Game-Theoretic Probability: Theory and Applications to Prediction, Science, and Finance 1st Edition by Glenn Shafer 1118548027 9781118548028

Game-Theoretic Probability: Theory and Applications to Prediction, Science, and Finance 1st Edition by Glenn Shafer – Ebook Instant Download/Delivery ISBN(s): 9781118548028, 1118548027

Product details:

• ISBN 10: 1118548027
• ISBN 13: 9781118548028
• Author: Glenn

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.

Table of contents:

Part I: Examples in Discrete Time

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

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

3 Some Basic Supermartingales

3.1 KOLMOGOROV’S MARTINGALE

3.2 DOL ANS’S SUPERMARTINGALE

3.3 HOEFFDING’S SUPERMARTINGALE

3.4 BERNSTEIN’S SUPERMARTINGALE

3.5 EXERCISES

3.6 CONTEXT

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

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

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

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

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

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

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

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

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

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

14 A Game‐Theoretic It Calculus

14.1 Martingale Spaces

14.2 Conservatism of Continuous Martingales

14.3 It Integration

14.4 Covariation and Quadratic Variation

14.5 It’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

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

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

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

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