Introductory Mathematical Analysis for Quantitative Finance

✍ Scribed by Daniele Ritelli; Giulia Spaletta

• Publisher: CRC Press
• Year: 2020
• Tongue: English
• Leaves: 322
• Series: Chapman and Hall/CRC Financial Mathematics
• Category: Library

✦ Synopsis

Introductory Mathematical Analysis for Quantitative Finance is a textbook designed to enable students with little knowledge of mathematical analysis to fully engage with modern quantitative finance. A basic understanding of dimensional Calculus and Linear Algebra is assumed.

The exposition of the topics is as concise as possible, since the chapters are intended to represent a preliminary contact with the mathematical concepts used in Quantitative Finance. The aim is that this book can be used as a basis for an intensive one-semester course.
Features:

Written with applications in mind, and maintaining mathematical rigor.

Suitable for undergraduate or master's level students with an Economics or Management background.

Complemented with various solved examples and exercises, to support the understanding of the subject.

✦ Table of Contents

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