1st Edition

Foundations of Quantitative Finance: Book V General Measure and Integration Theory

By Robert R. Reitano Copyright 2024
    256 Pages
    by Chapman & Hall

    256 Pages
    by Chapman & Hall

    Every finance professional wants and needs a competitive edge. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not—and that is the competitive edge these books offer the astute reader.

    Published under the collective title of Foundations of Quantitative Finance, this set of ten books develops the advanced topics in mathematics that finance professionals need to advance their careers. These books expand the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.

    As an investment executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial services industry and two decades in academia where he taught in highly respected graduate programs.

    Readers should be quantitatively literate and familiar with the developments in the earlier books in the set. While the set offers a continuous progression through these topics, each title can be studied independently.


    • Extensively referenced to materials from earlier books
    • Presents the theory needed to support advanced applications
    • Supplements previous training in mathematics, with more detailed developments
    • Built from the author's five decades of experience in industry, research, and teaching

    Published and forthcoming titles in the Robert R. Reitano Quantitative Finance Series:

    Book I: Measure Spaces and Measurable Functions

    Book II: Probability Spaces and Random Variables

    Book III: The Integrals of Lebesgue and (Riemann-)Stieltjes

    Book IV: Distribution Functions and Expectations

    Book V: General Measure and Integration Theory

    Book VI: Densities, Transformed Distributions, and Limit Theorems

    Book VII: Brownian Motion and Other Stochastic Processes

    Book VIII: Itô Integration and Stochastic Calculus 1

    Book IX: Stochastic Calculus 2 and Stochastic Differential Equations

    Book X: Classical Models and Applications in Finance


    1 Measure Spaces                                                                                       
    1.1 Lebesgue and Borel Spaces on JR 
    1.1.2 LebesgueMeasureSpace
    1.1.3 BorelMeasureSpaces
    1.2 GeneralExtensionTheory
    1.3 MeasureSpaceConstructions
    1.3.2 Borel Measures on JRn 
    1.3.3 InfiniteProductsofProbabilitySpaces
    1.4 ContinuityofMeasures

    2. Measurable Functions                                                                           
    2.1 Properties of Measurable Functions
    2.2 Limits of Measurable Functions
    2.3  Results on Function Sequences
    2.4 Approximating o-(X)-Measurable Functions
    2.5 Monotone Class Theorems
    2.5.1 Monotone Class Theorem
    2.5.2 Functional Monotone Class Theorem

    3 General Integration Theory                                                                              
    3.1 Integrating Simple Functions
    3.2 Integrating Nonnegative Measurable Functions
    3.2.1 Fatou's Lemma
    3.2.2 Lebesgue's Monotone Convergence Theorem
    3.2.3 Properties of Integrals
    3.2.4 Product Space Measures Revisited 
    3.3 Integrating General Measurable Functions
    3.3.1 Properties of Integrals
    3.3.2 Beppo Levi's Theorem
    3.3.3 Lebesgue's Dominated Convergence Theorem
    3.3.4 Bounded Convergence Theorem
    3.3.5 Uniform Integrability Convergence Theorem
    3.4 Leibniz Integral Rule
    3.4.1 Riemann Integrals
    3.4.2 Lebesgue/Lebesgue-Stieltjes Integrals
    3.5 Lebesgue-Stieltjes vs. Riemann-Stieltjes Integrals
    3.5.1 Lebesgue-Stieltjes Integrals on JR
    3.5.2 Lebesgue-Stieltjes Integrals on JRn

    4 Change of Variables 
    4.1 Change of Measure-Special Case
    4.1.1 Measures Defined by Integrals
    4.1.2 Integrals and Change of Measure
    4.2 Transformations and Change of Measure
    4.2.1 Measures Induced by Transformations
    4.2.2 Change of Variables Under Transformations
    4.3 Special Cases of Change of Variables
    4.3.1 Lebesgue Integrals on JR
    4.3.2 Linear Transformations on JRn
    4.3.3 Differentiable Transformations on JRn

    5 Integrals in Product Spaces                                                                
    5.1 Product Space Sigma Algebras
    5.1.1 Sigma Algebra Constructions
    5.1.2  Implications for Chapter Results
    5.2  Preliminary Results
    5.2.1 Introduction to Fubini/Tonelli Theorems
    5.2.2 Integrals of Characteristic Functions
    5.3 Fubini's Theorem
    5.3.1 Generalizing Fubini's Theorem
    5.3.2  Fubini's Theorem in o-1 (X x Y )
    5.4  Tonelli's Theorem
    5.4.1 Tonelli's Theorem in o-1 (X x Y )
    5.5  Examples

    6  Two Applications of Fubini/Tonelli                                                    

    6.1 Lebesgue-Stieltjes Integration by Parts
    6.1.1 Functions of Bounded Variation
    6.1.2 Lebesgue-Stieltjes Integration by Parts
    6.2 Convolution of Integrable Functions

    7  The Fourier Transform                 
    7.1 Integration of Complex Valued Functions
    7.2 Fourier Transforms
    7.3 Properties of Fourier Transforms
    7.4 Fourier-Stieltjes Inversion of ¢F (t)
    7.5 Fourier Inversion of Integrable ¢F (t)
    7.5.1 Integrability Versus Decay at ±oo
    7.5.2 Fourier Inversion: From Integrable ¢F (t) to J (x)
    7.6 Continuity Theorem for Fourier Transforms

    8 General Measure Relationships                                                            

    8.1 Decomposition of Borel Measures on (JR, B(JR), m)
    8.2   Decomposition of o--Finite Measures
    8.2.1 Signed Measures
    8.2.2 The Hahn and Jordan Decompositions
    8.2.3  The Radon-Nikodym Theorem
    8.2.4   The Lebesgue Decomposition Theorem

    9 The Lp Spaces                                                                                        
    9.1 Introduction to Banach Spaces
    9.2 The Lp(X)-Spaces
    9.3 Approximating Lp(X)-Functions
    9.4 Bounded Linear Functionals on Lp(X)-Spaces
    9.5 Hilbert Space: A Special Case of p = 2



    Robert R. Reitano is Professor of the Practice in Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University's Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management, was Chief Investment Officer of Controlled Risk Insurance Company (CRICO), and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M. Redington Prizes awarded biennially by the Investment Section of the Society of the Actuaries. Dr. Reitano has served as Vice-Chair of the Board of Directors of the Professional Risk Managers International Association (PRMIA) and on the Executive Committee of the PRMIA Board, and is currently a member of the PRMIA Boston Steering Committee, the Financial Research Committee of the Society of Actuaries, and other not-for-profit boards and investment committees.