A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

A groundbreaking and practical treatment of probability andstochastic processes

A Modern Theory of Random Variation presents a new andradical reformulation of the mathematical underpinnings of subjectsas diverse as investment, communication engineering, and quantummechanics. Setting aside the classical theory of probabilitymeasure spaces, the book utilizes a mathematically rigorous versionof the theory of random variation based exclusively on finitelyadditive probability distribution functions.

In place of twentieth–century Lebesgue integration and measuretheory, the author uses the simpler concept of Riemann sums and thenon–absolute Riemann–type integration of Henstock. Readers aresupplied with an accessible approach to standard elements ofprobability theory such as the central limit theorem and Brownianmotion as well as remarkable, new results on Feynman diagrams andstochastic integrals.

Detailed numerical examples and demonstrations guide the readerthrough the abstract mathematical exposition. In addition, numerousdiagrams and graphics provide vivid illustrations of the theory,from the elementary level to the more advanced.

A Modern Theory of Random Variation is suitable forcourses on mathematical analysis, probability theory, andmathematical finance at the upper–undergraduate and graduatelevels. The book is also an indispensable resource for researchersand practitioners who are seeking new concepts, techniques, andmethodologies in data analysis, numerical calculation, andfinancial asset valuation.

2 Introduction 37

3 Infinite–Dimensional Integration 83

4 Theory of the Integral 111

5 Random Variability 183

6 Gaussian Integrals 257

7 Brownian Motion 305

8 Stochastic Integration 383

9 Numerical Calculation 447

A Epilogue 491

Bibliography 505

Index 521

PATRICK MULDOWNEY, PhD, served as lecturer in the Magee Business School at 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.