Math 546 Continuous Time Stochastic Processes TTh 9:30-11:00, Math Annex 1118.

This is a rigorous course on finite dimensional continuous Markov processes. Most top-ics covered will be included in Chapters IV and V of Rogers and Williams ’ text. We will study stochastic integration with respect to continuous semimartingales, and Itô’s stochas-tic calculus. The focus of the course will be on finite-dimensional stochastic differential equations. After a brief review of Brownian motion we will study Itô’s pathwise uniqueness results and then introduce the weak solutions, martingale problems and the relationship with strong or pathwise solutions. Change of measure (Girsanov) formulae will be derived and applied to the well-posedness of the martingale problem for finite dimensional sde’s. Depending on the interest of the students and energy level of the instructor we will then study local times and one-dimensional diffusion theory or Stroock-Varadhan martingale problems. The course will assume familiarity with measure theoretic probability theory ∗ includ-ing discrete parameter martingale theory and Brownian motion although there will be a brief review of these last two topics. The text is self-contained for the most part but does refer to Volume 1 (cited below) on occasion. Prequisites: Math 545 or consent of the instructor. ∗Students from other Departments interested in learning about stochastic differential equa-tions from a mathematical perspective are encouraged–Measure theoretic prerequisites may be treated as “black boxes”. Instructor