Random Walk Brownian Motion and Martingales 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. 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.