Foundations of Quantitative Finance, Book I: Measure Spaces and Measurable Functions book cover

Foundations of Quantitative Finance, Book I: Measure Spaces and Measurable Functions book cover

1st Edition

Foundations of Quantitative Finance, Book I: Mensurate Spaces and Measurable Functions

Copyright Year 2022

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Book Description

This is the first book in a set of 10 written for professionals in quantitative finance. These books fill a gap between informal developments in mathematics constitute in introductory materials and more advanced materials that summarize without formal developing the important results professionals need to learn and employ.

Book 1 in the Foundations in Quantitative Finance Serial covers measure out theory, including the Riemann and Lebesgue Integral, measurable functions, Borel measures in R, and Littlewood'south three principles, a rarity in books on these topics. It lays the foundation for the subsequent volumes.

The overriding goal of these books is a consummate and detailed development of the many mathematical theories and results one finds in pop resource in finance and quantitative finance. Each book is dedicated to a specific surface area of mathematics or probability theory, with applications to finance that are relevant to the needs of professionals. A range of practitioners, students, and researchers will find these books valuable to their career development.

All ten volumes are completely self-referencing. The reader can enter the drove at any point or topic of interest, so work backwards to identify and make full in needed details. This approach also works for a course on a given volume, with earlier books used for reference. In addition, these volumes back up both class-based and self-report approaches to a multifariousness of sequential studies.

The field of quantitative finance books typically develop materials with an eye to comprehensiveness in the given subject matter, even so not with an centre toward efficiently curating and developing the theories needed for applications in quantitative finance. This book and serial of volumes fills this need.

Table of Contents

Preface
Introduction

1 The Notion of Measure 0
1.1 The Riemann Integral
i.2 The Lebesgue Integral

2 Lebesgue Measure on R xiii
2.1 Sigma Algebras and Borel Sets
2.2 Definition of a Lebesgue Measure
2.3 At that place is No Lebesgue Measure out on _(P(R)
ii.4 Lebesgue Measurable Sets: ML(R) $ _(P(R))
2.v Calculating Lebesgue Measures
2.6 Approximating Lebesgue Measurable Sets
ii.seven Properties of Lebesgue Measure out
2.7.1 Regularity
2.7.2 Continuity
2.eight 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
three.iv Properties of Measurable Functions
3.4.1 Elementary Role Combinations
3.4.2 Function Sequences
Function Sequence Behaviors
Role Sequence Measurability Properties
3.five Approximating Lebesgue Measurable Functions
3.vi Distribution Functions of Measurable Functions

4 Littlewood.south Iii Principles
4.one Measurable Sets
four.ii Convergent Sequences of Measurable Functions
iv.3 Measurable Functions

5 Borel Measures on R
five.i Functions Induced by Borel Measures
5.2 Borel Measures from Distribution Functions
five.iii Consistency of Borel Measure Constructions
v.four Approximating Borel Measurable Sets
v.5 Properties of Borel Measures
5.6 Differentiable F-Length and Lebesgue Measure

vi Generating Measures by Extension
6.1 Recap of Lebesgue and Borel Constructions
6.two Extension Theorems
6.3 Summary - Construction of Measure Spaces
6.4 Approaches to Countable Additivity
half dozen.5 Completion of a Measure Infinite

7 Finite Products of Measure Spaces
7.1 Product Space Semi-Algebras
vii.2 Properties of the Semi-Algebra
7.3 Measure on the Algebra A
7.four Extension to a Measure on the Product Space
7.5 Well-Definedness of _-Finite Production Measure out Spaces
7.6 Products of Lebesgue and Borel Measure Spaces

 viii Borel Measures on Rn
viii.ane Rectangle Collections that Generate B(Rn)
8.2 Borel Measures and Induced Functions
eight.three Backdrop of General Borel Measures on Rn

9 Space Products of Probability Spaces
ix.i A Naive Attempt at a First Step
9.ii A Semi-Algebra A0
9.iii 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
ix.6 Extension to a Probability Mensurate on RN
9.7 Probability of General Rectangles

References

Writer(southward)

Biography

Robert R. Reitano is Professor of the Practise in Finance at the Brandeis International Business School where he specializes in chance 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 nugget/liability run a 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 take won an Annual Prize of the Society of Actuaries and ii F.Yard. Redington Prizes awarded biennially by the Investment Section of the Guild of the Actuaries. Dr. Reitano has served as Vice-Chair of the Board of Directors of the Professional person Adventure 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 Commission of the Society of Actuaries, and other non-for-profit boards and investment committees.