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Math 418-419

This pair of undergraduate courses is ideal for a graduate student who wishes to learn some serious probability but does not have a background in measure theory. Math 418 studies random variables and limit theorems, and Math 419 studies stochastic processes including Markov chains, martingales and Brownian motion.

Probability: Math 544-545

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

Stochastic Analysis: Math 546 (listed as 608 in 2010)

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 (listed as 608 in 2011)

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.