Course Information
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Lecture note
lecture note (updated regularly)
references
We will loosely follow Le Gall’s presentation at the beginning, and then switch to KS, Chapters 5 and 6 for SDEs and Lévy’s theory. KS chapters 1-4 can be a substitute for 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
biweekly
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. |