Math 735 Stochastic Analysis
|Meetings: MWF 9:55-10:45 Van Vleck B119
|Instructor: Timo Seppäläinen
|Office: Van Vleck 425. Office Hours: MW after class, other times by appointment. |
|Phone: 263-3624 |
seppalai math wisc edu|
Stochastic analysis is a term that refers to stochastic integration and stochastic differential equations and related themes. Here is a list of topics we expect to cover. The amount of time devoted
to the fundamentals in the beginning will depend on the level of
background that the audience possesses.
- Foundations of probability theory, especially conditional
- Generalities about stochastic processes, Brownian motion,
- Stochastic integral with respect to Brownian motion (quick
overview of the Math 635 stochastic integral)
- Predictable processes
and stochastic integral with respect to cadlag martingales
- Itô's formula
- Stochastic differential equations
- Local time for Brownian motion, Girsanov's theorem
- White noise integrals and a stochastic partial differential equation
This course has flexible prerequisites.
The ideal background would be one or two semesters of
graduate measure-theoretic probability theory, such as
our 733 or 733-734.
An essential prerequisite is a certain degree of mathematical
maturity, so familiarity with advanced probability is not absolutely
necessary. The course will rely on modern integration theory (measure theory
covered in Math 629 and 721) and
advanced probability, and we can cover some of these points
quickly in the beginning.
Course grades will be based on take-home work and a possible in-class exam. Homework will be posted on Learn@UW. You can also see your score record on Learn@UW.
The course is based on lecture notes written by the instructor, available on Learn@UW. No textbook purchase is required.
Realized schedule for Fall 2014.
- Week 1. Measures and integration, BV functions.
- Week 2. Probability spaces,
σ-algebras as information, conditional expectations.
- Week 3. Stochastic processes, filtrations, stopping times, quadratic variation.
- Week 4. Quadratic variation, path spaces, Markov
- Week 5. Strong Markov property. Brownian motion.
- Week 6. Brownian motion, Poisson process, martingales. Friday September 10: no class on account of Midwest Probability Colloquium.
- Week 7. Martingales.
- Week 8. Stochastic integral with respect to Brownian motion.
- Week 9. Stochastic integral with respect to cadlag
L2 martingales and local L2 martingales.
- Week 10. Stochastic integral with respect to cadlag
local L2 martingales and semimartingales.
- Week 11. Itô's formula for cadlag semimartingales: proof
of the single variable case. Applications of Itô's formula. Lévy's characterization of Brownian motion.
- Week 12. Bessel process. Part of Burkholder-Davis-Gundy inequalities. SDEs, first example Ornstein-Uhlenbeck process.
- Week 13. Geometric Brownian motion.
Strong existence and uniqueness for Itô equations. (Thanksgiving week.)
- Week 14. Weak uniqueness and strong Markov property for
Itô equations. Local time for Brownian motion.
- Week 15. Local time for Brownian motion. Tanaka's formula. Skorohod reflection problem. In-class exam on Wednesday.
- A modern, rather deep treatment of the subject
can be found in
P. Protter: Stochastic Integration and Differential Equations,
- An easier read is K. Chung and R. Williams: Introduction to Stochastic Integration, Birkhäuser.
- A carefully written book is
Y. Karatzas and S. Shreve: Brownian Motion and Stochastic
This book covers integrals with respect to continuous martingales.
- Concise lecture notes are available on T. Kurtz's
Instructions for homework
- Homework must be handed in by the due date, either in
class or by 3 PM in the instructor's office or mailbox. Late submissions
cannot be accepted.
- Neatness and clarity are essential. Write one problem per page
except in cases of very short problems. Staple you sheets together.
- It is not trivial to learn to write solutions. You have
to write enough to show that you understand the flow of
ideas and that you are not jumping to unjustified conclusions,
but not too much to get lost in details. If you are unsure
of the appropriate level of detail to include, you can separate
some of the technical details as "Lemmas" and put them at the
end of the solution.
A good rule of thumb is
if the grader needs to pick up a pencil to check your
assertion, you should have proved it. The grader can
deduct points in such cases.
- You can use basic facts from analysis and measure theory
in your homework, and the theorems we cover in class
without reproving them. If you do use other
literature for help, cite your sources
properly. However, it is better to
attack the problems
with your own resources instead of searching the literature.
- It is extremely valuable, maybe essential,
to discuss ideas for homework problems with other
students. But it is not acceptable to
write solutions together or to copy another person's
solution. In the end you have to hand in your
own personal work. Similarly, finding solutions on the internet is tantamount to cheating. It is the same as copying someone else's solution.
Check out the
probability seminar for talks on topics that
might interest you.