Basics of Stochastic Processes

**Instructor:**Richard Sowers**Office:**347 Illini Hall**Phone:**(217) 333-6246**email:**r-sowers@math.uiuc.edu**Home page:**`https://math.uiuc.edu/~r-sowers`(this syllabus can be found there)**Office Hours:**MWF 11-12 and by appointment**Class meets:**MWF 12-12:50 P.M. in 143 Henry**Text:**Oksendal,*Stochastic Differential Equations: An Introduction with Applications*, 2003, Springer-Verlag

**Outline:** This is a short course (January 16th to March 9th) which quickly introduces the basics
of the modern theory of stochastic processes. We shall emphasize
those parts of the theory which are useful in a wide variety
of disciplines. Our intended audience is not only mathematicians,
but students from engineering, physics, and finance. We assume that
the students will either be willing to accept, *ex cathedra*, basic aspects
of measure theory, or have the ability to understand them on their own.

- Overview
- Measure Theory
- Brownian Motion
- Construction
- Regularity
- Scaling
- Ito theory
- Stochastic Integration
- Ito's formula
- Feynman-Kac formulae
- Levy's characterization of Brownian motion
- Girsanov's formula
- Lecture 4: Stochastic differential equations
- Construction, Existence and Uniqueness
- 1-dimensional SDE's (Feller's test for explosions)
- Markov processes
- Lecture 5: Filtering

Grades will be determined on the basis of homework (30%), a midterm (30%) and a final (40%).

Department of Mathematics

University of Illinois
at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu
(217) 333-6246