Math 735 Stochastic Analysis

Fall 2003

Meetings: 9:30-10:45 TR, B211 Van Vleck
Instructor: Timo Seppäläinen
Office: 419 Van Vleck, Office Hours: by appointment
Phone: 263-2812
E-mail: seppalai@math.wisc.edu
Course homepage: http://www.math.wisc.edu/~seppalai/735/735home.html

Prerequisites. Formally 431 and consent of instructor. The true prerequisite is a certain degree of mathematical maturity. Consent is automatic for students who have had advanced undergraduate or graduate analysis or probability.

Textbook. No textbook is required. The lectures will be based on a number of sources. Lecture notes will be available on the web, as the instructor gets material ready. If students wish to acquire a book, a modern treatment of the subject can be found in
P. Protter: Stochastic Integration and Differential Equations, Springer.
According to the Springer website, this book should be available in October. A good, carefully written book is
Y. Karatzas and S. Shreve: Brownian Motion and Stochastic Calculus, Springer.
This book covers integrals with respect to continuous martingales. A third source are the concise lecture notes on T. Kurtz's homepage: http://www.math.wisc.edu/~kurtz/m735.htm

Description. 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.

  1. Sort out the different integrals in analysis (Lebesgue, Lebesgue-Stieltjes, Riemann-Stieltjes)
  2. Foundations of probability theory, especially conditional expectation
  3. Generalities about stochastic processes, Brownian motion, Poisson process
  4. Martingales
  5. Stochastic integral with respect to Brownian motion (quick overview of the Math 635 stochastic integral)
  6. Predictable processes and stochastic integral with respect to cadlag martingales and semimartingales
  7. Ito's formula
  8. Stochastic differential equations
  9. Feynman-Kac formula
  10. Local time for Brownian motion
  11. Other topics where stochastic analysis appears (Girsanov's theorem perhaps)

Grades. Course grades will be based on homework sets.


Timo Seppalainen
Last modified: Mon Sep 1 14:34:18 CDT 2003