S T O C H A S T I C A N A L Y S I S , F a l l 2 0 1 7

**Lectures**: Monday, 7.10pm - 9pm, in Warren Weaver Hall 109.

** Lecturer**: Paul Bourgade, office hours Wednesday 2-3pm (Warren Weaver Hall 603), you also can email me (bourgade@cims.nyu.edu)
to set up an appointment.

**Optional problem session
**: Wednesdy, 5.30pm - 7pm, in Warren Weaver Hall 202, by Guillaume Dubach.
Guillaume also holds office hours on Tuesday, 3.30-4.30pm.

** Material.** For the most theoretical part: chapters 1, 2, 3, 4, 5 from my lecture notes.
Please let me know about the inaccuracies and typos you will find.

For some examples and motivations from mathematical finance:
Stochastic Calculus and Financial Applications by Michael Steele.

**Course description**: introduction to continuous stochastic processes,
connections with partial differential equations and emphasis on examples from mathematical finance.

**Prerequisites**: you need to be familiar with basic probability theory
(random variables, conditional expectation, convergence types).

**Grading**:
problem sets (40%), midterm (20%) and a final exam (40%).

**Academic integrity**:
Unless explicitly stated in writing on the assignment, all homework in this class is individual. If assignments from different students have similarities that show one was copied from the other, both students will be penalized.

A tentative schedule for this course is:

- Sep. 11. Conditional expectation, convergence types and discrete martingales.
- Sep. 18. Discrete time processes, a multiperiod model.
- Sep. 25. Brownian motion I: Gaussian vectors, construction, invariance properties.
- Oct. 2. Brownian motion II: Donsker's principle. Exponential Brownian motion.
- Oct. 9. No class, Columbus day.
- Oct. 16. Semimartingales I: filtrations, processes, stopping times.
- Oct. 23. Semimartingales II: local martingales, bracket.
- Oct. 30. Midterm exam.
- Nov. 6. Stochastic integrals.
- Nov. 13. Itô formula.
- Nov. 20. Change of measure, Girsanov's theorem.
- Nov. 27. Stochastic differential equations I.
- Dec. 4. Stochastic differential equations II.
- Dec. 11. Link with partial differential equations I. Back to the discrete world: Markov chains.
- Dec. 12. Link with partial differential equations II. Kolmogorov, Feynman and Kac.
- Dec. 18. Final exam.