# Part III - Advanced Probability

## Lectured by M. Lis, Michaelmas 2017

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Some measure theory
- 2 Martingales in discrete time
- 2.1 Filtrations and martingales
- 2.2 Stopping time and optimal stopping
- 2.3 Martingale convergence theorems
- 2.4 Applications of martingales
- 3 Continuous time stochastic processes
- 4 Weak convergence of measures
- 5 Brownian motion
- 5.1 Basic properties of Brownian motion
- 5.2 Harmonic functions and Brownian motion
- 5.3 Transience and recurrence
- 5.4 Donsker's invariance principle
- 6 Large deviations