Course Information
- Time: Tue 7 - 10 PM
- Location: 三教 308
- Office hour: M622, Tue 3-5 PM or by appointment
- TA: 陈梓艺 (12231268 at mail dot sustech dot edu dot cn)
Lecture note
lecture note (updated Apr 21)
references
We will begin with Le Gall’s presentation, and then switch to KS, Chapters 5 and 6 for SDEs and Lévy’s theory. KS chapters 1-4 can serve as an substitute to Le Gall, but they are more technically involved.
- Le Gall, Jean-François. Brownian Motion, Martingales, and Stochastic Calculus. Vol. 274. Graduate Texts in Mathematics. Cham: Springer International Publishing, 2016. https://doi.org/10.1007/978-3-319-31089-3.
- Karatzas, Ioannis, and Steven Shreve. Brownian Motion and Stochastic Calculus. 2nd ed. Graduate Texts in Mathematics. New York: Springer-Verlag, 1998. https://www.springer.com/us/book/9780387976556.
- Revuz, Daniel, and Marc Yor. Continuous Martingales and Brownian Motion. 3rd ed. Grundlehren Der Mathematischen Wissenschaften. Berlin Heidelberg: Springer-Verlag, 1999. https://doi.org/10.1007/978-3-662-06400-9.
HW
bi-weekly
| Assignments | Due date |
|---|---|
| HW1 | Mar 11th |
| HW2 | Mar 25th |
| HW3: Le Gall 3.26,27,28,29 | Apr 8th |
| HW4: Le Gall 4.23,24,25,27 | Apr 22nd |
| HW5: Le Gall 5.26,31,32,33 | May 6th |
Grading scheme
| % | ||
|---|---|---|
| HW assignments | 40% | bi-weekly |
| Final | 60% |
Schedule
| Week | Content |
|---|---|
| 1 | Preliminaries: stochastic processes, Gaussian spaces and Gaussian processes, measure theory on infinite-dimensional spaces. |
| 2-4 | Brownian motion and continuous martingales: construction of Brownian motions, path properties; stopping times, continuous-time martingales, Optional Sampling Theorem, maximal inequality; the Doob-Meyer decomposition; filtration, augmentation and usual condition. |
| 5-9 | Stochastic integrals: continuous local martingales, quadratic and cross variation; Construction of the Itô integral; technique of localization; the change-of-variable formula (Itô’s Formula), semi-martingales; Lévy’s characterization; representations of continuous martingales as time-change Brownian motion; continuous martingale as Brownian integrals; Girsanov theorem, exponential martingales, Novikov condition. |
| 10-14 | Stochastic differential equations: Feller semi-groups, generators; strong and weak solutions; Lipschitz case; pathwise uniqueness; Yamada–Watanabe Theorem; martingale problem, existence and uniqueness; strong Markov property for diffusion. |
| 12-13 | Connection with partial differential equations: representation of solutions via diffusion; Feynman–Kac Formula; Brownian motion and harmonic functions; regular boundary points; recurrence of Brownian motions, study of hitting time; Doob’s \(h\)-transform and conditioned diffusion. |
| 14-15 | Local time: Tanaka’s Formula, generalized Itô’s Formula, Ray–Knight Theorem, Lévy’s Theory. |