Preface xi
About the Authors xvii
CHAPTER 1 Basic Concepts: Sets, Functions, and Variables 1
Introduction 2
Sets and Set Operations 2
Distances and Quantities 6
Functions 10
Variables 10
Key Points 11
CHAPTER 2 Differential Calculus 13
Introduction 14
Limits 15
Continuity 17
Total Variation 19
The Notion of Differentiation 19
Commonly Used Rules for Computing Derivatives 21
Higher-Order Derivatives 26
Taylor Series Expansion 34
Calculus in More Than One Variable 40
Key Points 41
CHAPTER 3 Integral Calculus 43
Introduction 44
Riemann Integrals 44
Lebesgue-Stieltjes Integrals 47
Indefinite and Improper Integrals 48
The Fundamental Theorem of Calculus 51
Integral Transforms 52
Calculus in More Than One Variable 57
Key Points 57
CHAPTER 4 Matrix Algebra 59
Introduction 60
Vectors and Matrices Defined 61
Square Matrices 63
Determinants 66
Systems of Linear Equations 68
Linear Independence and Rank 69
Hankel Matrix 70
Vector and Matrix Operations 72
Finance Application 78
Eigenvalues and Eigenvectors 81
Diagonalization and Similarity 82
Singular Value Decomposition 83
Key Points 83
CHAPTER 5 Probability: Basic Concepts 85
Introduction 86
Representing Uncertainty with Mathematics 87
Probability in a Nutshell 89
Outcomes and Events 91
Probability 92
Measure 93
Random Variables 93
Integrals 94
Distributions and Distribution Functions 96
Random Vectors 97
Stochastic Processes 100
Probabilistic Representation of Financial Markets 102
Information Structures 103
Filtration 104
Key Points 106
CHAPTER 6 Probability: Random Variables and Expectations 107
Introduction 109
Conditional Probability and Conditional Expectation 110
Moments and Correlation 112
Copula Functions 114
Sequences of Random Variables 116
Independent and Identically Distributed Sequences 117
Sum of Variables 118
Gaussian Variables 120
Appproximating the Tails of a Probability Distribution: Cornish-Fisher Expansion and Hermite Polynomials 123
The Regression Function 129
Fat Tails and Stable Laws 131
Key Points 144
CHAPTER 7 Optimization 147
Introduction 148
Maxima and Minima 149
Lagrange Multipliers 151
Numerical Algorithms 156
Calculus of Variations and Optimal Control Theory 161
Stochastic Programming 163
Application to Bond Portfolio: Liability-Funding Strategies 164
Key Points 178
CHAPTER 8 Difference Equations 181
Introduction 182
The Lag Operator L 183
Homogeneous Difference Equations 183
Recursive Calculation of Values of Difference Equations 192
Nonhomogeneous Difference Equations 195
Systems of Linear Difference Equations 201
Systems of Homogeneous Linear Difference Equations 202
Key Points 209
CHAPTER 9 Differential Equations 211
Introduction 212
Differential Equations Defined 213
Ordinary Differential Equations 213
Systems of Ordinary Differential Equations 216
Closed-Form Solutions of Ordinary Differential Equations 218
Numerical Solutions of Ordinary Differential Equations 222
Nonlinear Dynamics and Chaos 228
Partial Differential Equations 231
Key Points 237
CHAPTER 10 Stochastic Integrals 239
Introduction 240
The Intuition behind Stochastic Integrals 243
Brownian Motion Defined 248
Properties of Brownian Motion 254
Stochastic Integrals Defined 255
Some Properties of Itoˆ Stochastic Integrals 259
Martingale Measures and the Girsanov Theorem 260
Key Points 266
CHAPTER 11 Stochastic Differential Equations 267
Introduction 268
The Intuition behind Stochastic Differential Equations 269
Itoˆ Processes 272
Stochastic Differential Equations 273
Generalization to Several Dimensions 276
Solution of Stochastic Differential Equations 278
Derivation of Itoˆ ’s Lemma 282
Derivation of the Black-Scholes Option Pricing Formula 284
Key Points 291
Index 293
