MATH 672: Stochastic Processes (Spring 2006)

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 squareintegrable martingales.
    
5. Ito's stochastic calculus.
   Stochastic integrals. Stochastic differential equations. Ito's differentiation rule. Diffusions. Elements of general stochastic calculus.

Last modified: October 3, 2005