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- Wiley
More About This Title Game-Theoretic Foundations for Probability and Finance
English
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
English
Glenn Shafer is University Professor at Rutgers University.
Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London. They 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.
English
Preface xi
Acknowledgments xv
Part I Examples in Discrete Time
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 15
1.4 Slackenings and Supermartingales 17
1.5 Calibration 18
1.6 The Computation of Strategies 19
1.7 Exercises 20
1.8 Context 22
2 Bernoulli’s and De Moivre’s Theorems 29
2.1 Game-Theoretic Expected Value and Probability 31
2.2 Bernoulli’s Theorem for Bounded Forecasting 34
2.3 A Central Limit Theorem 36
2.4 Global Upper Expected Values for Bounded Forecasting 42
2.5 Exercises 43
2.6 Context 45
3 Some Basic Supermartingales 49
3.1 Kolmogorov’s Martingale 50
3.2 Doléans’s Supermartingale 50
3.3 Hoeffding’s Supermartingale 52
3.4 Bernstein’s Supermartingale 56
3.5 Exercises 59
3.6 Context 60
4 Kolmogorov’s Law of Large Numbers 61
4.1 Stating Kolmogorov’s Law 62
4.2 Supermartingale Convergence Theorem 64
4.3 How Skeptic Forces Convergence 70
4.4 How Reality Forces Divergence 71
4.5 Forcing Games 72
4.6 Exercises 76
4.7 Context 78
5 The Law of the Iterated Logarithm 81
5.1 Validity of the Iterated-Logarithm Bound 82
5.2 Sharpness of the Iterated-Logarithm Bound 86
5.3 Additional Recent Game-Theoretic Results 87
5.4 Connections with Large Deviation Inequalities 91
5.5 Exercises 91
5.6 Context 93
Part II Abstract Theory in Discrete Time
6 Betting on a Single Outcome 99
6.1 Upper and Lower Expectations 101
6.2 Upper and Lower Probabilities 103
6.3 Upper Expectations with Smaller Domains 105
6.4 Offers 107
6.5 Dropping the Continuity Axiom 112
6.6 Exercises 113
6.7 Context 116
7 Abstract Testing Protocols 121
7.1 Terminology and Notation 122
7.2 Supermartingales 123
7.3 Global Upper Expected Values 127
7.4 Lindeberg’s Central Limit Theorem for Martingales 130
7.5 General Abstract Testing Protocols 131
7.6 Making the Results of Part I Abstract 136
7.7 Exercises 137
7.8 Context 139
8 Zero-One Laws 141
8.1 Lévy’s Zero-One Law 142
8.2 Global Upper Expectation 144
8.3 Global Upper and Lower Probabilities 145
8.4 Global Expected Values and Probabilities 146
8.5 Other Zero-One Laws 148
8.6 Exercises 151
8.7 Context 152
9 Relation to Measure-Theoretic Probability 155
9.1 Ville’s Theorem 157
9.2 Measure-Theoretic Representation of Upper Expectations 159
9.3 Embedding Game-Theoretic Martingales in Probability Spaces 167
9.4 Exercises 169
9.5 Context 170
Part III Applications in Discrete Time
10 Using Testing Protocols in Science and Technology 175
10.1 Signals in Open Protocols 176
10.2 Cournot’s Principle 179
10.3 Daltonism 180
10.4 Least Squares 185
10.5 Parametric Statistics with Signals 188
10.6 Quantum Mechanics 191
10.7 Jeffreys’s Law 193
10.8 Exercises 200
10.9 Context 201
11 Calibrating Lookbacks and p-Values 205
11.1 Lookback Calibrators 206
11.2 Lookback Protocols 210
11.3 Lookback Compromises 216
11.4 Lookbacks in Financial Markets 216
11.5 Calibrating p-Values 219
11.6 Exercises 222
11.7 Context 224
12 Defensive Forecasting 227
12.1 Defeating Strategies for Skeptic 229
12.2 Calibrated Forecasts 233
12.3 Proving the Calibration Theorems 237
12.4 Using Calibrated Forecasts for Decision Making 242
12.5 Proving the Decision Theorems 246
12.6 From Theory to Algorithm 257
12.7 Discontinuous Strategies for Skeptic 262
12.8 Exercises 265
12.9 Context 269
Part IV Game-theoretic Finance
13 Emergence of Randomness in Idealized Financial Markets 277
13.1 Capital Processes and Instant Enforcement 278
13.2 Emergence of Brownian Randomness 280
13.3 Emergence of Brownian Expectation 287
13.4 Applications of Dubins-Schwarz 291
13.5 Getting Rich Quick with the Axiom of Choice 296
13.6 Exercises 298
13.7 Context 299
14 A Game-Theoretic Itô Calculus 303
14.1 Martingale Spaces 304
14.2 Conservatism of Continuous Martingales 312
14.3 Itô Integration 313
14.4 Covariation and Quadratic Variation 317
14.5 Itô’s Formula 319
14.6 Doléans Exponential and Logarithm 320
14.7 Game-Theoretic Expectation and Probability 322
14.8 Game-Theoretic Dubins-Schwarz Theorem 322
14.9 Coherence 324
14.10 Exercises 324
14.11 Context 326
15 Numeraires in Market Spaces 331
15.1 Market Spaces 332
15.2 Martingale Theory in Market Spaces 334
15.3 Girsanov’s Theorem 335
15.4 Exercises 340
15.5 Context 341
16 Equity Premium and CAPM 343
16.1 Three Fundamental Continuous I-martingales 345
16.2 Equity Premium 346
16.3 Capital Asset Pricing Model 349
16.4 Theoretical Performance Deficit 352
16.5 Sharpe Ratio 353
16.6 Exercises 354
16.7 Context 354
17 Game-Theoretic Portfolio Theory 359
17.1 Stroock-Varadhan Martingales 361
17.2 Boosting Stroock-Varadhan Martingales 362
17.3 Outperforming the Market with Dubins-Schwarz 368
17.4 Jeffreys’s Law in Finance 369
17.5 Exercises 370
17.6 Context 371
Terminology and Notation 373
List of Symbols 379
References 383
Index 409
