Contents
Preface ix
Chapter 1. One-Dimensional Brownian Motion 1
§1.1. Some motivation 1
§1.2. The multivariate Gaussian distribution 2
§1.3. Processes with stationary independent increments 5
§1.4. Definition of Brownian motion 5
§1.5. The construction 9
§1.6. Path properties 15
§1.7. The Markov property 21
§1.8. The strong Markov property and applications 28
§1.9. Continuous time martingales and applications 38
§1.10. The Skorokhod embedding 47
§1.11. Donsker’s theorem and applications 51
Chapter 2. Continuous Time Markov Chains 57
§2.1. The basic setup 57
§2.2. Some examples 59
§2.3. From Markov chain to infinitesimal description 61
§2.4. Blackwell’s example 65
§2.5. From infinitesimal description to Markov chain 68
§2.6. Stationary measures, recurrence, and transience 79
§2.7. More examples 86
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