Foundations of Quantitative Finance Book II: Probability Spaces and Random Variables

Description

Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Author Introduction 1. Probability Spaces 1.1. Probability Theory: A Very Brief History 1.2. A Finite Measure Space with a “Story” 1.2.1. Bond Loss Example 1.3. Some Probability Measures on R 1.3.1. Measures from Discrete Probability Theory 1.3.2. Measures from Continuous Probability Theory 1.3.3. More General Probability Measures on R 1.4. Independent Events 1.4.1. Independent Classes and Associated Sigma Algebras 1.5. Conditional Probability Measures 1.5.1. Law of Total Probability 1.5.2. Bayes' Theorem 2. Limit Theorems on Measurable Sets 2.1. Introduction to Limit Sets 2.2. The Borel-Cantelli Lemma 2.3. Kolmogorov's Zero-One Law 3. Random Variables and Distribution Functions 3.1. Introduction and Definitions 3.1.1. Bond Loss Example (Continued) 3.2. “Inverse” of a Distribution Function 3.2.1. Properties of F* 3.2.2. The Function F** 3.3. Random Vectors and Joint Distribution Functions 3.3.1. Marginal Distribution Functions 3.3.2. Conditional Distribution Functions 3.4. Independent Random Variables 3.4.1. Sigma Algebras Generated by R.V.s 3.4.2. Independent Random Variables and Vectors 3.4.3. Distribution Functions of Independent R.V.s 3.4.4. Independence and Transformations 4. Probability Spaces and i.i.d. RVs 4.1. Probability Space (S',E',µ') and i.i.d. {Xj}Nj=1 4.1.1. First Construction: (S′F, E′F, µ′F) 4.2. Simulation of Random Variables - Theory 4.2.1. Distributional Results 4.2.2. Independence Results 4.2.3. Second Construction: (S′U, E′U, µ′U) 4.3. An Alternate Construction for Discrete Random Variables 4.3.1. Third Construction: (S′p, E′p, µ′p) 5. Limit Theorems for RV Sequences 5.1. Two Limit Theorems for Binomial Sequences 5.1.1. The Weak Law of Large Numbers 5.1.2. The Strong Law of Large Numbers 5.1.3. Strong Laws versus Weak Laws 5.2. Convergence of Random Variables 1 5.2.1. Notions of Convergence 5.2.2. Convergence Relationships 5.2.3. Slutsky's Theorem 5.2.4. Kolmogorov's Zero-One Law 6. Distribution Functions and Borel Measures 6.1. Distribution Functions on R 6.1.1. Probability Measures from Distribution Functions 6.1.2. Random Variables from Distribution Functions 6.2. Distribution Functions on Rn 6.2.1. Probability Measures from Distribution Functions 6.2.2. Random Vectors from Distribution Functions 6.2.3. Marginal and Conditional Distribution Functions 7. Copulas and Sklar's Theorem 7.1. Fréchet Classes 7.2. Copulas and Sklar's Theorem 7.2.1. Identifying Copulas 7.3. Partial Results on Sklar's Theorem 7.4. Examples of Copulas 7.4.1. Archimedean Copulas 7.4.2. Extreme Value Copulas 7.5. General Result on Sklar's Theorem 7.5.1. The Distributional Transform 7.5.2. Sklar's Theorem - The General Case 7.6. Tail Dependence and Copulas 7.6.1. Bivariate Tail Dependence 7.6.2. Multivariate Tail Dependence and Copulas 7.6.3. Survival Functions and Copulas 8. Weak Convergence 8.1. Definitions of Weak Convergence 8.2. Properties of Weak Convergence 8.3. Weak Convergence and Left Continuous Inverses 8.4. Skorokhod's Representation Theorem 8.4.1. Mapping Theorem on R 8.5. Convergence of Random Variables 2 8.5.1. Mann-Wald Theorem on R 8.5.2. The Delta-Method 9. Estimating Tail Events 1 9.1. Large Deviation Theory 1 9.2. Extreme Value Theory 1 9.2.1. Introduction and Examples 9.2.2. Extreme Value Distributions 9.2.3. The Fisher-Tippett-Gnedenko Theorem 9.3. The Pickands-Balkema-de Haan Theorem 9.3.1. Quantile Estimation 9.3.2. Tail Probability Estimation 9.4. γ in Theory: von Mises' Condition 9.5. Independence vs. Tail Independence 9.6. Multivariate Extreme Value Theory 9.6.1. Multivariate Fisher-Tippett-Gnedenko Theorem 9.6.2. The Extreme Value Distribution G 9.6.3. The Extreme Value Copula CG References Index