product iamge

Foundations of Quantitative Finance: Book III. The Integrals of Riemann, Lebesgue and (Riemann-)Stieltjes

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

Every financial professional wants and needs an advantage. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not―and that is the advantage these books offer the astute reader.

Published under the collective title of Foundations of Quantitative Finance, this set of ten books presents the advanced mathematics finance professionals need to advance their careers. These books develop the theory most do not learn in Graduate Finance programs, or in most Financial Mathematics undergraduate and graduate courses.

As a high-level industry executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial industry and two decades in education where he taught in highly respected graduate programs.

Readers should be quantitatively literate and familiar with the developments in the first books in the set. The set offers a linear progression through these topics, though each title can be studied independently since the collection is extensively self-referenced.

Book III: The Integrals of Lebesgue and (Riemann-) Stieltjes, develops several approaches to an integration theory. The first two approaches were introduced in the Chapter 1 of Book I to motivate measure theory. The general theory of integration on measure spaces will be developed in Book V, and stochastic integrals then studies on Book VIII.

Book III Features:

    • Extensively referenced to utilize materials from earlier books.

    • Presents the theory needed to better understand 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 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 10: Applications and Classic Models

    Author(s): Robert R. Reitano
    Series: Chapman & Hall/CRC Finance Series
    Publisher: CRC Press/Chapman & Hall
    Year: 2023

    Language: English
    Pages: 213
    City: Boca Raton

    Cover
    Half Title
    Series Page
    Title Page
    Copyright Page
    Dedication
    Contents
    Preface
    Author
    Introduction
    1. The Riemann Integral
    1.1. The Integral in ℝ
    1.1.1. The Riemann Approach
    1.1.2. The Darboux Approach
    1.1.3. Riemann and Darboux Equivalence
    1.1.4. On Existence of the Integral
    1.1.5. Properties of the Integral
    1.1.6. Evaluating Integrals
    1.2. The Integral and Function Limits
    1.2.1. Positive Results on Limits
    1.2.2. Generalization Counterexamples
    1.3. The Integral in ℝn
    1.3.1. The Riemann Approach
    1.3.2. The Darboux Approach
    1.3.3. Riemann and Darboux Equivalence
    1.3.4. On Existence of the Integral
    1.3.5. Properties of the Integral
    1.3.6. Evaluating Integrals
    2. The Lebesgue Integral
    2.1. Integrating Simple Functions
    2.2. Integrating General Functions
    2.3. Bounded Measurable Functions
    2.3.1. Definition of the Integral
    2.3.2. Riemann Implies Lebesgue
    2.3.3. Properties of the Integral
    2.3.4. Bounded Convergence Theorem
    2.3.5. Evaluating Integrals
    2.4. Nonnegative Measurable Functions
    2.4.1. Definition of the Integral
    2.4.2. Properties of the Integral
    2.4.3. Fatou's Lemma
    2.4.4. Lebesgue's Monotone Convergence Theorem
    2.4.5. Evaluating Integrals
    2.5. General Measurable Functions
    2.5.1. Definition of the Integral
    2.5.2. Properties of the Integral
    2.5.3. Lebesgue's Dominated Convergence Theorem
    2.5.4. Evaluating Integrals
    2.5.5. Absolute Riemann Implies Lebesgue
    2.6. Summary of Convergence Results
    2.6.1. Lebesgue Integration to the Limit
    2.6.2. The Riemann Integral: A Discussion
    3. Lebesgue Integration and Differentiation
    3.1. Derivative of a Lebesgue Integral
    3.1.1. Monotonic Functions
    3.1.2. Functions Differentiable a.e.
    3.1.3. The Lebesgue FTC – Version II
    3.2. Lebesgue Integral of a Derivative
    3.2.1. Examples of FTC Failures
    3.2.2. The Lebesgue FTC – Version I
    3.3. Lebesgue Integration by Parts
    4. Stieltjes Integration
    4.1. Introduction
    4.1.1. The Riemann-Stieltjes Integral
    4.1.2. The Lebesgue-Stieltjes Integral
    4.2. Riemann-Stieltjes Integral on ℝ
    4.2.1. Definition of the Integral
    4.2.2. Riemann and Darboux Equivalence
    4.2.3. On Existence of the Integral
    4.2.4. Integrators of Bounded Variation
    4.2.5. Properties of the Integral
    4.2.6. Evaluating Riemann-Stieltjes Integrals
    4.3. A Result on R-S Integrators
    4.3.1. Banach Spaces
    4.3.2. Baire's Category Theorem
    4.3.3. Banach-Steinhaus Theorem
    4.3.4. Final Result on R-S Integrators
    4.4. Riemann-Stieltjes Integrals on ℝn
    4.4.1. Multivariate Integrators
    4.4.2. Definition of the Integral
    4.4.3. Riemann and Darboux Equivalence
    4.4.4. On Existence of the Integral
    4.4.5. Properties of the Integral
    4.4.6. Evaluating Riemann-Stieltjes Integrals
    References
    Index