Math 735 Stochastic Analysis

Fall 2003

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 topics:

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

Grades. Course grades will be based on homework sets.