You are previewing A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration.
O'Reilly logo
A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Book Description

A ground-breaking and practical treatment of probability and stochastic processes

A Modern Theory of Random Variation is a new and radical re-formulation of the mathematical underpinnings of subjects as diverse as investment, communication engineering, and quantum mechanics. Setting aside the classical theory of probability measure spaces, the book utilizes a mathematically rigorous version of the theory of random variation that bases itself exclusively on finitely additive probability distribution functions.

In place of twentieth century Lebesgue integration and measure theory, the author uses the simpler concept of Riemann sums, and the non-absolute Riemann-type integration of Henstock. Readers are supplied with an accessible approach to standard elements of probability theory such as the central limmit theorem and Brownian motion as well as remarkable, new results on Feynman diagrams and stochastic integrals.

Throughout the book, detailed numerical demonstrations accompany the discussions of abstract mathematical theory, from the simplest elements of the subject to the most complex. In addition, an array of numerical examples and vivid illustrations showcase how the presented methods and applications can be undertaken at various levels of complexity.

A Modern Theory of Random Variation is a suitable book for courses on mathematical analysis, probability theory, and mathematical finance at the upper-undergraduate and graduate levels. The book is also an indispensible resource for researchers and practitioners who are seeking new concepts, techniques and methodologies in data analysis, numerical calculation, and financial asset valuation.

Patrick Muldowney, PhD, served as lecturer at the Magee Business School of the UNiversity of Ulster for over twenty years. Dr. Muldowney has published extensively in his areas of research, including integration theory, financial mathematics, and random variation.

Table of Contents

  1. Coverpage
  2. Titlepage
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface
  7. Symbols
  8. 1 Prologue
    1. 1.1 About This Book
    2. 1.2 About the Concepts
    3. 1.3 About the Notation
    4. 1.4 Riemann, Stieltjes, and Burkill Integrals
    5. 1.5 The -Complete Integrals
    6. 1.6 Riemann Sums in Statistical Calculation
    7. 1.7 Random Variability
    8. 1.8 Contingent and Elementary Forms
    9. 1.9 Comparison With Axiomatic Theory
    10. 1.10 What Is Probability?
    11. 1.11 Joint Variability
    12. 1.12 Independence
    13. 1.13 Stochastic Processes
  9. 2 Introduction
    1. 2.1 Riemann Sums in Integration
    2. 2.2 The -Complete Integrals in Domain ]0,1]
    3. 2.3 Divisibility of the Domain ]0, 1]
    4. 2.4 Fundamental Theorem of Calculus
    5. 2.5 What Is Integrability?
    6. 2.6 Riemann Sums and Random Variability
    7. 2.7 How to Integrate a Function
    8. 2.8 Extension of the Lebesgue Integral
    9. 2.9 Riemann Sums in Basic Probability
    10. 2.10 Variation and Outer Measure
    11. 2.11 Outer Measure and Variation in [0, 1]
    12. 2.12 The Henstock Lemma
    13. 2.13 Unbounded Sample Spaces
    14. 2.14 Cauchy Extension of the Riemann Integral
    15. 2.15 Integrability on ]0, ∞[
    16. 2.16 “Negative Probability”
    17. 2.17 Henstock Integration in Rn
    18. 2.18 Conclusion
  10. 3 Infinite-Dimensional Integration
    1. 3.1 Elements of Infinite-Dimensional Domain
    2. 3.2 Partitions of RT
    3. 3.3 Regular Partitions of RT
    4. 3.4 δ-Fine Partially Regular Partitions
    5. 3.5 Binary Partitions of RT
    6. 3.6 Riemann Sums in RT
    7. 3.7 Integrands in RT
    8. 3.8 Definition of the Integral in RT
    9. 3.9 Integrating Functions in RT
  11. 4 Theory of the Integral
    1. 4.1 The Henstock Integral
    2. 4.2 Gauges for RT
    3. 4.3 Another Integration System in RT
    4. 4.4 Validation of Gauges in RT
    5. 4.5 The Burkill-Complete Integral in RT
    6. 4.6 Basic Properties of the Integral
    7. 4.7 Variation of a Function
    8. 4.8 Variation and Integral
    9. 4.9 RT ×N(T)-Variation
    10. 4.10 Introduction to Fubini’s Theorem
    11. 4.11 Fubini’s Theorem
    12. 4.12 Limits of Integrals
    13. 4.13 Limits of Non-Absolute Integrals
    14. 4.14 Non-Integrable Functions
    15. 4.15 Conclusion
  12. 5 Random Variability
    1. 5.1 Measurability of Sets
    2. 5.2 Measurability of Random Variables
    3. 5.3 Representation of Observables
    4. 5.4 Basic Properties of Random Variables
    5. 5.5 Inequalities for Random Variables
    6. 5.6 Joint Random Variability
    7. 5.7 Two or More Joint Observables
    8. 5.8 Independence in Random Variability
    9. 5.9 Laws of Large Numbers
    10. 5.10 Introduction to Central Limit Theorem
    11. 5.11 Proof of Central Limit Theorem
    12. 5.12 Probability Symbols
    13. 5.13 Measurability and Probability
    14. 5.14 The Calculus of Probabilities
  13. 6 Gaussian Integrals
    1. 6.1 Fresnel’s Integral
    2. 6.2 Evaluation of Fresnel’s Integral
    3. 6.3 Fresnel’s Integral in Finite Dimensions
    4. 6.4 Fresnel Distribution Function in Rn
    5. 6.5 Infinite-Dimensional Fresnel Integral
    6. 6.6 Integrability on RT
    7. 6.7 The Fresnel Function Is VBG*
    8. 6.8 Incremental Fresnel Integral
    9. 6.9 Fresnel Continuity Properties
  14. 7 Brownian Motion
    1. 7.1 c-Brownian Motion
    2. 7.2 Brownian Motion With Drift
    3. 7.3 Geometric Brownian Motion
    4. 7.4 Continuity of Sample Paths
    5. 7.5 Introduction to Continuous Modification
    6. 7.6 Continuous Modification
    7. 7.7 Introduction to Marginal Densities
    8. 7.8 Marginal Densities in RT
    9. 7.9 Regular Partitions
    10. 7.10 Step Functions in RT
    11. 7.11 c-Brownian Random Variables
    12. 7.12 Introduction to u-Observables
    13. 7.13 Construction of Step Functions in RT
    14. 7.14 Estimation of E[fU(XT)]
    15. 7.15 u-Observables in c-Brownian Motion
    16. 7.16 Diffusion Equation
    17. 7.17 Feynman Path Integrals
    18. 7.18 Feynman’s Definition of Path Integral
    19. 7.19 Convergence of Binary Sums
    20. 7.20 Feynman Diagrams
    21. 7.21 Interpretation of the Perturbation Series
    22. 7.22 Validity of Feynman Diagrams
    23. 7.23 Conclusion
  15. 8 Stochastic Integration
    1. 8.1 Introduction to Stochastic Integrals
    2. 8.2 Varieties of Stochastic Integral
    3. 8.3 Strong Stochastic Integral
    4. 8.4 Weak Stochastic Integral
    5. 8.5 Definition of Weak Stochastic Integral
    6. 8.6 Properties of Weak Stochastic Integral
    7. 8.7 Evaluating Stochastic Integrals
    8. 8.8 Stochastic and Observable Integrals
    9. 8.9 Existence of Weak Stochastic Integrals
    10. 8.10 Itô’s Formula
    11. 8.11 Proof of Itô’s Formula
    12. 8.12 Application of Itô’s Formula
    13. 8.13 Derivative Asset Valuation
    14. 8.14 Risk-Neutral Pricing
    15. 8.15 Comments on Risk-Neutral Pricing
    16. 8.16 Pricing a European Call Option
    17. 8.17 Call Option as Contingent Observable
    18. 8.18 Black-Scholes Equation
    19. 8.19 Construction of Risk-Neutral Model
  16. 9 Numerical Calculation
    1. 9.1 Introduction
    2. 9.2 Random Walk
    3. 9.3 Calculation of Strong Stochastic Integrals
    4. 9.4 Calculation of Weak Stochastic Integrals
    5. 9.5 Calculation of Itô’s Formula
    6. 9.6 Calculating with Binary Partitions of RT
    7. 9.7 Calculation of Observable Process in RT
    8. 9.8 Other Joint-Contingent Observables
    9. 9.9 Empirical Data
    10. 9.10 Empirical Distributions
    11. 9.11 Calculation of Empirical Distribution
  17. A Epilogue
    1. A.1 Measurability
    2. A.2. Historical Note
  18. Bibliography
  19. Index