Game-Theoretic Foundations for Probability and Finance

Game-Theoretic Foundations for Probability and Finance

by Glenn Shafer, Vladimir Vovk
Released May 2019
Publisher(s): Wiley
ISBN: 9780470903056

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Book description

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

Table of contents

  1. Cover
  2. WILEY SERIES IN PROBABILITY AND STATISTICS
  3. Preface
  4. Acknowledgments
  5. Part I: Examples in Discrete Time
    1. 1 Borel's Law of Large Numbers
      1. 1.1 A PROTOCOL FOR TESTING FORECASTS
      2. 1.2 A GAME‐THEORETIC GENERALIZATION OF BOREL'S THEOREM
      3. 1.3 BINARY OUTCOMES
      4. 1.4 SLACKENINGS AND SUPERMARTINGALES
      5. 1.5 CALIBRATION
      6. 1.6 THE COMPUTATION OF STRATEGIES
      7. 1.7 Exercises
      8. 1.8 CONTEXT
    2. 2 Bernoulli's and De Moivre's Theorems
      1. 2.1 Game‐Theoretic Expected value and Probability
      2. 2.2 Bernoulli's Theorem for Bounded Forecasting
      3. 2.3 A Central Limit Theorem
      4. 2.4 Global Upper Expected Values for Bounded Forecasting
      5. 2.5 Exercises
      6. 2.6 Context
    3. 3 Some Basic Supermartingales
      1. 3.1 KOLMOGOROV'S MARTINGALE
      2. 3.2 DOL ANS'S SUPERMARTINGALE
      3. 3.3 HOEFFDING'S SUPERMARTINGALE
      4. 3.4 BERNSTEIN'S SUPERMARTINGALE
      5. 3.5 EXERCISES
      6. 3.6 CONTEXT
    4. 4 Kolmogorov's Law of Large Numbers
      1. 4.1 STATING KOLMOGOROV'S LAW
      2. 4.2 SUPERMARTINGALE CONVERGENCE THEOREM
      3. 4.3 HOW SKEPTIC FORCES CONVERGENCE
      4. 4.4 HOW REALITY FORCES DIVERGENCE
      5. 4.5 FORCING GAMES
      6. 4.6 EXERCISES
      7. 4.7 CONTEXT
    5. 5 The Law of the Iterated Logarithm
      1. 5.1 VALIDITY OF THE ITERATED‐LOGARITHM BOUND
      2. 5.2 SHARPNESS OF THE ITERATED‐LOGARITHM BOUND
      3. 5.3 ADDITIONAL RECENT GAME‐THEORETIC RESULTS
      4. 5.4 CONNECTIONS WITH LARGE DEVIATION INEQUALITIES
      5. 5.5 EXERCISES
      6. 5.6 CONTEXT
  6. Part II: Abstract Theory in Discrete Time
    1. 6 Betting on a Single Outcome
      1. 6.1 Upper and Lower Expectations
      2. 6.2 Upper and Lower Probabilities
      3. 6.3 Upper Expectations with Smaller Domains
      4. 6.4 Offers
      5. 6.5 Dropping the Continuity Axiom
      6. 6.6 Exercises
      7. 6.7 Context
    2. 7 Abstract Testing Protocols
      1. 7.1 Terminology and Notation
      2. 7.2 Supermartingales
      3. 7.3 Global Upper Expected Values
      4. 7.4 Lindeberg's Central Limit Theorem for Martingales
      5. 7.5 General Abstract Testing Protocols
      6. 7.6 Making the Results of Part I Abstract
      7. 7.7 Exercises
      8. 7.8 Context
    3. 8 Zero‐One Laws
      1. 8.1 LÉvy's Zero‐One Law
      2. 8.2 Global Upper Expectation
      3. 8.3 Global Upper and Lower Probabilities
      4. 8.4 Global Expected Values and Probabilities
      5. 8.5 Other Zero‐One Laws
      6. 8.6 Exercises
      7. 8.7 Context
    4. 9 Relation to Measure‐Theoretic Probability
      1. 9.1 VILLE'S THEOREM
      2. 9.2 MEASURE‐THEORETIC REPRESENTATION OF UPPER EXPECTATIONS
      3. 9.3 EMBEDDING GAME‐THEORETIC MARTINGALES IN PROBABILITY SPACES
      4. 9.4 EXERCISES
      5. 9.5 CONTEXT
  7. Part III: Applications in Discrete Time
    1. 10 Using Testing Protocols in Science and Technology
      1. 10.1 SIGNALS IN OPEN PROTOCOLS
      2. 10.2 COURNOT'S PRINCIPLE
      3. 10.3 DALTONISM
      4. 10.4 LEAST SQUARES
      5. 10.5 PARAMETRIC STATISTICS WITH SIGNALS
      6. 10.6 QUANTUM MECHANICS
      7. 10.7 JEFFREYS'S LAW
      8. 10.8 EXERCISES
      9. 10.9 Context
    2. 11 Calibrating Lookbacks and p‐Values
      1. 11.1 LOOKBACK CALIBRATORS
      2. 11.2 LOOKBACK PROTOCOLS
      3. 11.3 LOOKBACK COMPROMISES
      4. 11.4 LOOKBACKS IN FINANCIAL MARKETS
      5. 11.5 CALIBRATING p‐VALUES
      6. 11.6 EXERCISES
      7. 11.7 CONTEXT
    3. 12 Defensive Forecasting
      1. 12.1 DEFEATING STRATEGIES FOR SKEPTIC
      2. 12.2 CALIBRATED FORECASTS
      3. 12.3 PROVING THE CALIBRATION THEOREMS
      4. 12.4 USING CALIBRATED FORECASTS FOR DECISION MAKING
      5. 12.5 PROVING THE DECISION THEOREMS
      6. 12.6 FROM THEORY TO ALGORITHM
      7. 12.7 DISCONTINUOUS STRATEGIES FOR SKEPTIC
      8. 12.8 Exercises
      9. 12.9 CONTEXT
  8. Part IV: Game‐Theoretic Finance
    1. 13 Emergence of Randomness in Idealized Financial Markets
      1. 13.1 CAPITAL PROCESSES AND INSTANT ENFORCEMENT
      2. 13.2 EMERGENCE OF BROWNIAN RANDOMNESS
      3. 13.3 EMERGENCE OF BROWNIAN EXPECTATION
      4. 13.4 APPLICATIONS OF DUBINS–SCHWARZ
      5. 13.5 GETTING RICH QUICK WITH THE AXIOM OF CHOICE
      6. 13.6 Exercises
      7. 13.7 CONTEXT
    2. 14 A Game‐Theoretic It Calculus
      1. 14.1 Martingale Spaces
      2. 14.2 Conservatism of Continuous Martingales
      3. 14.3 It Integration
      4. 14.4 Covariation and Quadratic Variation
      5. 14.5 It's Formula
      6. 14.6 DOLÉANS EXPONENTIAL AND LOGARITHM
      7. 14.7 GAME‐THEORETIC EXPECTATION AND PROBABILITY
      8. 14.8 Game‐Theoretic Dubins–Schwarz Theorem
      9. 14.9 Coherence
      10. 14.10 Exercises
      11. 14.11 Context
    3. 15 Numeraires in Market Spaces
      1. 15.1 MARKET SPACES
      2. 15.2 MARTINGALE THEORY IN MARKET SPACES
      3. 15.3 GIRSANOV'S THEOREM
      4. 15.4 EXERCISES
      5. 15.5 CONTEXT
    4. 16 Equity Premium and CAPM
      1. 16.1 Three Fundamental Continuous I‐Martingales
      2. 16.2 Equity Premium
      3. 16.3 Capital Asset Pricing Model
      4. 16.4 Theoretical Performance Deficit
      5. 16.5 Sharpe Ratio
      6. 16.6 Exercises
      7. 16.7 Context
    5. 17 Game‐Theoretic Portfolio Theory
      1. 17.1 STROOCK–VARADHAN MARTINGALES
      2. 17.2 BOOSTING STROOCK–VARADHAN MARTINGALES
      3. 17.3 OUTPERFORMING THE MARKET WITH DUBINS–SCHWARZ
      4. 17.4 JEFFREYS'S LAW IN FINANCE
      5. 17.5 EXERCISES
      6. 17.6 CONTEXT
  9. Terminology and Notation
  10. List of Symbols
  11. References
  12. Index
  13. End User License Agreement

Product information

  • Title: Game-Theoretic Foundations for Probability and Finance
  • Author(s): Glenn Shafer, Vladimir Vovk
  • Release date: May 2019
  • Publisher(s): Wiley
  • ISBN: 9780470903056