Math 735 Stochastic Analysis

Fall 2014

Meetings: MWF 9:55-10:45 Van Vleck B119

Office: Van Vleck 425. Office Hours: MW after class, other times by appointment.

Phone: 263-3624

E-mail: seppalai math wisc edu

Course description

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 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
Itô's formula
Stochastic differential equations
Local time for Brownian motion, Girsanov's theorem
White noise integrals and a stochastic partial differential equation

Prerequisites

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.

Evaluation

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.

Lecture notes

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 processes.
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.

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

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.