Introductory Mathematical Analysis for Quantitative Finance

Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
1: Euclidean Space
	1.1 Vectors
	1.2 Topology of ℝn
	1.3 Limits of Functions
2: Sequences and Series of Functions
	2.1 Sequences and Series of Real or Complex Numbers
	2.2 Sequences of Functions
	2.3 Uniform Convergence
	2.4 Series of Functions
	2.5 Power Series: Radius of Convergence
	2.6 Taylor-Maclaurin series
		2.6.1 Binomial Series
		2.6.2 The Error Function
		2.6.3 Abel Theorem and Series Summation
	2.7 Basel Problem
	2.8 Extension of Elementary Functions to the Complex Eld
		2.8.1 Complex Exponential
		2.8.2 Complex Goniometric Hyperbolic Functions
		2.8.3 Complex Logarithm
	2.9 Exercises
		2.9.1 Solved Exercises
		2.9.2 Unsolved Exercises
3: Multidimensional Differential Calculus
	3.1 Partial Derivatives
	3.2 Differentiability
	3.3 Maxima and Minima
	3.4 Suffcient Conditions
	3.5 Lagrange Multipliers
	3.6 Mean-Value Theorem
	3.7 Implicit Function Theorem
	3.8 Proof of Theorem 3.22
	3.9 Suffcient Conditions
4: Ordinary Differential Equations of First Order: General Theory
	4.1 Preliminary Notions
		4.1.1 Systems of ODEs: Equations of High Order
	4.2 Existence of Solutions: Peano Theorem
	4.3 Existence and Uniqueness: Picard-Lindelöhf Theorem
		4.3.1 Interval of Existence
		4.3.2 Vector-Valued Differential Equations
		4.3.3 Solution Continuation
5: Ordinary Differential Equations of First Order: Methods for Explicit Solutions
	5.1 Separable Equations
		5.1.1 Exercises
	5.2 Singular Integrals
	5.3 Homogeneous Equations
		5.3.1 Exercises
	5.4 Quasi Homogeneous Equations
	5.5 Exact Equations
		5.5.1 Exercises
	5.6 Integrating Factor for Non-Exact Equations
		5.6.1 Exercises
	5.7 Linear Equations of First Order
		5.7.1 Exercises
	5.8 Bernoulli Equation
		5.8.1 Exercises
	5.9 Riccati Equation
		5.9.1 Cross-Ratio Property
		5.9.2 Reduced form of the Riccati Equation
		5.9.3 Connection with the Linear Equation of Second Order
		5.9.4 Exercises
	5.10 Change of Variable
		5.10.1 Exercises
6: Linear Differential Equations of Second Order
	6.1 Homogeneous Equations
		6.1.1 Operator Notation
		6.1.2 Wronskian Determinant
		6.1.3 Order Reduction
		6.1.4 Constant-Coeffcient Equations
		6.1.5 Cauchy-Euler Equations
		6.1.6 Invariant and Normal Form
	6.2 Non-Homogeneous Equation
		6.2.1 Variation of Parameters
		6.2.2 Non-Homogeneous Equations with Constant coeffcients
		6.2.3 Exercises
7: Prologue to Measure Theory
	7.1 Set Theory
		7.1.1 Sets
		7.1.2 Indexes and Cartesian Product
		7.1.3 Cartesian Product
		7.1.4 Functions
		7.1.5 Equivalences
		7.1.6 Real Intervals
		7.1.7 Cardinality
		7.1.8 The Real Number System
		7.1.9 The Extended Real Number System
	7.2 Topology
		7.2.1 Closed Sets
		7.2.2 Limit
		7.2.3 Closure
		7.2.4 Compactness
		7.2.5 Continuity
8: Lebesgue Integral
	8.1 Measure Theory
		8.1.1 σ-algebras
		8.1.2 Borel Sets
		8.1.3 Measures
		8.1.4 Exercises
	8.2 Translation Invariance
		8.2.1 Exercises
	8.3 Simple Functions
		8.3.1 Integral of Simple Functions
	8.4 Measurable Functions
	8.5 Lebesgue Integral
	8.6 Almost Everywhere
	8.7 Connection with Riemann Integral
		8.7.1 The Riemann Integral
		8.7.2 Lebesgue-Vitali Theorem
		8.7.3 An interesting Example
	8.8 Non Lebesgue Integrals
		8.8.1 Dirac Measure
		8.8.2 Discrete Measure
	8.9 Generation of Measures
	8.10 Passage to the Limit
		8.10.1 Monotone Convergence
			8.10.1.1 Analytic Functions
		8.10.2 Exercises
		8.10.3 Dominated Convergence
		8.10.4 Exercises
		8.10.5 A property of Increasing Functions
	8.11 Differentiation Under the Integral Sign
		8.11.1 The Probability Integral (1)
		8.11.2 The Probability Integral (2)
		8.11.3 Exercises
	8.12 Basel Problem Again
	8.13 Debye Integral
9: Radon-Nikodym Theorem
	9.1 Signed Measures
	9.2 Radon-Nikodym Theorem
10: Multiple Integrals
	10.1 Integration in ℝ2
		10.1.1 Smart Applications of Fubini Theorem
			10.1.1.1 Fresnel Integrals
			10.1.1.2 Cauchy Formula for Iterated Integration
			10.1.1.3 A Challenging Definite Integral
			10.1.1.4 Frullani Integral
			10.1.1.5 Basel Problem, Once More
	10.2 Change of Variable
	10.3 Integration in ℝn
		10.3.1 Exercises
	10.4 Product of σ-algebras
11: Gamma and Beta functions
	11.1 Gamma Function
	11.2 Beta Function
		11.2.1 Г (1\\2) and the Probability Integral
		11.2.2 Legendre Duplication Formula
		11.2.3 Euler Reflexion Formula
	11.3 Definite Integrals
	11.4 Double Integration Techniques
12: Fourier Transform on the Real Line
	12.1 Fourier Transform
		12.1.1 Examples
	12.2 Properties of the Fourier Transform
		12.2.1 Linearity
		12.2.2 The Shift Theorem
		12.2.3 The Stretch Theorem
		12.2.4 Combining Shifts and Stretches
	12.3 Convolution
	12.4 Linear Ordinary Differential Equations
	12.5 Exercises
13: Parabolic Equations
	13.1 Partial Differential Equations
		13.1.1 Classification of Second-Order Linear Partial Differential Equations
	13.2 The Heat Equation
		13.2.1 Uniqueness of Solution: Homogeneous Case
		13.2.2 Fundamental Solutions: Heat Kernel
		13.2.3 Initial data on (0;∞)
	13.3 Parabolic Equations with Constant Coefficients
		13.3.1 Exercises
	13.4 Black-Scholes Equation
		13.4.1 Exercises
	13.5 Non-Homogeneous Equation: Duhamel Integral
Bibliography
Analytic Index