Math 624- Advanced Probability (Spring 2018)

Instructor: Xia Chen

Office: Ayres 202, 974-4284.



Class: MWF 10:10a.m.-11:00a.m. (Ayress 404)

Office Hours: MWF 9:00 a.m.- 10:00 a.m.

Brownian motion-- an introduction to stochastic processes. By Rene L. Schilling and Lothar Partzsch (2012)

Reference Books:

    Continuous martingales and Brownian motion. By Daniel Revuz and Marc Yor (1991)

    Brownian motion. By Yuval Peres and Peter Morters (2010)

    Course Description:
    As the most important stochastic process, Brownian motion appears as intersection of three fundamental classes of processes: It is a martingale, a Gaussian process and a Markov process. Since observed by physicists, Brownian motion has been in the center of the investigation for both mathematicians and the people in many other disciplines. Its fascinating link to the partial differential equations and harmonic analysis symbols the modern day of probability. As the scaling limit of the random walks, Brownian motion serves a bridge between analysis and some hard problems of discrete structure. This course aims to provide a systematic account for the sample properties of Brownian motions. The course is the continuation of Math 623 of the last semester. The main topics in this semester include Ito and other types of stochstic calculus, Cameron-Martin formula, Brownian fractal and Brownian local times.

    Grading policy:
    There will be no test and exams. Your performance in the classroom and in homework will decide the grade you receive.

    Homework #1
    Chapter 15: 1, 2, 4, 6. 9, 10, 12, 13. (Suplemental problem).

    Homework #1 solution (pdf)

    Homework #2
    Chapter 17: 1, 5, 7, 8, 13, 14.

    Homework #2 solution (pdf)

    Homework #3
    Chapter 18: 2, 3, 4, 5, 6, 11.

    Homework #3 solution (pdf)

    Homework #4
    (Suplemental problem). Chapter 19: 3, 5, 9, 14. Chapter 21: 6.


    Homework #4 solution (pdf)