Brownian motion, Spring 2021
Course outline
- Definition, construction and basic path properties.
- Brownian motion as a Markov process: Blumenthal's 0-1 law and martingales.
- Brownian motion as a scaling limit of random walks: Skorohod embedding and Donsker's theorem.
- Local times and the Ray-Knight theorem
- Stochastic calculus: construction of the stochastic integral, Itô's formula, Girsanov's theorem, Tanaka's formula, applications.
Further reading
- Probability: theory and examples, 4th edition or higher, Durrett.
- Stochastic calculus, Durrett.
- Brownian motion, Mörters and Peres.
- Foundations of modern probability, Olav Kallenberg.
- Brownian motion, martingales, and stochastic calculus, Jean-Francois Le Gall.
Homework