Instructor: Xia Chen
Office: Ayres 241, 974-4284.
Email: xchen@math.utk.edu
Website: http://www.math.utk.edu/~xchen
Class: TR 11:20a.m.-12:35p.m. Ayres G013
Office Hours: TR 9:00-10:00pm.
Textbook: Brownian motion-- A Guide to Random Processes and Stochastic Calculus By Rene L. Schilling (2021) (3th edtion)
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:
The topic for the school year 2023-2024 is “Brownian motion”.
The text book is
“Brownian motion--- a guide to random processes and stochastic calculus”
by Rene L. Schilling (2021).
As a single most important stochastic process, Brownian motion
appears as intersection of three fundamental classes of
stochastic processes: martingales, Gaussian processes
and Markov processes. Brownian motion is named after the botanist
Robert Brown, who
first described the phenomenon in 1827,
while looking through a microscope at pollen of
the plant Clarkia pulchella immersed in water.
Since then, 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 and
Lévy processes, Brownian motion serves a bridge between analysis and
some hard problems of discrete structure. Its central role in
mathematics is also
matched by numerous
applications in science, engineering and mathematical finance.
This course aims to provide a systematic account for the subject of
Brownian motions
and to develop the basic tools known as stochastic analysis.
This course is designed for students who finished Math523-524 or
any other equivalent courses on probability at graduate level, and who are interested in
probability or its
applications.
If the situation allowed, we will continue on the
same topic in Spring, 2023 (Math 624).
Grading policy:
There will be no tests and exams. Your
performance in the classroom (50%) and in homework (50%)
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 (Due: Thursday, Sept. 14)
Chapter 2: 1, 8, 16, 17, 18, 22, 24, 26
Solution to Homework #1 (pdf)
Homework #2 (Due: Tuesday, Oct. 3)
Chapter 5: 5, 6, 10, 11, 20,21, 22.
Solution to Homework #2 (pdf)
Homework #3 (Due: Thursday, Oct. 26)
Chapter 6: 1, 5, 8, 11, 12, 15, 16.
Solution to Homework #3 (pdf)
Homework #4
Chapter 7: 3, 6, 13, 14, 18, 19. Chapter 8: 2, 5, 6, 7.
Solution to Homework #4 (pdf)