A Modern Theory of Random Variation - Patrick Muldowney - ebook

A Modern Theory of Random Variation ebook

Patrick Muldowney

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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.

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Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Symbols

Chapter 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

Chapter 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

Chapter 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

Chapter 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 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 RT × (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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 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

Chapter 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 Joint-Contingent Observables

9.9 Empirical Data

9.10 Empirical Distributions

9.11 Calculation of Empirical Distribution

Appendix A: Epilogue

A.1 Measurability

A.2 Historical Note

Bibliography

Index

A Modern Theory of Random Variation

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

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Library of Congress Cataloging-in-Publication Data:

Muldowney, P. (Patrick), 1946–  A modern theory of random variation : with applications in stochastic calculus, financial mathematics, and Feynman integration / Patrick Muldowney.     p. cm.    Includes index.  ISBN 978-1-118-16640-6 (hardback)  1. Random variables. 2. Calculus of variations. 3. Path integrals. 4. Mathematical analysis. I. Title.  QA273.M864 2012  519.2’3—dc232012002023

Preface

The theory of probability is one of the success stories of twentieth century mathematics. Its success was founded on advances in the theory of integration associated with Henri Lebesgue, which, in turn, are based on the mathematical theory of measure.

But twentieth century probability theory is constrained by certain features of the Lebesgue integral. Lebesgue integration cannot safely be used without first mastering the underlying theory of measure—a subtle and difficult subject.

Furthermore, in Lebesgue integration, as in the Riemann integration that it superseded, a function is integrable only if its absolute value is integrable. Consequently, some perfectly straightforward functions cannot be integrated by Lebesgue’s method. This limitation meant that Richard Feynman’s mid-twentieth century discoveries in quantum mechanics, including the theory of light for which he received the Nobel Prize, could not be expressed in probability terms to which his theory bears a strong formal resemblance.

A further limitation is manifested in the Itô calculus used in financial mathematics and the theory of communication. This is because the Lebesgue version of stochastic calculus is relatively complicated and difficult to apply in practice.

This book overcomes these limitations by formulating probability theory in terms of the Stieltjes-complete integral instead of the Lebesgue integral. The roots of the Stieltjes-complete integral of this book are in developments in mathematical analysis in the 1950s and ‘60s.

In the 1950s a new method of integration, using Riemann sums, was discovered independently by Ralph Henstock and Jaroslav Kurzweil. The best-known version is the Henstock–Kurzweil integral. In this book the Henstock–Kurzweil integral is referred to as the Riemann-complete integral.

In contrast to Lebesgue theory, Riemann-complete integration uses non-absolute convergence. In other words, functions may be integrable even if their absolute value is not integrable.

What does this mean? The series converges if the terms are added up in the order in which they appear. But if positive and negative terms are first added up separately the series diverges. This is the essential difference between non-absolute and absolute convergence. The non-absolute method of summation enables cancelations to occur, and many important integrals converge only if their Riemann sums are calculated in this way.

This is also a key feature of the Stieltjes-complete integral of this book, which enables it to open up new vistas in the theory of probability and random variation. These vistas constitute a formulation of probability theory in which Lebesgue integration and traditional measure theory are absent. It is a Riemann sum approach to the theory of random variation. The concept of measurability of sets and functions is expressed in terms of Riemann sums rather than measure theory.

The book opens with elementary numerical calculations of means and variances which demonstrate the role of Riemann sums, and it closes with similar numerical Riemann sum calculations of stochastic integrals, Itô’s formula, and Feynman integrals. The meat in this sandwich is a Riemann sum-based theory of random variation using the Stieltjes-complete integral.

The book has introductory chapters describing the key points of absolute and non-absolute integration, and how these differ in dealing with random variation and probability. It presents in a new, simpler, and more efficient way the standard results of probability theory including the laws of large numbers, the central limit theorem, and the theory of Brownian motion.

It provides an account of Feynman integrals within a framework of Brownian motion. One of the striking features of Feynman’s theory is its diagrammatic calculus for analyzing fundamental quantum mechanical phenomena and processes—the Feynman diagrams. This book provides an explanation of the Feynman diagrammatic calculus and gives conditions which ensure convergence of the underlying perturbation series of which Feynman diagrams are a graphical representation.

Also included are a new approach to the Black–Scholes option pricing formula, and a new and simpler formulation of stochastic calculus, including Itô’s formula. It is shown that many stochastic integrals can be defined in the way that ordinary integrals of calculus courses are defined. The reason for this is the basic point that the Brownian sample paths, though they have infinite variation, have finite integrals when the non-absolute summation method is applied to the Brownian increments.

The main themes of the text are illustrated by numerical calculations using the Maple computer program. The gist of the book can be grasped by reading the introductory chapters and the concluding numerical calculations.

The book is fully self-contained in regard to both probability and integration theory, and it aspires to be accessible and useful to readers who are not expert in either field. The title “A Modern Theory of Random Variation” mirrors the title A Modern Theory of Integration, a book on Riemann-complete integration by Robert Bartle, whose paper Return to the Riemann integral (http://mathdl.maa.org/images/upload_library/22/Ford/Bartle625-632.pdf) is an introduction to the subject.

There is a website for commentary on technical issues arising in this book: https://sites.google.com/site/StieltjesComplete/

Technical communications can be addressed to:[email protected]

Pat Muldowney

Symbols

: Predicative relations for gauges γ in RT.

1s : Indicator function of a set S.

(α, θ)-continuity : Continuity of x(s)sT relative to moduli α and θ.

BM1–BM7 : Properties of c-Brownian motion.

Cταθ : Set of xTRT that are (α, θ)-continuous at τ.

CTαθ : Set of xTRT that are (α, θ)-continuous for all tT.

Dταθ : The set RT \ Cταθ.

DTαθ : The set RT \ CTαθ.

: Differentials for partial differentiation.

: Abbreviated partial differentiation operators.

, ε : Division of a domain or figure.

δ : Gauge for finite-dimensional domain.

δ,εδ : δ-fine division.

γ,εγ : γ-fine division.

d(x): Dirichlet point function.

D(I) : Dirichlet increment function.

E, EF : Expectation with respect to distribution function F.

, marginal density of expectation.

E : A figure, or finite union of cells.

: The union of a figure E and the associated points I* of each I ⊂ E.

: Contingent observable.

: The set .

f(xM), fM(x) : Function in RM, and corresponding cylinder function.

: Binary step function values for a function f in RT.

: Versions of .

F : Distribution function, defined on cells.

FX,FXt, FXT, : Distribution function of an observable.

: The characteristic function of X, or Fourier transform of FX.

: Martingale distribution function.

φρt : Discounting function.

: n-Dimensional Fresnel or Gaussian density function.

: n-Dimensional Fresnel or Gaussian cell function.

: n-Dimensional Fresnel cell function in unbounded cell.

: Stochastic integrands

gc(x(N)) : Incremental Fresnel density function in RT.

: Incremental Fresnel distribution function in RT.

: Values taken by GC on binary cells of RT.

: Fresnel density function in infinite-dimensional RT.

Gc(I[N]) : Fresnel distribution function in infinite-dimensional RT.

: Geometric Brownian distribution, growth rate ρ, volatility σ.

γ-fineAs in γ-fine associated triple in RT.

: h-equivalence, variational equivalence.

: h-convergence, weak convergence.

: The square root of – 1.

I, J : Cells or intervals in R.

I : The class of cells in a domain.

I* : The set of points x associated with a cell I.

: The union of a cell I and its associated points I*.

I(N) : A cell I1 × · · · × In of RN.

I[N] : A cell I1 × · · · × In × RT\N of RT.

k : Permutation of indices for binary partition points in RT.

Kq|k, Krq|k, rq : Binary partitioning of RT.

: Likelihood function.

(z) : Lognormal density function.

L(J) : Lognormal distribution function.

(ρ)(z) : Lognormal density function with growth rate ρ

Lρ(J) : Lognormal distribution function with growth rate ρ

N : A finite set {t1,…, tn} of indices tT; a dimension set.

: The class of finite subsets of T.

N(I) : Standard normal distribution function, equal to .

, normal distribution, mean µ, standard deviation σ.

nµσ(y) : Normal density function, mean µ, standard deviation σ.

: c-normal distribution function, with parameters and µ and σ.

: Fourier transform of the standard normal distribution function.

Ω : Sample space for random variability.

: Expresses in terms of binary partition of RT.

: Binary Riemann sum for .

P : Probability function in Kolmogorov probability space (P, , Ω).

P : Probability function defined by a Stieltjes-complete integral.

PX : Probability function derived from random variable X.

: Partition of a domain of integration.

: Projection function.

: Binary partition points in domain R.

πt : Risk-free portfolio.

: Wave function; marginal density of expectation.

Q : Set of rational numbers.

R : The set of all real numbers.

R+ : The set of positive real numbers ]0, ∞[.

RT : Cartesian product of R.

: Rieman sum functional in stochastic integration.

ρ : Riskless interest rate.

s, S, S : Observable, strong stochastic, and weak stochastic integrals.

: Integral on E with division points labeled x.

: Sets of labels or indices; .

.

.

-data.

: Step function values .

Urq, Urq : Alternative construction of step function .

V : Variation function, corresponding to outer measure.

: The variation of h.

: The variation of h in S relative to E; infγ.

Vh(S) : The variation of h in S; infγ.

: The variance of the random variable f (X).

: Discounted derivative price martingale.

(x, N, I[N]) : Associated triple in RT.

: Discrete representation in RM of xTRT.

χ : Random variable in the Kolmogorov sense; P-random variable.

: Joint-basic observable.

: Elementary observable.

: Discounted price martingale.

Chapter 1

Prologue

1.1 About This Book

This is a self-contained study of a Riemann sum approach to the theory of random variation, assuming only some familiarity with probability or statistical analysis, basic Riemann integration, and mathematical proof. The primary idea of the book, and the reason why it is different from other treatments of random variation, is its use of non-absolute convergence. The series diverges to infinity. On the other hand, the oscillating series converges—but only on condition that the terms are added up in the order in which they are written, without rearranging them. This convergence is called conditional or non-absolute.

What has this got to do with the theory of random variation? Any conception or understanding of the random variation phenomenon hinges on the notions of probability and its mathematical representation in the form of probability distribution functions. The central, recurring theme of this book is that, provided a non-absolute method of summation is used, every finitely additive function of disjoint intervals is integrable. In other words, every distribution function is integrable.

In contrast, more traditional methods in probability theory exclude significant classes of such functions whose integrability cannot be established whenever only absolute convergence is considered. Examples of this include:

The Feynman “probability measure” (which is not a measure and not a probability)—the probability amplitudes used in the Feynman path integrals of quantum mechanics. This book presents a framework in which the Feynman path integrals are actual integrals. In effect, the missing pieces of Feynman’s original paper [64] are provided here; and then used to express Feynman diagrams as convergent series of integrals—as they were originally conceived.
The increments in the sample paths of Brownian motion—these have infinite variation in every interval, and their integrals (in the usual absolute sense) are therefore divergent. But these increments are integrable in the non-absolute sense, so the stochastic calculus of Brownian motion can be put on a simpler footing.

Incorporating these innovations in the theory of random variation entails a radical reformulation of the subject. It turns out that the standard theory of probability or random variation can be simplified and extended provided non-absolute summation procedures are used.

Reformulation and extension of the theory involves some changes and reinterpretations in the standard concepts and notations. Unnecessary changes have been avoided, and as far as possible the text is consistent with more traditional versions. Therefore, with due caution and attention to definitions of terminology and notation, the text can be read in that spirit. An outline and overview are presented in Chapters 1 and 2.

Chapter 7 is the main part of this book, with Chapter 6 providing introductory material, and Chapter 8 some consequences. The book presents a new sphere of application of probability theory by means of the conception of random variation which is elaborated in Chapter 5.

Ralph Henstock’s general theory of integration, as extended in [162] (Muldowney, 1987), is the basis for this reformulation of the traditional theory of probability and random variation, and is presented in Chapter 4.

Even though Henstock’s theory is different from standard integration theory, many of the results are similar. Therefore Chapter 4 can be regarded as a kind of appendix to subsequent chapters, providing technical background in the manner of many books on probability theory in which measure and integration are appended to the main part of the text. Included in this chapter are results for non-absolutely integrable functions which are not available in traditional integration theory.

A fundamental modification and extension of the Riemann integral was introduced by R. Henstock and, independently, by J. Kurzweil in the 1950s. In Henstock [93] this was designated as the Riemann-complete1 integral.

The work of Kurzweil has transformed the theory of differential equations—see, for instance, Schwabik [129, 207]. Henstock went on to develop a general theory of integration [85, 93, 94, 103, 105], which includes as special cases the integrals of Riemann, Stieltjes, Lebesgue, Perron, Denjoy, Ward, Burkill, Henstock-Kurzweil, and McShane (see [82]). This is the Henstock integral on which this book is based.

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