting 45 2.5 Exercises 46 2.6 Context 49 3 Some Basic Supermartingales 55 3.1 Kolmogorov's Martingale 56 3.2 Doléans's Supermartingale 56 3.3 Hoeffding's Supermartingale 58 3.4 Bernstein's Supermartingale 63 3.5 Exercises 66 3.6 Context 67 4 Kolmogorov's Law of Large Numbers 69 4.1 Stating...

Autorentext

Glenn Shafer is University Professor at Rutgers University. Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London. Shafer and Vovk are the authors of Probability and Finance: It's Only a Game, published by Wiley and co-authors of Algorithmic Learning in a Random World. Shafer's other previous books include A Mathematical Theory of Evidence and The Art of Causal Conjecture.

Klappentext

Game-theoretic probability and finance come of age Glenn Shafer and Vladimir Vovk's Probability and Finance, published in 2001, showed that perfect-information games can be used to define mathematical probability. Based on fifteen years of further research, Game-Theoretic Foundations for Probability and Finance presents a mature view of the foundational role game theory can play. Its account of probability theory opens the way to new methods of prediction and testing and makes many statistical methods more transparent and widely usable. Its contributions to finance theory include purely game-theoretic accounts of Ito's stochastic calculus, the capital asset pricing model, the equity premium, and portfolio theory. Game-Theoretic Foundations for Probability and Finance is a book of research. It is also a teaching resource. Each chapter is supplemented with carefully designed exercises and notes relating the new theory to its historical context. Praise from early readers "Ever since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure-theoretic. In this ground-breaking work, Shafer and Vovk give a game-theoretic foundation instead. While being just as rigorous, the game-theoretic approach allows for vast and useful generalizations of classical measure-theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades." - Peter Grünwald, CWI and University of Leiden "Shafer and Vovk have thoroughly re-written their 2001 book on the game-theoretic foundations for probability and for finance. They have included an account of the tremendous growth that has occurred since, in the game-theoretic and pathwise approaches to stochastic analysis and in their applications to continuous-time finance. This new book will undoubtedly spur a better understanding of the foundations of these very important fields, and we should all be grateful to its authors." - Ioannis Karatzas, Columbia University

Inhalt

Preface xi

Acknowledgments xv

Part I Examples in Discrete Time 1

1 Borel's Law of Large Numbers 5

1.1 A Protocol for Testing Forecasts 6

1.2 A Game-Theoretic Generalization of Borel's Theorem 8

1.3 Binary Outcomes 16

1.4 Slackenings and Supermartingales 18

1.5 Calibration 19

1.6 The Computation of Strategies 21

1.7 Exercises 21

1.8 Context 24

2 Bernoulli's and De Moivre's Theorems 31

2.1 Game-Theoretic Expected Value and Probability 33

2.2 Bernoulli's Theorem for Bounded Forecasting 37

2.3 A Central Limit Theorem 39

2.4 Global Upper Expected Values for Bounded Forecasting 45

2.5 Exercises 46

2.6 Context 49

3 Some Basic Supermartingales 55

3.1 Kolmogorov's Martingale 56

3.2 Doléans's Supermartingale 56

3.3 Hoeffding's Supermartingale 58

3.4 Bernstein's Supermartingale 63

3.5 Exercises 66

3.6 Context 67

4 Kolmogorov's Law of Large Numbers 69

4.1 Stating Kolmogorov's Law 70

4.2 Supermartingale Convergence Theorem 73

4.3 How Skeptic Forces Convergence 80

4.4 How Reality Forces Divergence 81

4.5 Forcing Games 82

4.6 Exercises 86

4.7 Context 89

5 The Law of the Iterated Logarithm 93

5.1 Validity of the Iterated-Logarithm Bound 94

5.2 Sharpness of the Iterated-Logarithm Bound 99

5.3 Additional Recent Game-Theoretic Results 100

5.4 Connections with Large Deviation Inequalities 104

5.5 Exercises 104

5.6 Context 106

Part II Abstract Theory in Discrete Time 109

6 Betting on a Single Outcome 111

6.1 Upper and Lower Expectations 113

6.2 Upper and …