MATH 672: Stochastic Processes (spring 2008)

Instructor: Eugene Dynkin

Meeting Time & Room

1. Theory of stochastic interaction.

Gibbs formula. Conditional independence. Markov chains. Markov fields. Infinite particle systems. Gaussian fields.

2. Markov chains in an arbitrary state space: asymptotic behavior at large time.

Ergodic property of Markov chains. Strong Markov property. Doeblin's method.

3. Brownian motion.

Three views: limit of random walks, Markov process, Gaussian system. Construction of a continuous Brownian motion. Invariance propertyies and self-similarity. Strong Markov property. Blumenthal's 0-1 law. Probabilistic solution of the Dirichlet problem. Probabilistic approach to nonlinear PDEs.

4. Martingales

Doob-Meyer decomposition of a supermartingale. Optional sampling. Doob's upcrossing inequality. Kolmogorov's inequality. Hilbert space of continuous square-integrable martingales.

5. Ito's stochastic calculus.

Stochastic integrals. Stochastic differential equations. Ito's differentiation rule.

Diffusions. Elements of general stochastic calculus.

Last modified:October 1, 2007