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 561 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).

- Lecture 0: A scenic trip through measure theory
- Lecture 1: Brownian motion
- Review of Brownian motion
- Scaling properties
- Lecture 2: Stochastic integration and differentiation
- Construction and basic properties
- Ito's formula
- Girsanov's Formula
- Levy's Characterization of Brownian Motion
- Lecture 3: Stochastic differential equations
- Existence and Uniqueness
- 1-dimensional SDE's
- Feynman-Kac
- KPP equation
- Lecture 4: Markov processes
- Markov property
- Strong Markov property
- Generators
- Martingale problem
- Jump processes
- Invariant Measures
- Chapman-Kolmogorov
- Filtering Theory
- Lecture 5: Asymptotics

**Text**: Øksendal, Stochastic Differential Equations, Springer-Verlag, 6th ed., 2003 and my Notes

**Additional Reference: **Karatzas and Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag

**Instructor:** Richard Sowers

**Office:** 347 Illini Hall

**Phone:** (217) 333-6246

**email:** r-sowers@illinois.edu (note; this email may be forwarded to an off-campus email address)

**Home page:** https://faculty.math.illinois.edu/~r-sowers

**Class meets:** MWF 2-3 in 243 Altgeld Hall

**Office Hours:** MW 12:30-2