1st Edition
Foundations of Quantitative Finance Book IV: Distribution Functions and Expectations
Preface
Introduction
1 Distribution and Density Functions
l.l Summary of Book II Results
l.l.l DistributionFunctionsonJR
l.l.2 Distribution Functions on JRn
l.2 DecompositionofDistributionFunctionsonJR
l.3 DensityFunctionsonJR
l.3.l TheLebesgueApproach
l.3.2 RiemannApproach
l.3.3 Riemann-Stieltjes Framework
l.4 Examples of Distribution Functions on JR
l.4.l DiscreteDistributionFunctions
l.4.2 ContinuousDistributionFunctions
l.4.3 MixedDistributionFunctions
2 Transformed Random Variables-
2.l MonotonicTransformations
2.2 SumsofIndependentRandomVariables
2.2.l DistributionFunctionsofSums
2.2.2 Density Functions of Sums
2.3 Ratios of Random Variables
2.3.l Independent Random Variable
2.3.2 Example without Independence
3 Order Statistics
3.l-M -Samples and Order Statistics
3.2-Distribution Functions for kth Order Statistics
3.3-Density Functions for kth Order Statistics
3.4-Joint Distribution of all Order Statistics
3.5-Density Functions on JRn
3.6-Multivariate Density Functions
-3.6.l Joint Density of all Order Statistics
-3.6.2 Marginal Densities and Distributions
-3.6.3 Conditional Densities and Distributions
3.7-The Renyi Representation Theorem
4 EXpectationsofRandomVariables1
4.l General Definitions
4.l.l Is Expectation Well Defined?
4.l.2 Formal Resolution of Well-Definedness
4.2 Moments of Distributions
4.2.l 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 Moment Examples-Discrete Distributions
4.2.7 Moment Examples-Continuous Distributions
4.3 Moment Inequalities
4.3.l Chebyshev's Inequality
4.3.2 Jensen's Inequality
4.3.3 Kolmogorov's Inequality
4.3.4 Cauchy-Schwarz Inequality
4.3.5 Holder and Lyapunov Inequalities
4.4 Uniqueness of Moments
4.4.l Applications of Moment Uniqueness
4.5 Weak Convergence and Moment Limits
5 Simulating Samples of RVs - EXamples
5.l Random Samples
5.l.l Discrete Distributions
5.l.2 Simpler Continuous Distributions
5.l.3 Normal and Lognormal Distributions
5.l.4 Student T Distribution
5.2 Ordered Random Samples
5.2.l Direct Approaches
5.2.2 The Renyi Representation
6 Limit Theorems
6.l Introduction
6.2 Weak Convergence of Distributions
6.2.l Student T ⇒ 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 l
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.l Tail Events and Kolmogorov's 0-l 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.l 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.l Large Deviation Theory 2
7.l.l Chernoff Bound
7.l.2 Cramer-Chernoff Theorem
7.2 Extreme Value Theory 2
7.2.l Fisher-Tippett-Gnedenko theorem
7.2.2 The Hill Estimator, 1 > 0
7.2.3 F E D(G,) is Asymptotically Pareto for 1 > 0
7.2.4 F E D(G,), 1 > 0, then 1H � 1
7.2.5 F E D(G,), 1 > 0, then 1H -1 1
7.2.6 Asymptotic Normality of the Hill Estimator
7.2.7 The Pickands-Balkema-de Haan Theorem: 1 > 0
References
Biography
Robert R. Reitano is Professor of the Practice of Finance at the Brandeis International Business School where he specializes in risk 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 asset/liability risk management 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 have won an Annual Prize of the Society of Actuaries and two F.M.
