- Stochastic calculus for continuous processes. Martingales, local martingales, semi-martingales, quadratic variation and cross-variation, Itô's isometry, definition of the stochastic integral, Kunita-Watanabe theorem, and Itô's formula.
- Applications to Brownian motion and martingales. Lévy characterization of Brownian motion, Dubins-Schwarz theorem, martingale representation, Girsanov theorem, and Dirichlet problems.
- Stochastic differential equations. Strong and weak solutions, notions of existence and uniqueness, Yamada-Watanabe theorem, strong Markov property, and relation to second order partial differential equations.
- Applications and examples.
- Example Sheet 1 (posted January 18).
- Example Sheet 2 (posted February 1).
- Example Sheet 3 (posted February 17).
- Example Sheet 4 (posted March 1).
- Solution sketches for Example Sheets 1-3 and Example Sheet 4
The tentative dates and locations for the example classes are:
- We 13 Feb, 2-4pm, MR3
- We 27 Feb, 2-4pm, MR9
- We 13 Mar, 2-4pm, MR9
- Fr 3 May, 2-4pm, MR9
- Revision class: Mo 20 May, 2-4pm, MR9
[In the revision class, the exam questions from 2018 and 2017 will be discussed.]
I will mostly follow the following references:
- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
- N. Berestycki, Stochastic Calculus.
- M. Tehranchi, Stochastic Calculus.
- V. Silvestri, Stochastic Calculus.
- J. Miller, Stochastic Calculus.
- J. Norris, Advanced Probability.
- P. Sousi, Advanced Probability.
- J. Norris, Probability and Measure.