OLD HOMEPAGE FOR Math 735 Stochastic Analysis

Fall 2010

Meetings: MWF 12:05-12:55 Van Vleck B123

Instructor: Timo Seppäläinen

Office: Van Vleck 419. Office Hours: Wednesday 10-11, and any time by appointment

Phone: 263-2812

E-mail: seppalai@math.wisc.edu

Lecture Notes

Current version of the stochastic analysis course notes.

Prerequisites

This course has flexible prerequisites. The ideal background would be one or two semesters of graduate measure-theoretic probability theory, such as our 831 or 831-832. 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.

Homework

When you do homework be sure to check the latest version of the notes for the correct exercise number. But note: old homework assignments are not updated as the notes are updated, so the exercise numbers of old homework no longer match the problems.

Homework 1 due Wednesday, Sept. 15. Exercises 1.4, 1.7(a) and (c) from p. 32-33 of the notes.
Homework 2 due Wednesday, Sept. 22. Hand in Exercise 1.16. Suggested, but not to hand in: Exercises 1.9-1.15.
Homework 3 due Wednesday, Sept. 29. Exercises 2.5, 2.6.
Homework 4 due Wednesday, Oct. 6. Exercise 2.16. Try to do this as elegantly as possible, but without cutting corners.
Homework 5 due Friday, Oct. 15. Exercises 2.19 and 2.21.
Homework 6 due Wednesday, Oct. 27. Exercise 4.4.
Homework 7 due Wednesday, Nov. 3. Exercise 5.4 (the non-predictability of the Poisson process).
Homework 8 due Wednesday, Nov. 17. Exercises 6.3 and 6.6.
Bonus Problem only for those ready for a more interesting problem: Exercise 6.5. (Only neat and complete solutions get credit.)
Homework 9 due Wednesday, Nov. 23. Exercise 6.9 (the expectation of the exit time from a ball).
Homework 10 due Monday, Dec. 13. Exercises 7.2 and 7.4. Note that for the quadratic variation questions you do not need to start with sums of squares of increments. Use existing results from the notes, such as Lemma 5.40 from p. 171.

Fall 2010 Schedule

Week 1. Began overview of Lebesgue integration.
Week 3. BV functions and their Lebesgue-Stieltjes measures. Radon-Nikodym theorem. Independence. Definition and first example of conditional expectation.
Week 4. Properties of conditional expectation. Filtrations and stopping times.
Week 5. Quadratic variation process. Brownian motion.
Week 6. Markov property and quadratic variation of Brownian motion. Poisson processes. Beginning of martingales: optional stopping.
Week 7. Martingale inequalities. Local martingales and semimartingales. Quadratic variation for martingales and local martingales. Spaces of cadlag and continuous L² martingales.
Week 8. Stochastic integral with respect to Brownian motion.
Week 9. Stochastic integral with respect to cadlag martingales and local L² martingales.
Week 10. Stochastic integral with respect to cadlag semimartingales. Discussion of Itô's formula. Lévy's characterization of Brownian motion.
Week 12. Strong Markov property. Recurrence and transience of Brownian motion in dimensions 2 and higher as an application of Itô's formula.
Saturday Nov. 20 1-3 PM. Proof of the multidimensional Itô formula.
Week 13. SDEs, Itô equations, Ornstein-Uhlenbeck process, geometric Brownian motion, Brownian bridge, stochastic exponential.
Week 14. Strong existence and uniqueness for Itô equations.
Saturday Dec. 4 1-3 PM. Local time for Brownian motion.
Week 16. Cameron-Martin-Girsanov theorem and the reflection principle for Brownian motion. The distribution of the hitting time of a line.

Other material:

A modern, rather deep treatment of the subject can be found in P. Protter: Stochastic Integration and Differential Equations, Springer.
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 Calculus, Springer. This book covers integrals with respect to continuous martingales.
Concise lecture notes are available on T. Kurtz's homepage: http://www.math.wisc.edu/~kurtz/m735.htm

Grades. Course grades will be based on take-home work.

Course content. Here is a tentative list of possible topics. The amount of time devoted to the fundamentals in the beginning will depend on the level of background that the audience possesses.

Sort out the different integrals in analysis (Lebesgue, Lebesgue-Stieltjes, Riemann-Stieltjes)
Foundations of probability theory, especially conditional expectation
Generalities about stochastic processes, Brownian motion, Poisson process
Martingales
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 and semimartingales
Ito's formula
Stochastic differential equations
Local time for Brownian motion
Other topics where stochastic analysis appears (Girsanov's theorem perhaps)

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.
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 or the internet.
It is valuable 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.

Check out the probability seminar for talks on topics that might interest you.