Random Walk, Brownian Motion, and Martingales (Hardcover)

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. Bienyame?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 Bienyame?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: Bienyame?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.