Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics E-Book
Patrick Muldowney
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
GAUGE INTEGRAL STRUCTURES FOR STOCHASTIC CALCULUS AND QUANTUM ELECTRODYNAMICS A stand-alone introduction to specific integration problems in the probabilistic theory of stochastic calculus Picking up where his previous book, A Modern Theory of Random Variation, left off, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics introduces readers to particular problems of integration in the probability-like theory of quantum mechanics. Written as a motivational explanation of the key points of the underlying mathematical theory, and including ample illustrations of the calculus, this book relies heavily on the mathematical theory set out in the author's previous work. That said, this work stands alone and does not require a reading of A Modern Theory of Random Variation in order to be understandable. Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics takes a gradual, relaxed, and discursive approach to the subject in a successful attempt to engage the reader by exploring a narrower range of themes and problems. Organized around examples with accompanying introductions and explanations, the book covers topics such as: * Stochastic calculus, including discussions of random variation, integration and probability, and stochastic processes * Field theory, including discussions of gauges for product spaces and quantum electrodynamics * Robust and thorough appendices, examples, illustrations, and introductions for each of the concepts discussed within * An introduction to basic gauge integral theory (for those unfamiliar with the author's previous book) The methods employed in this book show, for instance, that it is no longer necessary to resort to unreliable "Black Box" theory in financial calculus; that full mathematical rigor can now be combined with clarity and simplicity. Perfect for students and academics with even a passing interest in the application of the gauge integral technique pioneered by R. Henstock and J. Kurzweil, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics is an illuminating and insightful exploration of the complex mathematical topics contained within.
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Table of Contents
Cover
Title Page
Copyright
Preface
Reading this BooknotesSet
Note
Introduction
Notes
Part I: Stochastic Calculus
Chapter 1: Stochastic Integration
Notes
Chapter 2: Random Variation
2.1 What is Random Variation?
2.2 Probability and Riemann Sums
2.3 A Basic Stochastic Integral
2.4 Choosing a Sample Space
2.5 More on Basic Stochastic Integral
Notes
Chapter 3: Integration and Probability
3.1 ‐Complete Integration
3.2 Burkill‐complete Stochastic Integral
3.3 The Henstock Integral
3.4 Riemann Approach to Random Variation
3.5 Riemann Approach to Stochastic Integrals
Notes
Chapter 4: Stochastic Processes
4.1 From to
4.2 Sample Space with Uncountable
4.3 Stochastic Integrals for Example 12
4.4 Example 12
4.5 Review of Integrability Issues
Notes
Chapter 5: Brownian Motion
5.1 Introduction to Brownian Motion
5.2 Brownian Motion Preliminaries
5.3 Review of Brownian Probability
5.4 Brownian Stochastic Integration
5.5 Some Features of Brownian Motion
5.6 Varieties of Stochastic Integral
Notes
Chapter 6: Stochastic Sums
6.1 Review of Random Variability
6.2 Riemann Sums for Stochastic Integrals
6.3 Stochastic Sum as Observable
6.4 Stochastic Sum as Random Variable
6.5 Introduction to
6.6 Isometry Preliminaries
6.7 Isometry Property for Stochastic Sums
6.8 Other Stochastic Sums
6.9 Introduction to Itô’s Formula
6.10 Itô’s Formula for Stochastic Sums
6.11 Proof of Itô’s Formula
6.12 Stochastic Sums or Stochastic Integrals?
Notes
Part II: Field Theory
Chapter 7: Gauges for Product Spaces
7.1 Introduction
7.2 Three‐dimensional Brownian Motion
7.3 A Structured Cartesian Product Space
7.4 Gauges for Product Spaces
7.5 Gauges for Infinite‐dimensional Spaces
7.6 Higher‐dimensional Brownian Motion
7.7 Infinite Products of Infinite Products
Notes
Chapter 8: Quantum Field Theory
8.1 Overview of Feynman Integrals
8.2 Path Integral for Particle Motion
8.3 Action Waves
8.4 Interpretation of Action Waves
8.5 Calculus of Variations
8.6 Integration Issues
8.7 Numerical Estimate of Path Integral
8.8 Free Particle in Three Dimensions
8.9 From Particle to Field
8.10 Simple Harmonic Oscillator
8.11 A Finite Number of Particles
8.12 Continuous Mass Field
Notes
Chapter 9: Quantum Electrodynamics
9.1 Electromagnetic Field Interaction
9.2 Constructing the Field Interaction Integral
9.3 ‐Complete Integral Over Histories
9.4 Review of Point‐Cell Structure
9.5 Calculating Integral Over Histories
9.6 Integration of a Step Function
9.7 Regular Partition Calculation
9.8 Integrand for Integral over Histories
9.9 Action Wave Amplitudes
9.10 Probability and Wave Functions
Notes
Part III: Appendices
Chapter 10: Appendix 1: Integration
10.1 Monstrous Functions
10.2 A Non‐monstrous Function
10.3 Riemann‐complete Integration
10.4 Convergence Criteria
10.5 “I would not care to fly in that plane”
Notes
Chapter 11: Appendix 2: Theorem 63
11.1 Fresnel's Integral
11.2 Theorem 188 of [MTRV]
11.3 Some Consequences of Theorem 63 Fallacy
Notes
Chapter 12: Appendix 3: Option Pricing
12.1 American Options
12.2 Asian Options
Notes
Chapter 13: Appendix 4: Listings
13.1 Theorems
13.2 Examples
13.3 Definitions
13.4 Symbols
Bibliography
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Distribution of payouts.
Table 2.2 Relative frequency table of distribution of weights.
Table 2.3 Calculation of mean and standard deviation.
Table 2.4 UD sample paths for processes
Table 2.5 Calculations for two UD sample paths for processes
Chapter 8
Table 8.1 List of four cells forming a regular partition
of
.
Table 8.2 List of representative co‐ordinates for step function calculation.
Chapter 9
Table 9.1 Elements of calculation of Riemann sum over a regular partition.
Chapter 12
Table 12.1 Glanbia share prices, 3 August 1991 to 3 September 2011.
List of Illustrations
Chapter 10
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Chapter 12
Figure 12.1
Figure 12.2
Figure 12.3
,
,
Figure 12.4
Figure 12.5
Figure 12.6
,
,
Figure 12.7 moving averages.
Figure 12.8 Exponential regression
Figure 12.9 All moving SD's
Figure 12.10 Option values
Guide
Cover
Table of Contents
Begin Reading
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Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics
Patrick Muldowney
This edition first published copyright
© 2021 John Wiley & Sons
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Library of Congress Cataloging‐in‐Publication Data
Names: Muldowney, P. (Patrick), 1946- author.
Title: Gauge integral structures for stochastic calculus and quantum electrodynamics / Patrick Muldowney.
Description: Hoboken, NJ :Wiley, [2020] | Includes bibliographical references and index.
Identifiers: LCCN 2020016333 (print) | LCCN 2020016334 (ebook) | ISBN 9781119595496 (cloth) | ISBN 9781119595502 (adobe pdf) | ISBN 9781119595526 (epub)
Subjects: LCSH: Stochastic analysis. | Henstock-Kurzweil integral. | Feynman integrals. | Quantum electrodynamics‐Mathematics.
Classification: LCC QA274.2 .M85 2020 (print) | LCC QA274.2 (ebook) | DDC 519.2/2‐dc23
LC record available at https://lccn.loc.gov/2020016333
LC ebook record available at https://lccn.loc.gov/2020016334
Cover Design: Wiley
Cover Image: © bannerwega/Getty Images
Set in 9.5/12.5pt STIXTwoText by SPi Global, Chennai, India
Preface
This book is about infinite‐dimensional integration in stochastic calculus and in quantum electrodynamics, using the gauge integral technique pioneered by R. Henstock and J. Kurzweil.
A link between stochastic calculus and quantum mechanics is provided in a previous book by the author ([121], A Modern Theory of Random Variation, or [MTRV] for short), which establishes a mathematical connection between large scale Brownian motion on the one hand and, on the other, small scale quantum level phenomena of particle motion subject to a conservative external mechanical force. In [MTRV] each of the two subjects is a special case of ‐Brownian motion.
The present book is a continuation of [MTRV], in the sense that it develops and extends some of the themes of that book. On the other hand this book is a stand‐alone introduction to particular problems of integration in the probabilistic theory of stochastic calculus, and in the probability‐like theory of quantum mechanics.
Between [MTRV] and this book there is a significant difference in style of exposition. Practically all the underlying mathematical theory is already set out in [MTRV]. The present book includes motivational explanation of the key points of the underlying mathematical theory, along with ample illustrations of the calculus—the routine procedures—of the gauge theory of integration.
But because the “mathematical heavy lifting” (or rigorous mathematical underpinning) is already accomplished in [MTRV], the present book can take a more gradual, relaxed, and discursive approach which seeks to engage the reader with the subject by exploring a much smaller range of chosen themes.
Thus there is hardly anything of the formal Theorem‐Proof structure in this book. Instead the text is organised around Examples with accompanying introductions and explanation, illustrating themes from probability and physics which can be difficult and taxing. Particular areas of interest in the book can be selected and read without engaging with other topics. Its relatively self‐contained component parts can easily be “dipped into”.
In addition to [MTRV], two principal physics sources for this book are [39], Space‐time approach to non‐relativistic quantum mechanics (cited as [F1] for short) by R. Feynman; and [46], Quantum Mechanics and Path Integrals (cited as [FH] for short) by R. Feynman and A. Hibbs.
Certain modes of expression used by physicist R. Feynman are highly illuminating—but from a physics perspective. For instance: Integrate [some expression] over all degrees of freedom [all variables] of the [physical] system. This statement does not specify the mathematical domain for this integration process, nor how (the integral on ) is to be actually calculated.
For quantum electrodynamics, and with denoting the real line and an interval of time, this book proposes a product space domain
along with methods of calculating integrals on product spaces such as
The examples, explanations, and illustrations of this book, though prompted by issues in physics and finance, are primarily about underlying mathematical problems. A reader whose primary purpose is to investigate finance or physics as such should seek out other, more relevant sources.
Also, it is inadvisable to approach the gauge or Riemann sum theory of integration as if it were unchallenging or easy compared with, say, Lebesgue's integration theory. That said, the general idea of gauge integration is more accessible (initially, at least) than the mathematical theory of measure which underpins other approaches to integration. (Gauge integration theory includes measure and measurability—see section A.1 of [MTRV]—but these are outcomes rather than prerequisites of the theory.)
Another central theme of this book is to devise new and better versions of stochastic integrals. Functionals of the form appear in both the theory of stochastic processes (as stochastic integral) and quantum theory (as integral of lagrangian function, or system action).
Such integral functionals either do not actually exist, or are not easy to define. In their place, this book proposes replacement or equivalent functionals which involve Riemann sums rather than integrals. In the case of stochastic processes such sums are called stochastic sums. In quantum theory they are designated sampling sums. (The former are a special case of the latter.)
The general idea of infinite‐dimensional integration on a Cartesian product domain can be found in chapter 3 of [MTRV]. The technical foundations of the subject are in chapter 4 of that book.
The core of this book consists of Chapters 6 and 9. Without tackling the companion book [MTRV], a reader who wants a quick introduction to basic gauge integral theory can get it in Chapter 10
Reading this Book
The term “gauge” in the title relates to gauge integration in mathematics (a generalized form of Riemann integration). It is not about gauge symmetry or gauge transformations in theoretical physics.
The following abbreviations are used because of frequent references to these sources:
[F1] for [
39
],
Space‐time approach to non‐relativistic quantum mechanics
, by R.P. Feynman;
[FH] for [
46
],
Quantum Mechanics and Path Integrals
, by R.P. Feynman and A.R. Hibbs;
[MTRV] for [
121
],
A Modern Theory of Random Variation
, by P. Muldowney.
[
website
] for [
122
],
https://sites.google.com/site/StieltjesComplete/
This is the website for this book, and for [MTRV].
References to chapters, sections, and figures in this book use a capital letter, “Chapter x”; but for material from other sources lower‐case is used: “figure y”.
This book develops themes in probability and quantum mechanics which were introduced in [MTRV]. The range of topics is smaller, and the range of notation is correspondingly smaller, with only a few new symbols. One such is the notation (denoting stochastic sum) on page below. Section 13.4 has a list of the main symbols used.
The subject of the predecessor book [MTRV] is the role of Cartesian product spaces , , in the theory of probability (including quantum mechanics), where is the set of real numbers, and is an interval of time.
The present book examines in more detail different kinds of Cartesian products of , as domains for integration of functions ,
Though the symbol will generally represent time, it is also used as an arbitrary finite or infinite set of labels , depending on the context.
The notation has two components: domain , and integrand . When is an infinite set, such as an interval of time, two different perspectives are present. These are the perspectives indicated in figures 3.1 and 3.2 on page 87 of [MTRV].
Figure 3.1 represents the graph of where is an argument of integrand . Figure 3.2 represents the domain whose elements are points, not graphs. Both of these perspectives should be kept in mind while using this book.
To see the significance of these alternate perspectives1, suppose and , so . In this case, for integrands which appear in this book, there may be little difference between the values of integrands in domain , and in if is continuous.
But domains and are very different; just as differs geometrically from , . The latter difference is the vehicle for the 19th century satire Flatland by Edwin A. Abbott [1], in which two‐dimensional beings struggle with the idea of a three‐dimensional universe.
The book can be read as a collection of standalone accounts of topics which are suggested, or introduced, or touched upon, in [MTRV]. Equally, it can be read as a supplement to [MTRV], developing the ideas of stochastic sums and path integrals which were introduced in [MTRV].
Introduction
The gauge integral is a version of the Riemann integral, with much improved convergence properties. Convergence properties are conditions which ensure integrability of a function; in particular, integrability of the limit of a convergent sequence of integrable functions, with integral of the limit equal to the limit of the integrals—the limit theorems.
Another notable property of the gauge integral in one dimension is that, if a function possesses a corresponding derivative function, the derivative is integrable, with indefinite integral equal to the original function. Curiously, this “schooldays meaning”—integration as the reverse of differentiation—does not hold universally for the more widely used integration systems. See Section 10.2 of Chapter 10, which provides an overview of this subject.
The gauge integral (called ‐complete integral in [MTRV], and in this book) is non‐absolute. Other kinds of integration, such as Lebesgue's or Riemann's, have restrictive requirements of absolute convergence. But existence of ‐complete integrals requires only that the Riemann sum approximations converge non‐absolutely to the value of the integral; and this is central to the present book.
In [MTRV], instead of the more familiar term gauge integral, the term ‐complete integral (as in Riemann‐complete) is found to be helpful.2 This is because there are a great many different integration techniques—Riemann, Lebesgue, Stieltjes, Burkill, and others—which are used in different situations; most of which can be subsumed or adapted into a gauge integral system. But to assign indiscriminately the blanket designation “gauge integral” to each of the adapted versions is to ignore, firstly, their considerable difference in usages and origins; and, secondly, the fact that “gauge integral” has become practically a synonym for the one‐dimensional generalized Riemann (or Riemann‐complete) integral—also known as the Kurzweil‐Henstock integral.
Also, the general or abstract integral—called Henstock integral in chapter 4 of [MTRV]—has diverged historically from the more mainstream gauge or Kurzweil integration which has “integral‐as‐antiderivative” as its driving force. This aspect of the subject is touched on in Chapter 10 below.
The integral‐as‐antiderivative feature of one‐dimensional Riemann‐complete integration was mentioned in passing in Henstock's 1962–63 exposition [70], which concentrated on other aspects of integration (such as limit theorems3 and Fubini's theorem).
As a student Henstock was attracted to the theory of divergent series. When in 1944 he applied to Paul Dienes to do research in this subject, he was steered towards integration theory [11]; and his subsequent work often focussed on the margins between divergence and convergence.4
The gauge idea made its first appearance in Henstock's published work in [69], in a scenario of extreme divergence in which the gauge method is “tested to destruction” in its first public outing. (This counter‐example is rehearsed in pages 178–181 of [MTRV], section 4.14, Non‐Integrable Functions. See also Example 13 below.) There is no mention in his 1955 paper [69] of the reversal of differentiation which many students of the subject have found so useful. Nor does it touch on the notion of random variation in which theories of integration and measure play a central role, and where integral convergence is much more important than differentiation.5
The emphasis on convergence is maintained in the present book, which can be read as a stand‐alone, self‐contained, or self‐explanatory volume expanding on certain themes in [MTRV]. Like [MTRV] this book aspires to simplicity and transparency. No prior knowledge of the subject matter is assumed, and simple numerical examples set the scene. There is a degree of repetitiveness which may be tedious for experts. But experts can cope with that; more consideration is owed to less experienced readers.
For reasons demonstrated in [MTRV], and amply confirmed in the present volume, non‐absolute convergence is one of the characteristics which, in comparison with other methods, makes the gauge (or ‐complete) integrals suitable for the two main themes of this book: stochastic calculus and Feynman integration.
Stochastic calculus is the branch of the theory of stochastic processes which deals with stochastic integrals, also known as stochastic differential equations. A landmark result is Itô’s lemma, or Itô’s formula.
Stochastic integration is part of the mathematical theory of probability or random variation. Broadly speaking, quantities or variables are random or non‐deterministic if they can assume various unpredictable values; and they are non‐random or deterministic if they can take only definite known values.
Classically, stochastic integrals are constructed by means of a procedure involving weak limits. The purpose of this book is two‐fold:
To treat
stochastic integrals as actual integrals
; so that the limit process which defines a stochastic integral is essentially the same as the limit of Riemann sums which defines the more familiar kinds of integral.
To provide an alternative theory of
stochastic sums
which achieves the same purposes as stochastic integrals, but in a simpler way.
Mathematically, integration is more complicated and more sophisticated than summation (or addition) of a finite number of terms. It is demonstrated that stochastic sums can achieve the same (or better) results as stochastic integrals do. In the theory of stochastic processes, stochastic sums can replace stochastic integrals.
Examples of concrete nature are used to illustrate aspects of stochastic integration and stochastic summation, starting with relatively elementary ideas about finite numbers of things or events, in which there is no difference between summation and integration. A basic calculation of financial mathematics (growth of portfolio value) is used as a reference concept, as a vehicle, and as an aid to intuition and motivation.
In a review [145] of a book [31], Laurent Schwartz stated:
Each of us [Schwartz and Emery] tried to help the probabilists absorb stochastic infinitesimal calculus of the second order “without tears”; I don't know whether any of us succeeded or will succeed.
This book is a further effort in this direction.
The action functionals of quantum mechanics (see (8.7), page below) are analogous to stochastic integrals. They appear as integrands in the infinite‐dimensional integrals used by R.P. Feynman in his theory of quantum mechanics and quantum electrodynamics.
In comparison with alternative approaches such as those of J. Schwinger ([147–150]) and S. Tomonaga ([88–92, 164, 165]), Feynman's method is said to be physically intuitive. It contrasts with the mathematics‐leaning approach of Paul Dirac [27]:
The present lectures, like those of Eddington, are concerned with unifying relativity and quantum theory, but they approach the question from a different point of view. Eddington's method is first to get the physical ideas clear and then gradually to build up a mathematical scheme. The present method is just the opposite—first to set up a mathematical scheme and then try to get its physical interpretation.
In reading [FH] it can be helpful to bear in mind that “[Feynman was] the outstanding intuitionist of our age …”, (attributed to Schwinger in [32]).
Feynman's first published paper on path integrals was [F1], Space‐time approach to non‐relativistic quantum mechanics [39]. In a long tradition of the relationship between physics and mathematics it entailed problems of a pure mathematical kind:
There are very interesting problems involved in the attempt to avoid the subdivision and limiting processes [in Feynman's construction of path integrals]. Some sort of complex measure is being associated with the space of functions. Finite results can be obtained under unexpected circumstances because the measure is not positive everywhere, but the contributions from most of the paths largely cancel out. These curious mathematical problems are largely side‐stepped by the subdivision process. However, one feels as Cavalieri must have felt calculating the volume of a pyramid before the invention of calculus. [39] (R.P. Feynman [F1]; also page 79 of [10].)
These are problems essentially of mathematics, not physics or quantum mechanics. And the solutions proposed in [MTRV], and here, are intended to be contributions to mathematics, not physics.
The space of functions
(for
) is
where
is the set of real numbers and
. It is likely that Feynman's reference to “measure” above relates to Lebesgue‐type measure on measurable subsets of
, which is not available in the form suggested by Feynman. Here are some mathematical issues:
Instead of measurable sets and measure of sets, [MTRV] provides a solution based on a structure of interval‐type subsets of
, with a “natural” volume function for such subsets, and using the ‐complete system of non‐absolute integration described in [MTRV].
Feynman's statement that “
the contributions from most of the paths largely cancel out
” suggests a non‐absolute convergence approach, and confirms the unsuitability of methods requiring absolute convergence.
Stochastic integrals sometimes have the form
where
is a stochastic process. Feynman's integrals often include expressions involving the integral of kinetic energy
. These are
action functionals
, integrals such as
where the latter has the form of a stochastic integral . Generally speaking, for , is non‐differentiable. So none of these functionals actually exists as an integral and, in order to give mathematical meaning to them, various devices have to be used, such as the weak integrals of classical stochastic calculus, or Feynman's subdivision and limiting processes.
Feynman's “
subdivision and limiting processes
” are described in [F1], and in [FH] (
Quantum Mechanics and Path Integrals
[
46
], by R.P. Feynman and A.R. Hibbs). They are also examined in section 7.18 of [MTRV], along with their relationship
6
to the ‐complete integral solution.
This book provides an alternative solution to these problems. Instead of integrals , or , sample times are used to form Riemann sums. These are called stochastic sums in the stochastic integral case, and sampling sums in the case of action integrals:
These functionals are finite sums, not integrals;
they are sample versions of stochastic integrals (or of action integrals in the case of quantum mechanics);
they always exist;
and their expected values and other properties are defined and calculated by a well defined system of ‐complete (or gauge) integration in
.
And, just as it is reasonable to estimate integrals by means of finite Riemann sums, it is equally reasonable to use finite samples to estimate the functional integrands by means of finite samples (or sampling sums).
This book considers mathematical aspects of the Feynman integral theory as it is expounded in [FH], which starts with
a single
particle
interacting with a conservative mechanical force,
and which progresses through to a system consisting of the interaction of a charged particle with an electromagnetic
field
.
For the latter system, [FH] declares that a certain action functional should be integrated over “all degrees of freedom” of the system—over all possible values of each of the variables.
This highly intuitive mode of expression is physically suggestive and resonant. But in mathematics a domain of integration must be defined and formulated as a definite mathematical set composed of definite mathematical elements or points.
In [FH] as in [MTRV] this is achieved for motion by translating “integration over all degrees of freedom” of the single particle motion into integration on a domain consisting of elements or points ; or, simply, . (This deals only with one‐dimensional particle motion. For physical realism elements of the domain should be points of , where
and where , , , are the particle position co‐ordinates in for each .)
For system the domain and its elements are less obvious. In this book the domain
is proposed. This involves a one‐dimensional simplification (like the simplification instead of for system ), and also other simplifications which are contrary to physical reality but which make the mathematical exposition a bit easier to follow. An element of this domain is
where is particle position at time ; and, at time , elements and correspond to electromagnetic field components7 at a point in space. (An element is called a history of the interaction.)
The reason for trailing advance notice of details such as these is to provide a sense of the mathematical challenges presented by quantum electrodynamics (system above), further to the challenges already posed by system .
Feynman's theory of system —or interaction of with —posits certain integrands in domain , the integration being carried out over “all degrees of freedom” of the physical system. But how is an integral on , , to be defined? Is there a theory of measurable sets and measurable functions for ? (Even if such a measure‐theoretic integration actually existed it would fail on the requirement for non‐absolute convergence in quantum mechanics.) And if integrands in “” involve action functionals of the form , we face the further problem of how to give meaning to “” as integrand in domain .
This is reminiscent of the stochastic integrals/stochastic sums issue mentioned above. The resolution in both cases uses the following feature of the ‐complete or gauge system of integration.
A Riemann‐type integral in a one‐dimensional bounded domain is defined by means of Riemann sum approximations where the subintervals of domain are formed from partitions such as
Riemann sums can be expressed as Cauchy8 sums where or . In fact the Riemann‐complete integral can be defined in terms of suitably chosen finite samples of the elements in the domain of integration, without resort to measurable functions or measurable subsets—or even without explicit mention of subintervals of the domain of integration.
To define ‐complete integration in “rectangular” or Cartesian product domains such as above—no matter how complex their construction—the only requirements are:
Exact specification of the elements or points of the domain, and
A structuring of finite samples of points consistent with Axioms DS1 to DS8 of chapter 4 of [MTRV].
In other words integration requires a domain and a process of selecting samples of points or elements of —without reference to measurable subsets, or even to intervals of at the most basic level.
This skeletal structuring of finite samples of points of the domain provides us with a system of integration (the ‐complete integral) with all the useful properties—limit theorems, Fubini's theorem, a theory of measure, and so on. More than that, it provides criteria for non‐absolute convergence (theorems 62, ,9 64, and 65 of [MTRV]) wwynman integrals.
Notes
2
The attachment “‐complete” was introduced by R. Henstock in [
70
], the first book‐length exposition of this kind of integration theory. A few of copies of this edition were printed in 1962. A replacement edition with different page size was printed and distributed in 1963. Up to that time J. Kurzweil and R. Henstock had worked independently on this subject from around the mid‐1950s, without knowledge of each other.
3
Henstock's introduction of the “‐complete” appendage is suggestive of “enhanced integrability of limits” rather than “completeness of a domain with respect to a norm”.
4
As part of the College Prize awarded by St. John's College, Cambridge, on the results of the 1943 Mathematics Tripos Part 2 examination, Henstock received a copy of Dienes’ book [
23
], which includes close analysis of convergence‐divergence issues. In a late, unfinished work [
78
], c. 1992–1993, Henstock used some notable ideas from Dienes’ book.
5
The final chapter of Henstock's 1962‐1963 book [
70
] has the title
Integration in Statistics
Part IStochastic Calculus
Chapter 1Stochastic Integration
The idea or purpose of stochastic integration is to define a random variable
where is a random or unpredictable quantity, depending in a particular manner on unpredictable entities and ; and where
are stochastic processes and depends on time t. In textbooks, the integrand is usually presented as , but is used here in order to emphasise that the integrand is intended to be random.
The integrand (or, when appropriate, ) is to be regarded as a measurable function—as is —with respect to a probability space .
If is a deterministic or non‐random function of s, its value at time s is a definite (non‐random) number which, whenever necessary, can be regarded as a degenerate random variable. If is the same random variable for each s in , each j, then the process is a step function. (In textbooks, the term elementary function is often applied to this.)
The most important kind of stochastic integral is where , is standard Brownian motion, and this particular case (called the Itô integral) is outlined here. The main steps are as follows.
I1
Suppose the integrand
is a step function, with constant random variable value
for
,
. Then define
In this case (that is, a step function), the Itô isometry holds for expected values:
I2
Suppose the process
(not necessarily a step function) satisfies
Then there exists a sequence of step functions (processes) , such that
I3
For such
, define its stochastic integral
with respect to the process
as
I4
If
is Brownian motion the latter limit exists.
An objective of this book is to provide an alternative to the classical theory, not develop it. Thus the commentary, interpretation, and speculation of this section can be safely omitted by anybody who is either already familiar with, or is not interested in, the standard theory of stochastic integration.
Regarding notation, many textbooks use the symbol B for Brownian motion, whereas is used above. Textbooks also use the symbol for the integrand, where is used above. The reason for using notation instead of ) is to emphasise that the value of the integrand function is generally a random variable depending on s, and not generally a single, definite real or complex number (such as the deterministic function , for instance) of the kind which occurs in ordinary integration.
In classical probability theory, an underlying mathematical probability measure space is assumed, such that, for all random variables and processes, the probability that any random variable has an outcome in a particular set can be calculated using the appropriate technical calculation1 relevant to each random variable. If the random variables or processes have a time structure, then mathematical properties of filtration and adaptedness ensure that sets A which qualify as ‐measurable events at earlier times will still qualify as such at subsequent times.
The integrator is a random variable. The integrand function or is also a random variable. And the (stochastic) integral , is a random variable. This point is sometimes illustrated in textbooks by means of examples such as the following.
Example 1
Suppose (a random quantity) is the price of an asset at time t. Then, for times , is the change in the price of the asset, the change or difference also being random. Suppose the quantity of asset holding (sometimes denoted as ) is unpredictable or random. The product of these two,
is then a random variable representing the change in the value of the total asset holding. The stochastic integral , represents the aggregate or sum of these changes over the period of time ; and is a random variable.
Here, use of the symbol for the integrand (instead of the usual ) indicates that while the integrand is a random variable dependent on s, it does not necessarily depend on the integrator random variable . If, in fact, there is such dependence, then an appropriate notation2 for the integrand is .
The notation and terminology of ordinary integration is used in I1, I2, I3, I4, and they provide a certain “feel” for what is going on. But the various elements of the system are clearly different from ordinary integration. Can we get some more precise idea of what is really going on?
The “integration‐like” construction in I1 suggests that the domain of integration is , and that the integrand takes values in a class of functions (—random variables; that is, functions which are measurable with respect to some probability space, or spaces).
How does this compare with more familiar integration scenarios? Basic integration (“”) has two elements: firstly, a domain of integration containing values of the integration variable s, and secondly, an integrand function which depends on the values s in the domain of integration. The more familiar integrand functions have values which are real or complex numbers ; and which are deterministic (that is, “definite”, not approximate or estimated).
The construction in I1, I2, I3 indicates an integration domain or . (There is nothing surprising in that.) But in I1, I2, I3 the integrand values are not real or complex numbers, but random variables—which may be a bit surprising.
But it is not unprecedented. For instance, the Bochner integration process in mathematical analysis deals with integrands whose values are functions, not numbers.
The construction and definition of the Bochner integral [105] is similar in some respects to the classical Itô integral. What is the end result of the construction in I1, I2, I3?
In general, the integral of a function f gives a kind of average or aggregation of all the possible values of f. So if each value of the integrand is a random variable, the integral of f should itself be a random variable—that is, a function which is measurable with respect to an underlying probability measure space.
If the notation is valid or justifiable for the stochastic integral, it suggests that the Itô integral construction derives a single random variable (or ) from many jointly varying random variables, such as , as varies between the values 0 and t. This is reminiscent of Norbert Wiener's construction in [169], which is in some sense a mathematical replication in one dimension of Brownian motion; even though the latter is essentially an infinite‐dimensional phenomenon with infinitely many variables. Without losing any essential information, a situation involving infinitely many variables is converted to a scenario involving only one variable.3
The proof of the Itô isometry relation (see I1) indicates that, as a stochastic process, must be independent of . Otherwise the construction I1, I2, I3 would seem to be inadequate as it stands, whenever the process is replaced by a process .
In I3 the integrand does not have step function form; and, on the face of it, indicates dependence of (or ) on random variables and for everys, . If the integrand were (which, in general, it is not), with joint random variability for , and if is Brownian motion, then the joint probability space for the processes and is given by the Wiener probability measure and its associated multi‐dimensional measure space. (The latter are described in Chapter 5 below.)
Returning to I1, the Itô integral of step function is defined as
where the are random variable values of . It is perfectly valid to combine finite numbers of random variables in this way, in order to produce, as outcome, a single random variable (—which may be a joint random variable depending on many underlying random variables).
This part of the formulation of the integral of a step function in I1 corresponds to the integral of a step function in basic integration, and does not require any passage to a limit of random variables.
Now suppose each is a fixed real number ; so, for , . (Accordingly, in I1, can be regarded as a “degenerate” random variable, with atomic probability value.) Suppose the integrator is the real‐valued ds instead of the random variable‐valued . Then4
Formally, at least, this looks like the definition in I1 of when is a step function. The factor equals for each j. This emerges naturally from the mathematical meaning of the length or distance variable s, and from the mathematical meaning of .
Can this be replicated in I1 when is a step function, or when each is a fixed real number ? Is it the case that
With each , this would imply
If this is unproblematical, it should be possible to deduce it from one or other of the various mathematical definitions of Brownian motion , along with some mathematical definition of the integral in this context.
But it appears that there is no such understanding of . So, as in I1, it seems that this formulation is to be regarded as a basic postulate or axiom of stochastic integration.
Returning to the definition of the classical Itô integral, I2 has the following condition on the expected value of the integral of the process :
The idea here is that, if is the random entity obtained by carrying out some form of weighted aggregation—denoted by —of all the individual random variables (), then
This formulation assumes that the aggregative operation , involving infinitely many random variables (), produces a single random entity whose expected value can be obtained by means of the operation .
Additionally, is said to be a Lebesgue integral‐type construction. The part of this statement should be unproblematical. The domain is a real interval, and has a distance or length function, which, in the context of Lebesgue integration on the domain, gives rise to Lebesgue measure on the space of Lebesgue measurable subsets of . So can also be expressed as .
However, the random variable‐valued integrand is less familiar in Lebesgue integration. Suppose, instead, that the integrand is a real‐number‐valued function . Then the Lebesgue integral , or , is defined if the integrand function f is Lebesgue measurable. So if J is an interval of real numbers in the range of f, the set is a member of the class of measurable sets; giving
That is, for each J, is a Lebesgue measurable subset of . This is valid if, for instance, f is a continuous function of s, or if f is the limit of a sequence of step functions.
How does this translate to a random variable‐valued integrand such as ? Two kinds of measurability arise here, because, in addition to being a ‐measurable function of , is a random variable (as is ), and is therefore a P‐measurable function on the sample space :
Likewise . For to be meaningful as a Lebesgue‐type integral, the integrand must be ‐measurable (or ‐measurable) in some sense. At least, for purpose of measurability there needs to be some metric in the space of ‐measurable functions , , with , :
For example, the “distance” between and could be
With such a metric at hand, it may then be possible to define , or , as the limit of the integrals of (integrable) step functions converging to for , as .
Unfortunately, most standard textbooks do not give this point much attention. But for relatively straightforward integrands such as , it should not be too difficult.
Continuing the discussion of I1, I2, I3, I4, it appears that the output of this definition of stochastic integral is a random entity ; perhaps a process which is some collection of random variables .
Again comparing this with basic integration of a real number‐valued function , the integral is some kind of average or weighted aggregate value for . This integral, if it exists, produces a single unique real number (depending on the value of t), denoted by .
For random variable‐valued integrand , suppose (for the purpose of speculation) that the stochastic integral
(if it exists) is equivalent (in some unspecified sense) to a single, unique random variable . Remember, a random variable is a function, usually real‐valued5, defined on a sample space . Two such functions, and , are the same function if and only if
Does the definition of stochastic integral in I1, I2, I3 yield such a unique value for ? I2 and I3 do not guarantee uniqueness: there may be different sequences in I2 which converge “in mean square” to . In effect, I4 asserts weak convergence of the integrals of the step functions to a value for the integral of , that value being not necessarily unique.
If the integral does not have a unique value, what connections may exist between alternative values? Suppose there is more than one candidate random variable, say and , for the value of the stochastic integral,
In that case, what is the relation between and ? For instance, is it the case that, for each real number a, the probabilities of corresponding measurable sets are equal (such as ):
The framework outlined above does not include the important case , where () is Brownian motion. Broadly speaking, means that the random variables represented by finite sums
converge as tend to zero, each j. In fact the convergence is weak, not point‐wise, with
and the weak limit t is a fixed real number which can be regarded as a degenerate random variable. This result is basic to the construction I1, I2, I3, I4.
A closer reading of source material may provide answers and/or corrections to some or all of the above comments and queries. Any misinterpretation, confusion, and errors may be dispelled by closer examination of the underlying ideas.
Aside from these issues, and looking beyond the classical mathematical theory, the general idea of stochastic integral is, in intuitive terms, a persuasive, natural and practical way of thinking about the underlying reality.
An alternative (and hopefully more understandable) mathematical way of representing this reality is presented in subsequent chapters of this book.
Example 2
In order to focus on the underlying ideas, here is a simple illustration. Suppose, at different times t, (or
