Random walk, Brownian motion, and martingales / Rabi Bhattacharya, Edward C. Waymire.
Bhattacharya, R. N. (Rabindra Nath), 1937- author.; Waymire, Edward C., author.
2021
QA274.73 .B53 2021
Available Online

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Title Random walk, Brownian motion, and martingales / Rabi Bhattacharya, Edward C. Waymire.
Author Bhattacharya, R. N. (Rabindra Nath), 1937- author.
ISBN 9783030789398 (electronic bk.)
303078939X (electronic bk.)
9783030789374
3030789373
DOI https://doi.org/10.1007/978-3-030-78939-8
Published Cham : Springer, [2021]
Copyright ©2021
Language English
Description 1 online resource : illustrations
Item Number 10.1007/978-3-030-78939-8 doi
Call Number QA274.73 .B53 2021
Dewey Decimal Classification 519.2/82
Summary 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.
Bibliography, etc. Note Includes bibliographical references and indexes.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed October 5, 2021).
Added Author Waymire, Edward C., author.
Series Graduate texts in mathematics ; 292. 2197-5612
Available in Other Form Random walk, Brownian motion, and martingales.
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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 for Brownian 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.

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