Book Description
A groundbreaking and practical treatment of probability and stochastic processes
A Modern Theory of Random Variation is a new and radical reformulation 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 nonabsolute Riemanntype 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 upperundergraduate 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
 Coverpage
 Titlepage
 Copyright
 Dedication
 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
 2.1 Riemann Sums in Integration
 2.2 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, ∞[
 2.16 “Negative Probability”
 2.17 Henstock Integration in Rn
 2.18 Conclusion
 3 InfiniteDimensional Integration

4 Theory of the Integral
 4.1 The Henstock Integral
 4.2 Gauges for RT
 4.3 Another Integration System in RT
 4.4 Validation of Gauges in RT
 4.5 The BurkillComplete Integral in RT
 4.6 Basic Properties of the Integral
 4.7 Variation of a Function
 4.8 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 NonAbsolute Integrals
 4.14 NonIntegrable 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 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 InfiniteDimensional 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 cBrownian 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 cBrownian Random Variables
 7.12 Introduction to uObservables
 7.13 Construction of Step Functions in RT
 7.14 Estimation of E[fU(XT)]
 7.15 uObservables in cBrownian 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 Itô’s Formula
 8.11 Proof of Itô’s Formula
 8.12 Application of Itô’s Formula
 8.13 Derivative Asset Valuation
 8.14 RiskNeutral Pricing
 8.15 Comments on RiskNeutral Pricing
 8.16 Pricing a European Call Option
 8.17 Call Option as Contingent Observable
 8.18 BlackScholes Equation
 8.19 Construction of RiskNeutral 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 Itô’s Formula
 9.6 Calculating with Binary Partitions of RT
 9.7 Calculation of Observable Process in RT
 9.8 Other JointContingent Observables
 9.9 Empirical Data
 9.10 Empirical Distributions
 9.11 Calculation of Empirical Distribution
 A Epilogue
 Bibliography
 Index
Product Information
 Title: A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration
 Author(s):
 Release date: October 2012
 Publisher(s): Wiley
 ISBN: 9781118166406