STAT 698G "STOCHASTIC DIFFERENTIAL EQUATIONS"
Fall semester 2010
Instructor: Mark Freidlin (email@example.com)
- Wiener process (Brownian motion) and its properties.
- Stochastic integral and Ito's formula.
- Stochastic differential equations; existence and uniqueness, Markov
- Langevin's equation and Kramers approximation.
- Semigroup related to a diffusion process, Generator of the process.
- Backward and forward Kolmogorov's equations. Invariant density.
- Long-time behavior of the process; recurrent and transient processes.
Convergence to the limiting distribution.
- Markov times. Exit problems. Elliptic boundary value problems for
expected values of various functionals of diffusion processes.
- Diffusion processes with reflection on the boundary. Other boundary
conditions preserving the Markov property.
- Introductions to asymptotic methods for stochastic differential
equations and related PDE's: expansion in a small parameter, averaging,
homogenization, large deviations.
TEXTBOOK: B.Oksendal, Stochastic Differential Equations,
I plan also to consider some examples and applications.
I assume that the students in the class know a course of
probability, calculus. I will try to minimize the use of measure
theory and pay the main attention to various methods of calculation
of characteristics of stochastic processes.