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.

- 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
- Feynman-Kac formula
- Local time for Brownian motion
- 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