T . Lyapunov Exponents for Stochastic

Primary audience. Ph.D. students and advanced M.S. students interested in stochastic processes. General Course Description. A basic course which introduces the topic of stochastic partial di¤erential equations (SPDEs) via some simple examples that are amenable to detailed calculations, focusing mainly on parabolic equations. The prerequisite for the course being only a graduate course in probability theory, our “Chapter Zero” will be a tailored crash course on stochastics. We will then cover: existence and uniqueness questions, Feynman-Kac-type and particle representation, local (Holder continuity) and long-term (Lyapunov exponents) behavior via the theory of Gaussian regularity, and the brand new topic of SPDEs driven by fractional Brownian motion. We will look at speci…c applications in …nance (theory of stochastic interest rates) and/or …ltering (optimal nonlinear stochastic …ltering theory). The …rst half of the semester will be spent covering the basics of the theory. In the second half, the instructor will continue to present lectures, now on more advanced material. Simultaneously each student will choose or will be assigned a short research article (or a portion of a longer article) to read and present in class. The list of articles to choose from are the most representative of the main recent contributions to the theory. If the class is large, groups will be allowed for the oral presentations. An important consequence of this course is that several students should be able to clearly identify PhD topics. Prerequisites: A non-measure-theoretic working knowledge of probability theory AND stochastic calculus, contained for example in a solid performance in 598F or in 532, or in the …rst few chapters of Oksendal’s textbook on Stochastic Di¤erential Equations. Because most of this course will be based heavily on explicit Gaussian tools, it will be accessible to a large audience of students with a working knowledge of stochastic calculus. List of topics. This list is non-exhaustive, and we reserve the right to skip some of the topics listed, depending on student interest. Although we will attempt to give detailed proofs for a number of topics, some proofs will be abridged and/or left as homework problems for the students. –Background material in stochastic analysis basic stochastic calculus (along the lines of [6], Chapter 3), stochastic di¤erential equations and Feynman-Kac formulas for PDEs (e.g. [6], Ch 4) extension to SPDEs, and evolution equations,