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Math 418-419

Starting Fall 2013, Math 418 will be crosslisted with Math 544, and Math 419 with Math 545.

Probability: Math 544-545 (crosslisted with Math 418-419)

This pair of courses provides a thorough introduction to measure-theoretic probability. No prior knowledge of probability is assumed. Results from measure theory are stated and used without proof. The focus is on discrete time and continuous time stochastic processes. Topics include: martingales, law of large numbers, central limit theorem, Brownian motion and special topics.

Stochastic Analysis: Math 546

This is a rigorous course on finite dimensional continuous stochastic processes, focusing on Markov processes. Topics include: stochastic integration with respect to continuous semimartingales, Itô's formula for continuous semimartingales and applications, stochastic differential equations, Girsanov's formula, martingale problems.
Additional topics depending on the interests of the class may then be chosen from: one-dimensional diffusion theory, local time, introduction to SLE, applications to areas such as filtering, stochastic control, genetics, mathematical finance, Stroock-Varadhan theory for finite dimensional diffusions.
Prerequisites: Math 545 or consent of the instructor. Students from other Departments interested in learning about stochastic analysis from a mathematical perspective are encouraged.

Discrete Probability: Math 548

This course covers more advanced topics in discrete probability. Some probability background is needed, including Markov chains and martingales. Measure theory may be used at some points. Topics include spectral analysis of Markov chains and mixing times, electrical networks and random walks, random graphs (Erdos-Renyi, random regular graphs, etc.), percolation (leading up to Smirnov's theorem on conformal invariance) and other statistical mechanics models (Ising, Potts).

Topics in Probability: Math 608

This is a topics course in probability which is offered when there is sufficient student interest. The topic of the course changes from year to year depending on the interests of students and instructor.

Topics in Mathematical Physics: Math 609

This topics course often studies a subject of interest to graduate students in probability.