9781118345948

1118345940

9781118345924

1118345924

9781118345931

1118345932

9781118345955

1118345959

111816640X

9781118166406

1283835002

9781283835008

9781118166406

A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration; Contents; Preface; Symbols; 1 Prologue; 1.1 About This Book; 1.2 About the Concepts; 1.3 About the Notation; 1.4 Riemann, Stieltjes, and Burkill Integrals; 1.5 The -Complete Integrals; 1.6 Riemann Sums in Statistical Calculation; 1.7 Random Variability; 1.8 Contingent and Elementary Forms; 1.9 Comparison With Axiomatic Theory; 1.10 What Is Probability?; 1.11 Joint Variability; 1.12 Independence; 1.13 Stochastic Processes; 2 Introduction.

Machine generated contents note: 1. Prologue

1.1. About This Book

1.2. About the Concepts

1.3. About the Notation

1.4. Riemann, Stieltjes, and Burkill Integrals

1.5. Complete Integrals

1.6. Riemann Sums in Statistical Calculation

1.7. Random Variability

1.8. Contingent and Elementary Forms

1.9. Comparison With Axiomatic Theory

1.10. What Is Probability

1.11. Joint Variability

1.12. Independence

1.13. Stochastic Processes

2. Introduction

2.1. Riemann Sums in Integration

2.2. Complete Integrals in Domain [0, 1]

2.3. Divisibility of the Domain [0, 1]

2.4. Fundamental Theorem of Calculus

2.5. What Is Integrability

2.6. Riemann Sums and Random Variability

2.7. How to Integrate a Function

2.8. Extension of the Lebesgue Integral

2.9. Riemann Sums in Basic Probability

2.10. Variation and Outer Measure

2.11. Outer Measure and Variation in [0, 1]

2.12. Henstock Lemma

2.13. Unbounded Sample Spaces

2.14. Cauchy Extension of the Riemann Integral

2.15. Integrability on]0, [∞][

2.16. "Negative Probability"

2.17. Henstock Integration in Rn

2.18. Conclusion

3. Infinite-Dimensional Integration

3.1. Elements of Infinite-Dimensional Domain

3.2. Partitions of RT

3.3. Regular Partitions of RT

3.4. δ-Fine Partially Regular Partitions

3.5. Binary Partitions of RT

3.6. Riemann Sums in RT

3.7. Integrands in RT

3.8. Definition of the Integral in RT

3.9. Integrating Functions in RT

4. Theory of the Integral

4.1. Henstock Integral

4.2. Gauges for RT

4.3. Another Integration System in RT

4.4. Validation of Gauges in RT

4.5. Burkill-Complete Integral in RT

4.6. Basic Properties of the Integral

4.7. Variation of a Function

4.8. Variation and Integral

4.9. RTxN(T)-Variation

4.10. Introduction to Fubini's Theorem

4.11. Fubini's Theorem

4.12. Limits of Integrals

4.13. Limits of Non-Absolute Integrals

4.14. Non-Integrable Functions

4.15. Conclusion

5. Random Variability

5.1. Measurability of Sets

5.2. Measurability of Random Variables

5.3. Representation of Observables

5.4. Basic Properties of Random Variables

5.5. Inequalities for Random Variables

5.6. Joint Random Variability

5.7. Two or More Joint Observables

5.8. Independence in Random Variability

5.9. Laws of Large Numbers

5.10. Introduction to Central Limit Theorem

5.11. Proof of Central Limit Theorem

5.12. Probability Symbols

5.13. Measurability and Probability

5.14. Calculus of Probabilities

6. Gaussian Integrals

6.1. Fresnel's Integral

6.2. Evaluation of Fresnel's Integral

6.3. Fresnel's Integral in Finite Dimensions

6.4. Fresnel Distribution Function in Rn

6.5. Infinite-Dimensional Fresnel Integral

6.6. Integrability on RT

6.7. Fresnel Function Is VBG*

6.8. Incremental Fresnel Integral

6.9. Fresnel Continuity Properties

7. Brownian Motion

7.1. c-Brownian Motion

7.2. Brownian Motion With Drift

7.3. Geometric Brownian Motion

7.4. Continuity of Sample Paths

7.5. Introduction to Continuous Modification

7.6. Continuous Modification

7.7. Introduction to Marginal Densities

7.8. Marginal Densities in RT

7.9. Regular Partitions

7.10. Step Functions in RT

7.11. c-Brownian Random Variables

7.12. Introduction to u-Observables

7.13. Construction of Step Functions in RT

7.14. Estimation of E [fu(XT)]

7.15. U-Observables in c-Brownian Motion

7.16. Diffusion Equation

7.17. Feynman Path Integrals

7.18. Feynman's Definition of Path Integral

7.19. Convergence of Binary Sums

7.20. Feynman Diagrams

7.21. Interpretation of the Perturbation Series

7.22. Validity of Feynman Diagrams

7.23. Conclusion

8. Stochastic Integration

8.1. Introduction to Stochastic Integrals

8.2. Varieties of Stochastic Integral

8.3. Strong Stochastic Integral

8.4. Weak Stochastic Integral

8.5. Definition of Weak Stochastic Integral

8.6. Properties of Weak Stochastic Integral

8.7. Evaluating Stochastic Integrals

8.8. Stochastic and Observable Integrals

8.9. Existence of Weak Stochastic Integrals

8.10. Ito's Formula

8.11. Proof of Ito's Formula

8.12. Application of Ito's Formula

8.13. Derivative Asset Valuation

8.14. Risk-Neutral Pricing

8.15. Comments on Risk-Neutral Pricing

8.16. Pricing a European Call Option

8.17. Call Option as Contingent Observable

8.18. Black-Scholes Equation

8.19. Construction of Risk-Neutral Model

9. Numerical Calculation

9.1. Introduction

9.2. Random Walk

9.3. Calculation of Strong Stochastic Integrals

9.4. Calculation of Weak Stochastic Integrals

9.5. Calculation of Ito's Formula

9.6. Calculating with Binary Partitions of RT

9.7. Calculation of Observable Process in RT

9.8. Other Joint-Contingent Observables

9.9. Empirical Data

9.10. Empirical Distributions

9.11. Calculation of Empirical Distribution

A. Epilogue

A.1. Measurability

A.2. Historical Note.

2.1 Riemann Sums in Integration2.2 The -Complete Integrals in Domain]0,1]; 2.3 Divisibility of the Domain]0,1]; 2.4 Fundamental Theorem of Calculus; 2.5 What Is Integrability?; 2.6 Riemann Sums and Random Variability; 2.7 How to Integrate a Function; 2.8 Extension of the Lebesgue Integral; 2.9 Riemann Sums in Basic Probability; 2.10 Variation and Outer Measure; 2.11 Outer Measure and Variation in [0,1]; 2.12 The Henstock Lemma; 2.13 Unbounded Sample Spaces; 2.14 Cauchy Extension of the Riemann Integral; 2.15 Integrability on]0, (infinity)[; 2.16 ""Negative Probability""

4.7 Variation of a Function4.8 Variation and Integral; 4.9 Rt×N(T)-Variation; 4.10 Introduction to Fubini's Theorem; 4.11 Fubini's Theorem; 4.12 Limits of Integrals; 4.13 Limits of Non-Absolute Integrals; 4.14 Non-Integrable Functions; 4.15 Conclusion; 5 Random Variability; 5.1 Measurability of Sets; 5.2 Measurability of Random Variables; 5.3 Representation of Observables; 5.4 Basic Properties of Random Variables; 5.5 Inequalities for Random Variables; 5.6 Joint Random Variability; 5.7 Two or More Joint Observables; 5.8 Independence in Random Variability; 5.9 Laws of Large Numbers.

5.10 Introduction to Central Limit Theorem5.11 Proof of Central Limit Theorem; 5.12 Probability Symbols; 5.13 Measurability and Probability; 5.14 The Calculus of Probabilities; 6 Gaussian Integrals; 6.1 Fresnel's Integral; 6.2 Evaluation of Fresnel's Integral; 6.3 Fresnel's Integral in Finite Dimensions; 6.4 Fresnel Distribution Function in Rn; 6.5 Infinite-Dimensional Fresnel Integral; 6.6 Integrability on Rt; 6.7 The Fresnel Function Is Vbg*; 6.8 Incremental Fresnel Integral; 6.9 Fresnel Continuity Properties; 7 Brownian Motion; 7.1 c-Brownian Motion; 7.2 Brownian Motion With Drift.

MATHEMATICS > Mathematical Analysis.