- Time: Mon 6-9 pm
- Location: 三教 309
- TA: 徐子舟 (12331003 at sustech dot edu dot cn)
Lecture note#
lecture note (updated regularly)
- weekly; usually posted by Tuesday and due next Monday 12pm at noon.
- Discussion and collaboration are encouraged, but names of the collaborators should be given in the submitted work.
Grading scheme#
| % | |
|---|
| Participation | 20% | |
| HW assignments | 30% | weekly |
| Final | 50% | |
| Bonus | 5% | TBA |
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 diffusions. |
| 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 diffusions. |
| 14-15 | Local time: Tanaka’s Formula, generalized Itô’s Formula, Ray–Knight Theorem, Lévy’s Theorey. |