This textbook offers an approachable introduction to stochastic processes that explores the four pillars of random walk, branching processes, Brownian motion, and martingales. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study.
Consisting of many short chapters, the book begins with a comprehensive account of the simple random walk in one dimension. From here, different paths may be chosen according to interest. Themes span Poisson processes, branching processes, the Kolmogorov–Chentsov theorem, martingales, renewal theory, and Brownian motion. Special topics follow, showcasing a selection of important contemporary applications, including mathematical finance, optimal stopping, ruin theory, branching random walk, and equations of fluids. Engaging exercises accompany the theory throughout.
Random Walk, Brownian Motion, and Martingales is an ideal introduction to the rigorous study of stochastic processes. Students and instructors alike will appreciate the accessible, example-driven approach. A single, graduate-level course in probability is assumed.
Author(s): Rabi Bhattacharya, Edward C. Waymire
Series: Graduate Texts in Mathematics 292
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 396
City: Cham, Switzerland
Tags: Random Walks, Poisson Process, Brownian Motion, Branching Process, Martingale
Preface
Contents
Symbol Definition List
1 What Is a Stochastic Process?
Exercises
2 The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence
Exercises
3 The Simple Random Walk II: First Passage Times
Exercises
4 Multidimensional Random Walk
Exercises
5 The Poisson Process, Compound Poisson Process, and Poisson Random Field
Exercises
6 The Kolmogorov–Chentsov Theorem and Sample Path Regularity
Exercises
7 Random Walk, Brownian Motion, and the Strong Markov Property
Exercises
8 Coupling Methods for Markov Chains and the Renewal Theorem for Lattice Distributions
Exercises
9 Bienaymé–Galton–Watson Simple Branching Process and Extinction
Exercises
10 Martingales: Definitions and Examples
Exercises
11 Optional Stopping of (Sub)Martingales
Exercises
12 The Upcrossings Inequality and (Sub)Martingale Convergence
Exercises
13 Continuous Parameter Martingales
Exercises
14 Growth of Supercritical Bienaymé–Galton–Watson Simple Branching Processes
Exercises
15 Stochastic Calculus for Point Processes and a Martingale Characterization of the Poisson Process
Exercises
16 First Passage Time Distributions for Brownian Motion with Drift and a Local Limit Theorem
Exercises
17 The Functional Central Limit Theorem (FCLT)
Exercises
18 ArcSine Law Asymptotics
Exercises
19 Brownian Motion on the Half-Line: Absorption and Reflection
Exercises
20 The Brownian Bridge
Exercises
21 Special Topic: Branching Random Walk, Polymers, and Multiplicative Cascades
Exercises
22 Special Topic: Bienaymé–Galton–Watson Simple Branching Process and Excursions
Exercises
23 Special Topic: The Geometric Random Walk and the Binomial Tree Model of Mathematical Finance
Exercises
24 Special Topic: Optimal Stopping Rules
Exercises
25 Special Topic: A Comprehensive Renewal Theory for General Random Walks
Exercises
26 Special Topic: Ruin Problems in Insurance
Exercises
27 Special Topic: Fractional Brownian Motion and/or Trends: The Hurst Effect
Exercises
28 Special Topic: Incompressible Navier–Stokes Equations and the Le Jan–Sznitman Cascade
Exercises
References
Author Index
Subject Index
