Math 623- Advanced Probability (Fall 2017)

Instructor: Xia Chen

Office: Ayres 241, 974-4284.

Email: xchen@math.utk.edu

Website: http://www.math.utk.edu/~xchen

Class: MWF 10:10a.m.-11a.m. (Ayress 114)

Office Hours: MWF 2:00-3:00pm.

Textbook:
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 Peter Morters and Yuval Peres (2010)

Course Description:
As the single 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. Its central rule in mathematics is also matched by numerous applications in science, engineering and mathematical finance. This course aims to provide a systematic account for the sample properties of Brownian motions. If the situation allowed, we will continue on the same topic in Spring, 2018 (Math 624).

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

Prerequisite:
Officially, it requires Math523-524 (probability at graduate level) or equivalent. If you are not sure whether or not this course is right for you, please stop by and discuss it with me

Homework #1
Chapter 1: 5. Chapter 2: 5, 8, 13, 19, 21, 23.

Homework #1 solution (pdf)

Homework #2
Chapter 3: 2. Chapter 5: 4, 6, 10, 11, 17, 18, 19.

Homework #2 solution (pdf)

Homework #3
Chapter 6: 1, 5, 6, 8, 11, 12.

Homework #3 solution (pdf)

Homework #4
Chapter 7: 3, 6, 9, 10, 13, 18. Chapter 8: 2, 4, 5.

Homework #4 solution (pdf)

Homework #5
Chapter 9: 5, 6. Chapter 10: 1, 5. Chapter 12: 2, 3. Chapter 14: (Suplemental problems)

Homework #5 solution (pdf). Chapter 19: 3, 5, 9, 10, 14.