A groundbreaking and practical treatment of probability and stochastic processes
A Modern Theory of Random Variation presents 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 based 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 limit theorem and Brownian motion as well as remarkable, new results on Feynman diagrams and stochastic integrals.
Detailed numerical examples and demonstrations guide the reader through the abstract mathematical exposition. In addition, numerous diagrams and graphics provide vivid illustrations of the theory, from the elementary level to the more advanced.
A Modern Theory of Random Variation is suitable for courses on mathematical analysis, probability theory, and mathematical finance at the upper–undergraduate and graduate levels. The book is also an indispensable resource for researchers and practitioners who are seeking new concepts, techniques, and methodologies in data analysis, numerical calculation, and financial asset valuation.
1 Prologue 1
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