Random Walk, Brownian Motion, and Martingales

Rabi N. Bhattacharya is Professor of Mathematics at 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 made significant contributions to the theory and application of Markov processes, and more recently, nonparametric statistical inference on manifolds. He has served on editorial boards of many international journals and has published several research monographs and graduate texts on probability and statistics. Edward C. Waymire is Emeritus Professor of Mathematics at Oregon State University. He received a PhD in mathematics from the University of Arizona in the theory of interacting particle systems. His primary research concerns applications of probability and stochastic processes to problems of contemporary applied mathematics pertaining to various types of flows, dispersion, and random disorder. He is a former chief editor of the Annals of Applied Probability, and past president of the Bernoulli Society for Mathematical Statistics and Probability. Both authors have co-authored numerous books, including A Basic Course in Probability Theory, which is an ideal companion to the current volume.

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 Kolmogorov-Chentsov 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. Bienyamé-Galton-Watson 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 Bienyamé-Galton-Watson 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: Bienyamé-Galton-Watson 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 Navier-Stokes Equations and the LeJan-Sznitman Cascade.- References.- Related Textbooks and Monographs.- Symbol Definition List.- Name Index.- Index.

"The present text does an outstanding job of presenting many complementary aspects of the subject in a unified and coherent way ... the subject matter and appreciation for the work of the authors in producing such an engaging and readable book. The authors suggest various models for graduate-level courses ... . There is a great deal in the book that will be interesting, stimulating, and enjoyable for readers with an interest in probability theory and stochiastic processes." (Andrew Wade, zbMATH 1489.60001, 2022)

“The present text does an outstanding job of presenting many complementary aspects of the subject in a unified and coherent way … the subject matter and appreciation for the work of the authors in producing such an engaging and readable book. The authors suggest various models for graduate-level courses … . There is a great deal in the book that will be interesting, stimulating, and enjoyable for readers with an interest in probability theory and stochiastic processes.” (Andrew Wade, zbMATH 1489.60001, 2022)