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Foundations of Quantitative Finance: 1. Measure Spaces and Measurable Functions

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Foundations of Quantitative Finance: 1. Measure Spaces and Measurable Functions Foundations of Quantitative Finance: 1. Measure Spaces and Measurable Functions Robert R. Reitano Brandeis International Business School Waltham, MA 02454 October, 2017 Copyright © 2017 by Robert R. Reitano Brandeis International Business School . This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivativ es 4.0 International License. To view a copy of the license, visit: https://creativecommons.org/licenses/by-nc-nd/4.0/ Contents Preface ix to Michael, David, and Jeffrey xi Introduction xiii 1 Motivation for the Notion of Measure 1 1.1 The Riemann Integral . 3 1.2 The Lebesgue Integral . 5 2 Measurable Sets and Lebesgue Measure 11 2.1 Sigma Algebras and Borel Sets . 11 2.2 Definition of Lebesgue Measure . 19 2.3 Lebesgue Measure on (P (R)): An Attempt and Failure . 22 2.4 Lebesgue Measurable Sets: L $ (P (R)) . 30 M 2.5 Calculating Lebesgue Measure . 37 2.6 Approximating Lebesgue Measurable Sets . 38 2.7 Important Properties of Lebesgue Measure . 40 2.7.1 Regularity . 40 2.7.2 Continuity . 41 2.8 Discussion on (R) & .................... 43 B ML 3 Measurable Functions 47 3.1 Extended Real-Valued Functions . 47 3.2 Alternative Definitions of Measurability . 48 3.3 Examples of Measurable Functions . 52 3.3.1 Continuous Functions . 53 3.3.2 Characteristic or Indicator Functions . 59 3.3.3 A Nonmeasurable Function . 62 v vi CONTENTS 3.4 Properties of Measurable Functions . 62 3.4.1 Elementary Function Combinations . 64 3.4.2 Function Sequences . 69 3.5 Approximating Lebesgue Measurable Functions . 75 3.6 Distribution Functions of Measurable Functions . 80 4 Littlewood’sThree Principles 89 4.1 Measurable Sets . 89 4.2 Convergent Sequences of Measurable Functions . 91 4.3 Measurable Functions . 93 5 General Borel Measures on R 95 5.1 Distribution Functions from Borel Measures . 98 5.2 Borel Measures from Distribution Functions . 100 5.2.1 From F -Length to a Measure on an Algebra . 102 5.2.2 To a Measure on a Sigma Algebra . 107 5.3 Consistency of Borel Constructions . 115 5.4 Important Properties of Borel Measures . 116 5.5 Approximating Measurable Sets under a Borel Measure . 117 5.6 Differentiable F -Length and Lebesgue Measure . 120 6 Generating Measures by Extension 125 6.1 Recap of Lebesgue and Borel Procedures . 125 6.2 Extension Theorems . 127 6.2.1 From Outer Measure to Complete Measure . 127 6.2.2 Measure on an Algebra to Outer Measure . 130 6.2.3 Approximating Carathéodory Measurable Sets . 131 6.2.4 Pre-Measure on Semi-Algebra to Measure on Algebra 133 6.2.5 Uniqueness Of Extensions . 137 6.3 Summary - Construction of Measure Spaces . 140 6.4 Approaches to Countable Additivity . 143 6.4.1 Countable Subadditivity on a Semi-Algebra . 143 6.4.2 Countable Additivity on an Algebra . 144 6.5 Completion of a Measure Space . 146 7 Finite Products of Measure Spaces 151 7.1 Product Space Semi-Algebras . 151 7.2 Properties of the Semi-Algebras . 156 7.3 Measure on the Algebra . 159 A 7.3.1 Finite Additivity on the Semi-Algebra . 160 A0 CONTENTS vii 7.3.2 Countable Additivity on the Algebra for -Finite Spaces . .A . 168 7.4 Extension to a Measure on the Product Space . 173 7.5 Well-Definedness of -Finite Product Measure Spaces . 174 7.6 Products of Lebesgue and Borel Measure Spaces . 180 7.7 Regularity of Lebesgue and Borel Product Measures . 182 8 More General Borel Measures on Rn 187 8.1 The Borel -Algebra on Rn . 187 8.2 Borel Measures and Induced Functions . 189 8.2.1 Functions Induced by Finite Borel Measures . 190 8.2.2 Borel Measures Induced by Functions . 195 9 Infinite Product of Probability Spaces 203 9.1 A Naive Attempt . 203 9.2 A Semi-Algebra Structure . 205 9.3 Finite Additivity on the Semi-Algebra 0 in Probability Spaces208 9.4 "Free" Countable Additivity on Finite PointA Probability Spaces210 9.5 Countable Additivity on Algebra + in Probability Spaces A on R ................................213 9.5.1 Outline of Countable Additivity Proof . 215 + 9.5.2 The Algebra and Finite Additivity of A . 217 9.5.3 Countable AdditivityA on + . 222 A 9.6 Extension to a Measure on the Product Space RN . 225 9.7 Probabilities of Infinite Dimensional Rectangles in (RN, (RN), N) .................................227 References 231 Preface The idea for a reference book on the mathematical foundations of quantita- tive finance has been with me throughout my career in this field. But the urge to begin writing it didn’tmaterialize until shortly after completing my first book, Introduction to Quantitative Finance: A Math Tool Kit, in 2010. The one goal I had for this reference book was that it would be complete and detailed in the development of the many materials one finds referenced in the various areas of quantitative finance. The one constraint I realized from the beginning was that I could not accomplish this goal, plus write a complete survey of the quantitative finance applications of these materials, in the 700 or so pages that I budgeted for myself for my first book. Little did I know at the time that this project would require a multiple of this initial page count budget even without detailed finance applications. I was never concerned about the omission of the details on applications to quantitative finance because there are already a great many books in this area that develop these applications very well. The one shortcoming I perceived many such books to have is that they are written at a level of mathematical sophistication that requires a reader to have significant formal training in mathematics, as well as the time and energy to fill in omitted details. While such a task would provide a challenging and perhaps welcome exercise for more advanced graduate students in this field, it is likely to be less welcome to many other students and practitioners. It is also the case that quantitative finance has grown to utilize advanced mathematical theories from a number of fields. While there are again a great many very good references on these subjects, most are again written at a level that does not in my experience characterize the backgrounds of most students and practitioners of quantitative finance. So over the past several years I have been drafting this reference book, accumulating the mathematical theories I have encountered in my work in this field, and then attempting to integrate them into a coherent collection of books that develops the necessary ideas in some detail. My target readers would be quantitatively literate to the extent of familiarity, indeed comfort, with the materials and formal developments in my first book, and suffi ciently motivated to identify and then navigate the details of the materials they were attempting to master. Unfortunately, adding these details supports learning but also increases the lengths of the various developments. But this book was never intended to provide a “cover-to-cover”reading challenge, but rather to be a reference book in which one could find detailed foundational materials in a variety of areas that support further studies in quantitative finance. ix x Preface Over these past years, one volume turned into two, which then became a work not likely publishable in the traditional channels given its unforgiving size and likely limited target audience. So I have instead decided to self- publish this work, converting the original chapters into stand-alone books, of which there are now nine. My goal is to finalize each book over the coming year or two. I hope these books serve you well. I am grateful for the support of my family: Lisa, Michael, David, and Jeffrey, as well as the support of friends and colleagues at Brandeis Interna- tional Business School. Robert R. Reitano Brandeis International Business School to Michael, David, and Jeffrey xi Introduction This is the first book in a series of several that will be self-published under the collective title of Foundations of Quantitative Finance. Each book in the series is intended to build from the materials in earlier books, alternating between books with a more foundational mathematical perspective, which is the case with this first book, and books which develop probability theory and quantitative applications to finance. But while providing many of the foundational theories underlying quantitative finance, this series of books does not provide a detailed development of these financial applications. Instead this series is intended to be used as a reference work for researchers and practitioners of quantitative finance who already have other sources for these detailed financial applications but find that such sources are written at a level which assume significant mathematical expertise, which if not possessed can be diffi cult to supplement. Because the goal of many books in quantitative finance is to develop fi- nancial applications from an advanced point of view, it is often the case that advanced foundational materials from mathematics and probability theory are introduced and summarized but without a complete and formal develop- ment that would lead the authors too far astray from their intended objec- tives. And while there are a great many excellent books on mathematics and probability theory, a number of which are cited in the references, such books typically develop materials with a eye to comprehensiveness in the subject matter, and not with an eye toward effi ciently curating and developing the theory needed for applications in quantitative finance. Thus the goal of this series is to introduce and develop in some detail a number of the foundational theories underlying quantitative finance, with topics curated from a vast mathematical and probability literature for the express purpose of supporting applications in quantitative finance.
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