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 , motivating the general extension theory of measure spaces that follows. This general theory is applied to finite product measure spaces, Borel measures on n, 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.
Inhaltsverzeichnis
Preface
Introduction
1 The Notion of Measure 0
1.1 The Riemann Integral
1.2 The Lebesgue Integral
2 Lebesgue Measure on R 13
2.1 Sigma Algebras and Borel Sets
2.2 Definition of a Lebesgue Measure
2.3 There is No Lebesgue Measure on _(P(R)
2.4 Lebesgue Measurable Sets: ML(R) $ _(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 on B(R) &ML(R)
3 Measurable Functions 55
3.1 Extended Real-Valued Functions
3.2 Equivalent Definitions of Measurability
3.3 Examples of Measurable Functions
3.4 Properties of Measurable Functions
3.4.1 Elementary Function Combinations
3.4.2 Function Sequences
Function Sequence Behaviors
Function Sequence Measurability Properties
3.5 Approximating Lebesgue Measurable Functions
3.6 Distribution Functions of Measurable Functions
4 Littlewood.s Three Principles
4.1 Measurable Sets
4.2 Convergent Sequences of Measurable Functions
4.3 Measurable Functions
5 Borel Measures on R
5.1 Functions Induced by Borel Measures
5.2 Borel Measures from Distribution Functions
5.3 Consistency of Borel Measure Constructions
5.4 Approximating Borel Measurable Sets
5.5 Properties of Borel Measures
5.6 Differentiable F-Length and Lebesgue Measure
6 Generating Measures by Extension
6.1 Recap of Lebesgue and Borel Constructions
6.2 Extension Theorems
6.3 Summary - Construction of Measure Spaces
6.4 Approaches to Countable Additivity
6.5 Completion of a Measure Space
7 Finite Products of Measure Spaces
7.1 Product Space Semi-Algebras
7.2 Properties of the Semi-Algebra
7.3 Measure on the Algebra A
7.4 Extension to a Measure on the Product Space
7.5 Well-Definedness of _-Finite Product Measure Spaces
7.6 Products of Lebesgue and Borel Measure Spaces
8 Borel Measures on Rn
8.1 Rectangle Collections that Generate B(Rn)
8.2 Borel Measures and Induced Functions
8.3 Properties of General Borel Measures on Rn
9 Infinite Products of Probability Spaces
9.1 A Naive Attempt at a First Step
9.2 A Semi-Algebra A0
9.3 Finite Additivity of _A on A for Probability Spaces
9.4 Free Countable Additivity on Finite Probability Spaces
9.5 Countable Additivity on A+ in Probability Spaces on R
9.6 Extension to a Probability Measure on RN
9.7 Probability of General Rectangles
References
