**Instructor:**Richard Sowers**Office:**107 Altgeld 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)**Class meets:**Mondays, Wednesdays, and Fridays 3-3:50 P.M. in 159 Altgeld Hall

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

This is a continuation of Math 451. We will here focus on some of the modern theory of stochastic processes. See also the Course Catalog Entry for Math 452 or the list of UIUC Probability Theory Classes.

Grades will be determined on the basis of homework questions.

- Lecture 1: Brownian motion
- Review of Brownian motion
- Canonical Brownian motion/Wiener measure
- Scaling properties
- Nondifferentiability
- Law of Iterated Logarithm
- Modulus of continuity

- Lecture 2: Technicalia for continuous-time processes
- Equality of continuous-time processes
- Continuous-time filtration
- Measurability of continuous-time processes
- Stopping times

- Lecture 3: Continuous-time martingales
- Continuous-time counterparts of discrete-time results
- Quadratic variation

- Lecture 4: Stochastic integration
- Construction and basic properties
- Ito's formula

- Lecture 5: Applications of stochastic integration and Ito's formula
- Burkholder-Davis-Gundy inequalities
- Levy's characterization of Brownian motion
- Girsanov's formula
- Martingales as time-changed Brownian motions

- Lecture 6: Stochastic differential equations
- Existence and Uniqueness
- Feynman-Kac

- Lecture 5: Markov processes
- Markov property
- Strong Markov property
- Brownian motion and SDE's have strong Markov property
- Generators
- Martingale problem
- Chapman-Kolmogorov
- Transition functions
- Stationary distributions
- Ergodicity

- Appendix: Prohorov's Theorem

**Additional References:**

- D. Stroock, Probability Theory: an Analytic View, Cambridge University Press, 1993.
- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1991.
- S. Ethier and T. Kurtz, Markov Processes: Characterization and Convergence, Wiley, 1986.

Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu

(217) 333-6246