STAT 698G "STOCHASTIC DIFFERENTIAL EQUATIONS"
Fall semester 2010
Instructor: Mark Freidlin (mif@math.umd.edu)
CONTENTS:
 Wiener process (Brownian motion) and its properties.
 Stochastic integral and Ito's formula.
 Stochastic differential equations; existence and uniqueness, Markov
property.
 Langevin's equation and Kramers approximation.
 Semigroup related to a diffusion process, Generator of the process.
 Backward and forward Kolmogorov's equations. Invariant density.
Gaussian
diffusion processes.
 Longtime 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,
Springer,
Fifth edition.
COMMENTS:
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.

