## COVID-19 Update on the Exam

The details of the online open book pass/fail exam in June are still being worked out and further information will be provided or linked here as it becomes available. I expect that the level of the questions will be comparable to that of the more straightforward problems in previous years.

## Schedule

*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 Sheets

- Example Sheet 1 and Solutions (posted January 18)
- Example Sheet 2 and Solutions (posted February 1)
- Example Sheet 3 and Solutions (posted February 17)
- Example Sheet 4 (posted March 1)

## Example Classes

- Example class 1: Mo 10 Feb, 3:30-5:30 MR3 / Tu 11 Feb, 1:30-3:30 MR3
- Example class 2: Mo 24 Feb, 3:30-5:30 MR11 / Tu 25 Feb, 1:30-3:30 MR3
- Example class 3: Mo 9 Mar,
**3:00-5:00**MR3 / Tu 10 Mar, 1:30-3:30 MR3 - Example class 4: Mo 20 Apr, 3:30:5:30 MR3 / Tu 21 Apr, 1:30-3:30 MR3
- Drop-in session 1: Mo 17 Feb, 4:00-5:00
- Drop-in session 2: Mo 2 Mar, 4:00-5:00
- Revision class: Mo 18 May / Tu 19 May

## Approximate list of lectures

- Introduction. The Wiener Integral.
- Lebesgue-Stieltjes integration and signed measures.
- Lebesgue-Stieltjes integration and signed measures.
- Finite variation processes.
- Local martingales.
- L^2 bounded martingales.
- Quadratic variation.
- Quadratic variation, covariation.
- Kunita-Watanabe inequality, semimartingales, Ito integral for simple processses
- Ito isometry
- Consequences of Ito isometry
- Integration by parts and Ito formula
- Levy's characterisation of Brownian motion. Dubins-Schwarz theorem.
- Dubins-Schwarz theorem. Girsanov's theorem.
- Girsanov's theorem and Cameron-Martin formula.
- SDEs
- Strong existence of solutions.
- Solution map.
- Examples of SDEs and local solutions.
- Local solutions and Dirichlet-Poisson problem.
- Dynkin's formula. Cauchy problem.
- Markov property.
- Generator and invariant measures.
- Convergence to equilibrium.

## References

I will mostly follow the following references:

- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer.

I will assume the content of these references:

- J. Norris, Probability and Measure
- J. Norris, Advanced Probability

Lecture notes from previous years:

- N. Berestycki, Stochastic Calculus
- M. Tehranchi, Stochastic Calculus
- V. Silvestri, Stochastic Calculus
- J. Miller, Stochastic Calculus
- P. Sousi, Advanced Probability