Brownian motion and stochastic calculus (Karatzas and Shreve)
16:642:591 - Topics in Probability & Ergodic Theory I
This course will cover topics in stochastic analysis at the graduate level. The topics will include continuous semimartingale theory: Brownian motion, continuous-time martingales, stochastic integration, Girsanov's theorem, stochastic differential equations, and diffusions. Time permitting, we will cover Levy processes, the Levy-Khintchine formula, and the semimartingale topology in stochastic integration.
Sections Taught This Semester:
Schedule of Sections:
- Notes on lectures 1 and 2 up to conditional expectation.
- Notes on the Dunford-Pettis compactness criterion
- Note: Exercise 5 of lecture 1 was misstated. For the correct statement and an extension of the problem, see the lecture 1 notes. I posted a corrected version on Feb. 6.
- Notes on lectures 2,3, and 4 on discrete-time martinagles.
- Notes on the definition of Brownian motion
- (Lecture Notes 4) Further notes on constructing Brownian motion
- (Lecture Notes 5) Notes on Levy processes
- (Lecture Notes 6) Filtrations and stopping times in continuous time; Applications to Levy processes
- (Lecture Notes 7) Introduction to Stochastic Integration