**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:

- Brownian motion
- Basics of Ito calculus for Brownian motion
- Poisson process
- Basic properties of martingales (optional sampling theorem, Doob's inequalities)

**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