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

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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 notand 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 advantage their careers, these books present the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.

As a high-level industry executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered in nearly three decades working in the financial industry and two decades teaching in highly respected graduate programs.

Readers should be quantitatively literate and familiar with the developments in the first book in the set, Foundations of Quantitative Finance Book I: Measure Spaces and Measurable Functions.
Preface xi
Author xiii
Introduction xv
1 Probability Spaces
1(34)
1.1 Probability Theory: A Very Brief History
1(1)
1.2 A Finite Measure Space with a "Story"
2(9)
1.2.1 Bond Loss Example
7(4)
1.3 Some Probability Measures on R
11(9)
1.3.1 Measures from Discrete Probability Theory
11(5)
1.3.2 Measures from Continuous Probability Theory "
16(4)
1.3.3 More General Probability Measures on K
20(1)
1.4 Independent Events
20(7)
1.4.1 Independent Classes and Associated Sigma Algebras
24(3)
1.5 Conditional Probability Measures
27(8)
1.5.1 Law of Total Probability
28(4)
1.5.2 Bayes' Theorem
32(3)
2 Limit Theorems on Measurable Sets
35(12)
2.1 Introduction to Limit Sets
35(3)
2.2 The Borel-Cantelli Lemma
38(5)
2.3 Kolmogorov's Zero-One Law
43(4)
3 Random Variables and Distribution Functions
47(30)
3.1 Introduction and Definitions
47(5)
3.1.1 Bond Loss Example (Continued)
50(2)
3.2 "Inverse" of a Distribution Function
52(10)
3.2.1 Properties of F*
54(6)
3.2.2 The Function F**
60(2)
3.3 Random Vectors and Joint Distribution Functions
62(7)
3.3.1 Marginal Distribution Functions
65(2)
3.3.2 Conditional Distribution Functions
67(2)
3.4 Independent Random Variables
69(8)
3.4.1 Sigma Algebras Generated by R.V.s
70(1)
3.4.2 Independent Random Variables and Vectors
71(3)
3.4.3 Distribution Functions of Independent R.V.s
74(1)
3.4.4 Independence and Transformations
75(2)
4 Probability Spaces and i.i.d. RVs
77(22)
4.1 Probability Space (S',E',μ) and i.i.d. {Xj}Nj=1
78(3)
4.1.1 First Construction: (SF, SS, μF)
79(2)
4.2 Simulation of Random Variables - Theory
81(10)
4.2.1 Distributional Results
81(5)
4.2.2 Independence Results
86(2)
4.2.3 Second Construction: (S'U, ε'U, μU)
88(3)
4.3 An Alternate Construction for Discrete Random Variables
91(8)
4.3.1 Third Construction: (S'p, ε'p, μp)
93(6)
5 Limit Theorems for RV Sequences
99(24)
5.1 Two Limit Theorems for Binomial Sequences
99(9)
5.1.1 The Weak Law of Large Numbers
100(3)
5.1.2 The Strong Law of Large Numbers
103(5)
5.1.3 Strong Laws versus Weak Laws
108(1)
5.2 Convergence of Random Variables 1
108(15)
5.2.1 Notions of Convergence
109(2)
5.2.2 Convergence Relationships
111(4)
5.2.3 Slutsky's Theorem
115(3)
5.2.4 Kolmogorov's Zero-One Law
118(5)
6 Distribution Functions and Borel Measures
123(14)
6.1 Distribution Functions on R
125(5)
6.1.1 Probability Measures from Distribution Functions
126(3)
6.1.2 Random Variables from Distribution Functions
129(1)
6.2 Distribution Functions on R
130(7)
6.2.1 Probability Measures from Distribution Functions
131(4)
6.2.2 Random Vectors from Distribution Functions
135(1)
6.2.3 Marginal and Conditional Distribution Functions
136(1)
7 Copulas and Sklar's Theorem
137(42)
7.1 Frechet Classes
137(3)
7.2 Copulas and Sklar's Theorem
140(5)
7.2.1 Identifying Copulas
144(1)
7.3 Partial Results on Sklar's Theorem
145(4)
7.4 Examples of Copulas
149(9)
7.4.1 Archimedean Copulas
150(4)
7.4.2 Extreme Value Copulas
154(4)
7.5 General Result on Sklar's Theorem
158(7)
7.5.1 The Distributional Transform
160(4)
7.5.2 Sklar's Theorem - The General Case
164(1)
7.6 Tail Dependence and Copulas
165(14)
7.6.1 Bivariate Tail Dependence
165(5)
7.6.2 Multivariate Tail Dependence and Copulas
170(3)
7.6.3 Survival Functions and Copulas
173(6)
8 Weak Convergence
179(22)
8.1 Definitions of Weak Convergence
180(4)
8.2 Properties of Weak Convergence
184(5)
8.3 Weak Convergence and Left Continuous Inverses
189(2)
8.4 Skorokhod's Representation Theorem
191(3)
8.4.1 Mapping Theorem on R
192(2)
8.5 Convergence of Random Variables 2
194(7)
8.5.1 Mann-Wald Theorem on R
194(1)
8.5.2 The Delta-Method
195(6)
9 Estimating Tail Events 1
201(48)
9.1 Large Deviation Theory 1
202(4)
9.2 Extreme Value Theory 1
206(17)
9.2.1 Introduction and Examples
206(4)
9.2.2 Extreme Value Distributions
210(2)
9.2.3 The Fisher-Tippett-Gnedenko Theorem
212(11)
9.3 The Pickands-Balkema-de Haan Theorem
223(6)
9.3.1 Quantile Estimation
223(1)
9.3.2 Tail Probability Estimation
224(5)
9.4 γ in Theory: von Mises' Condition
229(5)
9.5 Independence vs. Tail Independence
234(1)
9.6 Multivariate Extreme Value Theory
235(14)
9.6.1 Multivariate Fisher-Tippett-Gnedenko Theorem
236(2)
9.6.2 The Extreme Value Distribution G
238(3)
9.6.3 The Extreme Value Copula CG
241(8)
References 249(4)
Index 253
Robert R. Reitano is Professor of the Practice in Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance. 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. 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. Redington Prizes of the Investment Section of the Society of the Actuaries. Dr. Reitano serves on various not-for-profit boards and investment committees.
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