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Random Walk, Brownian Motion, and Martingales

Random Walk, Brownian Motion, and Martingales

Wydawnictwo Springer, Berlin
Data wydania
Liczba stron 396
Forma publikacji książka w twardej oprawie
Język angielski
ISBN 9783030789374
Kategorie Prawdopodobieństwo i statystyka
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Opis książki

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.

Random Walk, Brownian Motion, and Martingales

Spis treści

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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