## Syllabus

*Brownian motion.*Existence and sample path properties.*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, conformal invariance of planar Brownian motion, 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.*Stroock-Varadhan theory.*Diffusions, martingale problems, equivalence with SDEs, approximations of diffusions by Markov chains.

## Example Sheets

The example sheets will be posted here during the course of the term.

## References

I will mostly follow the following references:

- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
- N. Berestycki, Stochastic Calculus and Applications, Lecture Notes; available here.