Measure theory and integration, independence, laws of large numbers. Fourier analysis of distributions, central limit problem and infinitely divisible laws, conditional expectations, martingales. Prerequisite: either MATH 426 or MATH 576. Offered: jointly with MATH 522; W.
Instruction period: Jan 3 -- Mar 11
Week 1: Weak convergence on metric spaces. Week 2: Construction of Brownian motion, Donsker's invariance. Week 3: Basic properties of Brownian motion. Week 4: Continuous (local) martingales. Week 5: Stochastic integration with respect to Brownian motion. Week 6: Ito's formula for Brownian motion. Week 7: Markov processes. Strong Markov property. Week 8: Brownian motion as a strong Markov process. Week 9: Dirichlet problem. Feynman-Kac formula. Week 10: General Gaussian processes.
Student learning goals
General method of instruction
Class assignments and grading