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 KolmogorovChentsov 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 theorythroughout.

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.

Offers an accessible introduction to the rigorous study of stochastic processes Builds from simple examples to formal proofs, illuminating key ideas and computations Showcases a selection of important contemporary applications, including mathematical finance, optimal stopping, and ruin theory

Autorentext

Rabi Bhattacharya, PhD, has held regular faculty positions at UC, Berkeley; Indiana University; and the University of Arizona. He is a Fellow of the Institute of Mathematical Statistics and a recipient of the U.S. Senior Scientist Humboldt Award and of a Guggenheim Fellowship. He has served on editorial boards of many international journals and has published several research monographs and graduate texts on probability and statistics, including Nonparametric Inference on Manifolds, co-authored with A. Bhattacharya. Lizhen Lin, PhD, is Assistant Professor in the Department of Statistics and Data Science at the University of Texas at Austin. She received a PhD in Mathematics from the University of Arizona and was a Postdoctoral Associate at Duke University. Bayesian nonparametrics, shape constrained inference, and nonparametric inference on manifolds are among her areas of expertise. Vic Patrangenaru, PhD, is Professor of Statistics at Florida State University. He received PhDs in Mathematics from Haifa, Israel, and from Indiana University in the fields of differential geometry and statistics, respectively. He has many research publications on Riemannian geometry and especially on statistics on manifolds. He is a co-author with L. Ellingson of Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis.

Inhalt

1. What is a Stochastic Process?.- 2. The Simple Random Walk I: Associated Boundary Value Distributions, Transience and Recurrence.- 3. The Simple Random Walk II: First Passage Times.- 4. Multidimensional Random Walk.- 5. The Poisson Process, Compound Poisson Process, and Poisson Random Field.- 6. The KolmogorovChentsov Theorem and Sample Path Regularity.- 7. Random Walk, Brownian Motion and the Strong Markov Property.- 8. Coupling Methods for Markov Chains and the Renewal Theorem for Lattice Distributions.- 9. BienyaméGaltonWatson Simple Branching Process and Extinction.- 10. Martingales: Definitions and Examples.- 11. Optional Stopping of (Sub)Martingales.- 12. The Upcrossings Inequality and (Sub)Martingale Convergence.- 13.- Continuous Parameter Martingales.- 14. Growth of Supercritical BienyaméGaltonWatson Simple Branching Processes.- 15. Stochastic Calculus for Point Processes and a Martingale Characterization of the Poisson Process.- 16. First Passage Time Distributions forBrownian Motion with Drift and a Local Limit Theorem.- 17. The Functional Central Limit Theorem (FCLT).- 18. ArcSine Law Asymptotics.- 19. Brownian Motion on the Half-Line: Absorption and Reflection.- 20. The Brownian Bridge.- 21. Special Topic: Branching Random Walk, Polymers and Multiplicative Cascades.- 22. Special Topic: BienyaméGaltonWatson Simple Branching Process and Excursions.- 23. Special Topic: The Geometric Random Walk and the Binomial Tree Model of Mathematical Finance.- 24. Special Topic: Optimal Stopping Rules.- 25. Special Topic: A Comprehensive Renewal Theory for General Random Walks.- 26. Special Topic: Ruin Problems in Insurance.- 27. Special Topic: Fractional Brownian Motion and/or Trends: The Hurst Effect.- 28. Special Topic: Incompressible NavierStokes Equations and the LeJanSznitman Cascade.- References.- Related Textbooks and Monographs.- Symbol Definition List.- Name Index.- Index.