Syllabus for Math 452
Text: Karatzas and Shreve, Brownian
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
Course Catalog Entry for Math 452 or the list of UIUC
Probability Theory Classes.
Grades will be determined on the basis of homework questions.
Note: the above links are to jpeg versions of the lecture notes.
- Lecture 1: Brownian motion
- Review of Brownian motion
- Canonical Brownian motion/Wiener measure
- Scaling properties
- 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
- Lecture 5: Markov processes
- Markov property
- Strong Markov property
- Brownian motion and SDE's have strong Markov property
- Martingale problem
- Transition functions
- Stationary distributions
- Appendix: Prohorov's Theorem
- 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
Convergence, Wiley, 1986.
Go to exercise page
Old Classes Page for R. Sowers
Richard Sowers (Home Page)
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W Green St.
Urbana, IL 61801