**Instructor:**Richard Sowers**Office:**227B Illini Hall

**Phone:**(217) 333-6246**email:**r-sowers@math.uiuc.edu**Home page:**`https://faculty.math.illinois.edu/~r-sowers`

(this syllabus can be found there)**Office Hours:**MWF 1--2 and by appointment**Class meets:**Mondays, Wednesdays, and Fridays 3-3:50 P.M. in 341 Altgeld Hall

**Text:** Øksendal, Stochastic Differential
Equations, 5th ed., 1998, Springer-Verlag.

Roughly, this is a course about stochastic differential equations and the associated ideas of Ito calculus. We hope to understand both of these subjects and understand some of their structure. By structure, we mean some different perspectives and applications which reveal different perspectives on the material. Our goal will be to present this knowledge in a way which will be accessible to engineers who possess a certain level of mathematical maturity and at the same time to be precise enough that the mathematics student should have little trouble filling in the details. While Math 451 is not a prerequisite, you should be willing to dig through the notes for that class (which are online) to fill in as much detail as you need. Similarly, while you are not required per se to know measure theory, you should be able to become comfortable with the usage of measure theory (I will give a brief summary of the relevant portions of measure theory as needed).

Grades will be determined on the basis of homework questions.

- Lecture 0: Summary of some relevant mathematical facts
- Measure theory
- Conditional expectation

- Lecture 2: Brownian motion
- Review of Brownian motion
- Scaling properties

- Lecture 3: Stochastic integration and differentiation
- Construction and basic properties
- Ito's formula

- Lecture 4: Applications of stochastic integration and Ito's formula
- Burkholder-Davis-Gundy inequalities
- Girsanov's formula
- Feynman-Kac
- Hitting times
- Black-Scholes formula

- Lecture 5: Stochastic differential equations
- Existence and Uniqueness
- 1-dimensional SDE's

- Lecture 6: Markov processes
- Markov property
- Strong Markov property
- Generators
- Martingale problem
- Jump processes
- Chapman-Kolmogorov

**Additional References:**

- Karatzas and Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., 1994, Springer-Verlag.

Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu

(217) 333-6246