An Advanced Course in Probability and Stochastic Processes

✍ Scribed by Dirk P. Kroese, Zdravko Botev

• Publisher: Chapman and Hall/CRC
• Year: 2023
• Tongue: English
• Leaves: 378
• Edition: 1
• Category: Library

✦ Synopsis

An Advanced Course in Probability and Stochastic Processes provides a modern and rigorous treatment of probability theory and stochastic processes at an upper undergraduate and graduate level. Starting with the foundations of measure theory, this book introduces the key concepts of probability theory in an accessible way, providing full proofs and extensive examples and illustrations. Fundamental stochastic processes such as Gaussian processes, Poisson random measures, Lévy processes, Markov processes, and Itô processes are presented and explored in considerable depth, showcasing their many interconnections. Special attention is paid to martingales and the Wiener process and their central role in the treatment of stochastic integrals and stochastic calculus. This book includes many exercises, designed to test and challenge the reader and expand their skillset. An Advanced Course in Probability and Stochastic Processes is meant for students and researchers who have a solid mathematical background and who have had prior exposure to elementary probability and stochastic processes.

Key Features:

• Focus on mathematical understanding
• Rigorous and self-contained
• Accessible and comprehensive
• High-quality illustrations
• Includes essential simulation algorithms
• Extensive list of exercises and worked-out examples
• Elegant and consistent notation

✦ Table of Contents

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. Lévy 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. Doob’s Stopping Theorem
5.4. (Sub)Martingale Convergence
5.4.1. Upcrossings
5.5. Applications
5.5.1. Kolmogorov’s 0–1 Law
5.5.2. Strong Law of Large Numbers
5.5.3. Radon–Nikodym 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. Itô Calculus
7.1. Itô Integral
7.1.1. Itô Integral for Simple Processes
7.1.2. Itô Integral for Predictable Processes
7.1.3. Further Extensions of the Itô Integral
7.2. Itô Calculus
7.2.1. Itô’s Formula
7.2.2. Multivariate Itô’s 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. Euler’s 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. Sturm–Liouville Orthonormal Basis
B.5. Hermite Orthonormal Basis
B.6. Haar Orthonormal Basis
C. Existence of the Lebesgue Measure
Bibliography
Index