Foundations of Quantitative Finance, Book I: Measure Spaces and Measurable Functions (Chapman & Hall/CRC Finance Series)

Description

This is the first in a set of 10 books written for professionals in quantitative finance. These books fill the gap between informal mathematical developments found in introductory materials, and more advanced treatments that summarize without formally developing the important foundational results professionals need. Book I in the Foundations in Quantitative Finance Series develops topics in measure spaces and measurable functions and lays the foundation for subsequent volumes. Lebesgue and then Borel measure theory are developed on ℝ, motivating the general extension theory of measure spaces that follows. This general theory is applied to finite product measure spaces, Borel measures on ℝn, and infinite dimensional product probability spaces. The overriding goal of these books is a complete and detailed development of the many mathematical theories and results one finds in popular resources in finance and quantitative finance. Each book is dedicated to a specific area of mathematics or probability theory, with applications to finance that are relevant to the needs of professionals. Practitioners, academic researchers, and students will find these books valuable to their career development. All ten volumes are extensively self-referenced. The reader can enter the collection at any point or topic of interest, and then work backward to identify and fill in needed details. This approach also works for a course or self-study on a given volume, with earlier books used for reference. Advanced quantitative finance books typically develop materials with an eye to comprehensiveness in the given subject matter, yet not with an eye toward efficiently curating and developing the theories needed for applications in quantitative finance. This book and series of volumes fill this need. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface Author Bio Introduction 1. The Notion of Measure 0 1.1 Riemann Integrals 1.2 Lebesgue Integrals 2. Lebesgue Measure on R 2.1 Sigma Algebras and Borel Sets 2.2 Definition of a Lebesgue Measure 2.3 Is There a Lebesgue Measure on σ(P(R))? 2.4 Lebesgue Measurable Sets: ML(R) σ (P(R)) 2.5 Calculating Lebesgue Measures 2.6 Approximating Lebesgue Measurable Sets 2.7 Properties of Lebesgue Measure 2.7.1 Regularity 2.7.2 Continuity 2.8 Discussion of B(R) ML(R) 3. Measurable Functions 3.1 Extended Real-Valued Functions 3.2 Equivalent Definitions of Measurability 3.3 Examples of Measurable Functions 3.3.1 Continuous Functions 3.3.2 Characteristic or Indicator Functions 3.3.3 A Nonmeasurable Function 3.4 Properties of Measurable Functions 3.4.1 Elementary Function Combinations 3.4.2 Function Sequences 3.5 Approximating Measurable Functions 3.6 Distribution Functions 4. Littlewood’s Three Principles 4.1 Measurable Sets 4.2 Sequences of Measurable Functions 4.3 Measurable Functions 5. Borel Measures on R 5.1 Functions Induced by Measures 5.2 Measures from Distribution Functions 5.2.1 F-Length to a Measure on an Algebra 5.2.2 To a Borel Measure 5.3 Consistency of Borel Constructions 5.4 Approximating Borel Measurable Sets 5.5 Properties of Borel Measures 5.5.1 Continuity 5.5.2 Regularity 5.6 Differentiable F(x) 6. Measures by Extension 6.1 Recap of Lebesgue and Borel Constructions 6.2 Extension Theorems 6.2.1 From Outer Measure to Complete Measure 6.2.2 Measure on an Algebra to a Complete Measure 6.2.3 Approximating Carathéodory Measurable Sets 6.2.4 Pre-Measure on Semi-Algebra to Measure on Algebra 6.2.5 Uniqueness of Extensions 1 6.3 Summary of Construction Process 6.4 Approaches to Countable Additivity 6.4.1 Countable Additivity on a Semi-Algebra 6.4.2 Countable Additivity on an Algebra 6.5 Completion of a Measure Space 6.5.1 Uniqueness of Extensions 2 7. Finite Products of Measure Spaces 7.1 Product Space Semi-Algebras 7.2 Properties of the Semi-Algebra A' 7.3 Measure on the Algebra A 7.3.1 Finite Additivity on the Semi-Algebra A' 7.3.2 Countable Additivity on the Algebra A for σ-Finite Spaces 7.4 Extension to a Measure on the Product Space 7.5 Well-Definedness of Product Measure Spaces 7.6 Lebesgue and Borel Product Spaces 8. Borel Measures on Rn 8.1 Rectangle Collections that Generate B(Rn) 8.2 Borel Measures and Induced Functions 8.2.1 Functions Induced by Finite Borel Measures 8.2.2 Functions Induced by General Borel Measures 8.2.3 Borel Measures Induced by Functions 8.3 Properties of Borel Measures on Rn 8.3.1 Uniqueness and Consistency 8.3.2 Approximating Borel Measurable Sets 8.3.3 Continuity and Regularity 9. Infinite Product Spaces 9.1 Naive Attempt at a First Step 9.2 Semi-Algebra A' 9.3 Finite Additivity of μA on A 9.4 Free Countable Additivity on Finite Spaces 9.5 Countable Additivity on A+ 9.5.1 Outline of Proof and Need for A+ 9.5.2 Algebra A+ and Finite Additivity of μA 9.5.3 Countable Additivity of μA on A+ 9.6 Extension to a Probability Measure on RN 9.7 Probability of General Rectangles References Index