Game-theoretic foundations for probability and finance / Glenn Ray Shafer, Rutgers University, New Jersey, USA, Vladimir Vovk, University of London, Surrey, UK.

Format:

Book

Author/Creator:

Shafer, Glenn, 1946- author.

Vovk, Vladimir, 1960- author.

Series:

Wiley series in probability and statistics.

Wiley series in probability and statistics

Standardized Title:

Probability and finance

Language:

English

Subjects (All):

Finance--Statistical methods.

Finance.

Finance--Mathematical models.

Game theory.

Physical Description:

1 online resource (483 pages).

Edition:

1st edition

Place of Publication:

Hoboken, NJ : Wiley, 2019.

System Details:

text file

Summary:

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

Contents:

Cover

Title Page

Copyright

Contents

Preface

Acknowledgments

Part I Examples in Discrete Time

Chapter 1 Borel's Law of Large Numbers

1.1 A Protocol for Testing Forecasts

1.2 A Game‐Theoretic Generalization of Borel's Theorem

1.3 Binary Outcomes

1.4 Slackenings and Supermartingales

1.5 Calibration

1.6 The Computation of Strategies

1.7 Exercises

1.8 Context

Chapter 2 Bernoulli's and De Moivre's Theorems

2.1 Game‐Theoretic Expected value and Probability

2.2 Bernoulli's Theorem for Bounded Forecasting

2.3 A Central Limit Theorem

2.4 Global Upper Expected Values for Bounded Forecasting

2.5 Exercises

2.6 Context

Chapter 3 Some Basic Supermartingales

3.1 Kolmogorov's Martingale

3.2 Doléans's Supermartingale

3.3 Hoeffding's Supermartingale

3.4 Bernstein's Supermartingale

3.5 Exercises

3.6 Context

Chapter 4 Kolmogorov's Law of Large Numbers

4.1 Stating Kolmogorov's Law

4.2 Supermartingale Convergence Theorem

4.3 How Skeptic Forces Convergence

4.4 How Reality Forces Divergence

4.5 Forcing Games

4.6 Exercises

4.7 Context

Chapter 5 The Law of the Iterated Logarithm

5.1 Validity of the Iterated‐Logarithm Bound

5.2 Sharpness of the Iterated‐Logarithm Bound

5.3 Additional Recent Game‐Theoretic Results

5.4 Connections with Large Deviation Inequalities

5.5 Exercises

5.6 Context

Part II Abstract Theory in Discrete Time

Chapter 6 Betting on a Single Outcome

6.1 Upper and Lower Expectations

6.2 Upper and Lower Probabilities

6.3 Upper Expectations with Smaller Domains

6.4 Offers

6.5 Dropping the Continuity Axiom

6.6 Exercises

6.7 Context

Chapter 7 Abstract Testing Protocols

7.1 Terminology and Notation

7.2 Supermartingales

7.3 Global Upper Expected Values.

7.4 Lindeberg's Central Limit Theorem for Martingales

7.5 General Abstract Testing Protocols

7.6 Making the Results of Part I Abstract

7.7 Exercises

7.8 Context

Chapter 8 Zero‐One Laws

8.1 LÉvy's Zero‐One Law

8.2 Global Upper Expectation

8.3 Global Upper and Lower Probabilities

8.4 Global Expected Values and Probabilities

8.5 Other Zero‐One Laws

8.6 Exercises

8.7 Context

Chapter 9 Relation to Measure‐Theoretic Probability

9.1 VILLE'S THEOREM

9.2 Measure‐Theoretic Representation of Upper Expectations

9.3 Embedding Game‐Theoretic Martingales in Probability Spaces

9.4 Exercises

9.5 Context

Part III Applications in Discrete Time

Chapter 10 Using Testing Protocols in Science and Technology

10.1 Signals in Open Protocols

10.2 Cournot's Principle

10.3 Daltonism

10.4 Least Squares

10.5 Parametric Statistics with Signals

10.6 Quantum Mechanics

10.7 Jeffreys's Law

10.8 Exercises

10.9 Context

Chapter 11 Calibrating Lookbacks and p‐Values

11.1 Lookback Calibrators

11.2 Lookback Protocols

11.3 Lookback Compromises

11.4 Lookbacks in Financial Markets

11.5 Calibrating p‐values

11.6 Exercises

11.7 Context

Chapter 12 Defensive Forecasting

12.1 Defeating Strategies for Skeptic

12.2 Calibrated Forecasts

12.3 Proving the Calibration Theorems

12.4 Using Calibrated Forecasts for Decision Making

12.5 Proving the Decision Theorems

12.6 From Theory to Algorithm

12.7 Discontinuous Strategies for Skeptic

12.8 Exercises

12.9 Context

Part IV Game‐Theoretic Finance

Chapter 13 Emergence of Randomness in Idealized Financial Markets

13.1 Capital Processes and Instant Enforcement

13.2 Emergence of Brownian Randomness

13.3 Emergence of Brownian Expectation

13.4 Applications of Dubins-Schwarz.

13.5 Getting Rich Quick with the Axiom of Choice

13.6 Exercises

13.7 Context

Chapter 14 A Game‐Theoretic Ito Calculus

14.1 Martingale Spaces

14.2 Conservatism of Continuous Martingales

14.3 Ito Integration

14.4 Covariation and Quadratic Variation

14.5 Ito's Formula

14.6 Doléans Exponential and Logarithm

14.7 Game‐Theoretic Expectation and Probability

14.8 Game‐Theoretic Dubins-Schwarz Theorem

14.9 Coherence

14.10 Exercises

14.11 Context

Chapter 15 Numeraires in Market Spaces

15.1 Market Spaces

15.2 Martingale Theory in Market Spaces

15.3 Girsanov's Theorem

15.4 Exercises

15.5 Context

Chapter 16 Equity Premium and CAPM

16.1 Three Fundamental Continuous I‐Martingales

16.2 Equity Premium

16.3 Capital Asset Pricing Model

16.4 Theoretical Performance Deficit

16.5 Sharpe Ratio

16.6 Exercises

16.7 Context

Chapter 17 Game‐Theoretic Portfolio Theory

17.1 Stroock-Varadhan Martingales

17.2 Boosting Stroock-Varadhan Martingales

17.3 Outperforming the Market with Dubins-Schwarz

17.4 Jeffreys's Law in Finance

17.5 Exercises

17.6 Context

Terminology and Notation

List of Symbols

References

Index

EULA.

Notes:

Earlier edition published in 2001 as: Probability and finance : it's only a game!

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781118548028

1118548027

9781118547939

1118547934

9781118548035

1118548035

OCLC:

1084619208