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 an investment 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 book in the set. While the set offers a continuous progression through these topics, each title can also be studied independently. Features Published and forthcoming titles in the Robert R. Reitano Quantitative Finance Book Measure Spaces and Measurable Functions Book Probability Spaces and Random Variables Book III : The Integrals of Lebesgue and (Riemann-)Stieltjes Book Distribution Functions and Expectations Book General Measure and Integration Theory Book Densities, Transformed Distributions, and Limit Theorems Book VII : Brownian Motion and Other Stochastic Processes Book Itô Integration and Stochastic Calculus 1 Book Stochastic Calculus 2 and Stochastic Differential Equations Book Classical Models and Applications in Finance
Author(s): Robert R. Reitano
Publisher: CRC Press
Year: 2023
Language: English
Pages: 269
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
Introduction
1. Distribution and Density Functions
1.1. Summary of Book II Results
1.1.1. Distribution Functions on R
1.1.2. Distribution Functions on Rn
1.2. Decomposition of Distribution Functions on R
1.3. Density Functions on R
1.3.1. The Lebesgue Approach
1.3.2. Riemann Approach
1.3.3. Riemann-Stieltjes Framework
1.4. Examples of Distribution Functions on R
1.4.1. Discrete Distribution Functions
1.4.2. Continuous Distribution Functions
1.4.3. Mixed Distribution Functions
2. Transformed Random Variables
2.1. Monotonic Transformations
2.2. Sums of Independent Random Variables
2.2.1. Distribution Functions of Sums
2.2.2. Density Functions of Sums
2.3. Ratios of Random Variables
2.3.1. Independent Random Variables
2.3.2. Example without Independence
3. Order Statistics
3.1. M-Samples and Order Statistics
3.2. Distribution Functions of Order Statistics
3.3. Density Functions of Order Statistics
3.4. Joint Distribution of All Order Statistics
3.5. Density Functions on Rn
3.6. Multivariate Order Functions
3.6.1. Joint Density of All Order Statistics
3.6.2. Marginal Densities and Distributions
3.6.3. Conditional Densities and Distributions
3.7. The Rényi Representation Theorem
4. Expectations of Random Variables 1
4.1. General Definitions
4.1.1. Is Expectation Well Defined?
4.1.2. Formal Resolution of Well-Definedness
4.2. Moments of Distributions
4.2.1. Common Types of Moments
4.2.2. Moment Generating Function
4.2.3. Moments of Sums – Theory
4.2.4. Moments of Sums – Applications
4.2.5. Properties of Moments
4.2.6. Examples–Discrete Distributions
4.2.7. Examples–Continuous Distributions
4.3. Moment Inequalities
4.3.1. Chebyshev's Inequality
4.3.2. Jensen's Inequality
4.3.3. Kolmogorov's Inequality
4.3.4. Cauchy-Schwarz Inequality
4.3.5. Hölder and Lyapunov Inequalities
4.4. Uniqueness of Moments
4.4.1. Applications of Moment Uniqueness
4.5. Weak Convergence and Moment Limits
5. Simulating Samples of RVs – Examples
5.1. Random Samples
5.1.1. Discrete Distributions
5.1.2. Simpler Continuous Distributions
5.1.3. Normal and Lognormal Distributions
5.1.4. Student T Distribution
5.2. Ordered Random Samples
5.2.1. Direct Approaches
5.2.2. The Rényi Representation
6. Limit Theorems
6.1. Introduction
6.2. Weak Convergence of Distributions
6.2.1. Student T to Normal
6.2.2. Poisson Limit Theorem
6.2.3. "Weak Law of Small Numbers"
6.2.4. De Moivre-Laplace Theorem
6.2.5. The Central Limit Theorem 1
6.2.6. Smirnov's Theorem on Uniform Order Statistics
6.2.7. A Limit Theorem on General Quantiles
6.2.8. A Limit Theorem on Exponential Order Statistics
6.3. Laws of Large Numbers
6.3.1. Tail Events and Kolmogorov's 0-1 Law
6.3.2. Weak Laws of Large Numbers
6.3.3. Strong Laws of Large Numbers
6.3.4. A Limit Theorem in EVT
6.4. Convergence of Empirical Distributions
6.4.1. Definition and Basic Properties
6.4.2. The Glivenko-Cantelli Theorem
6.4.3. Distributional Estimates for Dn(s)
7. Estimating Tail Events 2
7.1. Large Deviation Theory 2
7.1.1. Chernoff Bound
7.1.2. Cramér-Chernoff Theorem
7.2. Extreme Value Theory 2
7.2.1. Fisher-Tippett-Gnedenko Theorem
7.2.2. The Hill Estimator, γ > 0
7.2.3. F ∈ D(Gγ) is Asymptotically Pareto for γ > 0
7.2.4. F ∈ D(Gγ), γ > 0, then γH ≈ γ
7.2.5. F ∈ D(Gγ), γ > 0, then γH →1 γ
7.2.6. Asymptotic Normality of the Hill Estimator
7.2.7. The Pickands-Balkema-de Haan Theorem: γ > 0
Bibliography
Index
