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Random Walk, Brownian Motion, and Martingales

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Cover of 'Random Walk, Brownian Motion, and Martingales'

Table of Contents

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    Book Overview
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    Chapter 1 What Is a Stochastic Process?
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    Chapter 2 The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence
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    Chapter 3 The Simple Random Walk II: First Passage Times
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    Chapter 4 Multidimensional Random Walk
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    Chapter 5 The Poisson Process, Compound Poisson Process, and Poisson Random Field
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    Chapter 6 The Kolmogorov–Chentsov Theorem and Sample Path Regularity
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    Chapter 7 Random Walk, Brownian Motion, and the Strong Markov Property
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    Chapter 8 Coupling Methods for Markov Chains and the Renewal Theorem for Lattice Distributions
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    Chapter 9 Bienaymé–Galton–Watson Simple Branching Process and Extinction
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    Chapter 10 Martingales: Definitions and Examples
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    Chapter 11 Optional Stopping of (Sub)Martingales
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    Chapter 12 The Upcrossings Inequality and (Sub)Martingale Convergence
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    Chapter 13 Continuous Parameter Martingales
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    Chapter 14 Growth of Supercritical Bienaymé–Galton–Watson Simple Branching Processes
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    Chapter 15 Stochastic Calculus for Point Processes and a Martingale Characterization of the Poisson Process
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    Chapter 16 First Passage Time Distributions for Brownian Motion with Drift and a Local Limit Theorem
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    Chapter 17 The Functional Central Limit Theorem (FCLT)
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    Chapter 18 ArcSine Law Asymptotics
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    Chapter 19 Brownian Motion on the Half-Line: Absorption and Reflection
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    Chapter 20 The Brownian Bridge
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    Chapter 21 Special Topic: Branching Random Walk, Polymers, and Multiplicative Cascades
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    Chapter 22 Special Topic: Bienaymé–Galton–Watson Simple Branching Process and Excursions
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    Chapter 23 Special Topic: The Geometric Random Walk and the Binomial Tree Model of Mathematical Finance
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    Chapter 24 Special Topic: Optimal Stopping Rules
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    Chapter 25 Special Topic: A Comprehensive Renewal Theory for General Random Walks
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    Chapter 26 Special Topic: Ruin Problems in Insurance
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    Chapter 27 Special Topic: Fractional Brownian Motion and/or Trends: The Hurst Effect
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    Chapter 28 Special Topic: Incompressible Navier–Stokes Equations and the Le Jan–Sznitman Cascade
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Title
Random Walk, Brownian Motion, and Martingales
Published by
Springer International Publishing, August 2021
DOI 10.1007/978-3-030-78939-8
ISBNs
978-3-03-078937-4, 978-3-03-078939-8
Authors

Bhattacharya, Rabi, Waymire, Edward C.

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