Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Notation
1. Measure Theory
1.1. Measurable Spaces
1.2. Measurable Functions
1.3. Measures
1.4. Integrals
1.4.1. Definition of an Integral
1.4.2. Properties of an Integral
1.4.3. Indefinite Integrals, Image Measures, and Measures with Densities
1.4.4. Kernels and Product Spaces
Exercises
2. Probability
2.1. Modeling Random Experiments
2.2. Random Variables
2.3. Probability Distributions
2.4. Expectation
2.5. Lp Spaces
2.6. Integral Transforms
2.6.1. Moment Generating Functions
2.6.2. Characteristic Functions
2.7. Information and Independence
2.8. Important Stochastic Processes
2.8.1. Gaussian Processes
2.8.2. Poisson Random Measures and Poisson Processes
2.8.3. Compound Poisson Processes
2.8.4. Lรฉvy Processes
Exercises
3. Convergence
3.1. Motivation
3.2. Almost Sure Convergence
3.3. Convergence in Probability
3.4. Convergence in Distribution
3.5. Convergence in Lp Norm
3.5.1. Uniform Integrability
3.6. Relations Between Modes of Convergence
3.7. Law of Large Numbers and Central Limit Theorem
Exercises
4. Conditioning
4.1. A Basic Example
4.2. Conditional Expectation
4.3. Conditional Probability and Distribution
4.4. Existence of Probability Spaces
4.5. Markov Property
4.5.1. Time-homogeneous Markov Chains
4.5.2. Markov Jump Processes
4.5.3. Infinitesimal Generator
Exercises
5. Martingales
5.1. Stopping Times
5.2. Martingales
5.3. Optional Stopping
5.3.1. Stochastic Integration
5.3.2. Doobโs Stopping Theorem
5.4. (Sub)Martingale Convergence
5.4.1. Upcrossings
5.5. Applications
5.5.1. Kolmogorovโs 0โ1 Law
5.5.2. Strong Law of Large Numbers
5.5.3. RadonโNikodym Theorem
5.6. Martingales in Continuous Time
5.6.1. Local Martingales and Doob Martingales
5.6.2. Martingale Inequalities
5.6.3. Martingale Extensions
Exercises
6. Wiener and Brownian Motion Processes
6.1. Wiener Process
6.2. Existence
6.3. Strong Markov Property
6.4. Martingale Properties
6.5. Maximum and Hitting Time
6.6. Brownian Motion and the Laplacian Operator
6.7. Path Properties
Exercises
7. Itรด Calculus
7.1. Itรด Integral
7.1.1. Itรด Integral for Simple Processes
7.1.2. Itรด Integral for Predictable Processes
7.1.3. Further Extensions of the Itรด Integral
7.2. Itรด Calculus
7.2.1. Itรดโs Formula
7.2.2. Multivariate Itรดโs Formula
7.3. Stochastic Differential Equations
7.3.1. Existence of Solutions to SDEs
7.3.2. Markov Property of Diffusion Processes
7.3.3. Methods for Solving Simple SDEs
7.3.4. Eulerโs Method for Numerically Solving SDEs
Exercises
A. Selected Solutions
A.1. Chapter 1
A.2. Chapter 2
A.3. Chapter 3
A.4. Chapter 4
A.5. Chapter 5
A.6. Chapter 6
A.7. Chapter 7
B. Function Spaces
B.1. Metric Spaces
B.2. Normed Spaces
B.3. Inner Product Spaces
B.4. SturmโLiouville Orthonormal Basis
B.5. Hermite Orthonormal Basis
B.6. Haar Orthonormal Basis
C. Existence of the Lebesgue Measure
Bibliography
Index