Random Walk Brownian Motion and Martingales

Description

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. 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