LMS/EPSRC Short Course on Stochastic Analysis
Oxford University, March 20-24, 2000
Weak approximation for Markov processes
Thomas G. Kurtz
University of Wisconsin - Madison
kurtz@math.wisc.edu
http://www.math.wisc.edu/~kurtz/
Prerequisites: Familiarity with the following topics will be assumed:
References:
Øksendal, Bernt. Stochastic differential equations. An introduction with applications. Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. (Chapters 1 - 5)
Ethier, Stewart N.; Kurtz, Thomas G. Markov processes.
Characterization and convergence. John Wiley &
Sons, Inc., New York, 1986. (Chapter 2, Sections 1 - 2.)
Problems and references: postscript pdf
Outline of lectures
DVI file: Transparencies for lectures
1. Stochastic equations for Markov processes
a) Ito equations for diffusion processes
b) Poisson random measures
c) Ito equations for Markov processes with jumps
2. Martingale problems for Markov processes
a) Levy and Watanabe characterizations.
b) Ito's formula and martingales associated with solutions of stochastic equations
c) Generators and martingale problems for Markov processes
d) Equivalence between stochastic equations and martingale problems
3. Weak convergence for stochastic processes
a) General definition of weak convergence
b) Prohorov's theorem
c) Skorohod representation theorem
d) Skorohod topology
e) Conditions for tightness in the Skorohod topology
4. Convergence for Markov processes characterized by stochastic equations
a) Martingale central limit theorem
b) Convergence for stochastic integrals
c) Diffusion approximations for Markov chains
d) Limit theorems involving Poisson random measures and Gaussian white noise
5. Convergence for Markov processes characterized by martingale problems
a) Tightness estimates based on generators
b) Convergence of processes based on convergence of generators
c) Averaging
d) A measure-valued limit
6. To be determined at the whim of the lecturer. Some possibilities:
a) Wong-Zakai corrections
b) Averaging for stochastic equations