**Time and place:**MWF at 1 in MS6229**Instructor:**Thomas M. Liggett**Office hours:**MWF 2:30-3:30 in MS 7919**Text:**"Continuous Time Markov Processes: An Introduction" by T. M. Liggett. We will cover Chapters 1 (Brownian Motion), 5 (Stochastic Integration) and 6 (Multi-Dimensional Brownian Motion and the Dirichlet Problem).**Prerequisite:**Mathematics 275B (especially discrete time martingales) or equivalent.**Grading:**Grades will be based on homework. There will be no exams.

**Brownian motion and applications.**Brownian motion is surely the most important continuous time stochastic process. It is, for example, the main building block for the theory of stochastic calculus and mathematical finance. Among the topics we will discuss are: (a) its definition and construction, (b) path properties, (c) the strong Markov property and its use in performing explicit calculations, and (d) the Skorokhod representation, which permits reduction of problems involving iid random variables to Brownian motion problems. Brownian motion is the main topic of the course.**Stochastic integration.**Brownian paths are not of bounded variation, so integrals with respect to them cannot be defined in the Stieltjes sense. The Ito integral that is used in this new context is central to much of the general theory of stochastic processes, and in particular to financial mathematics. We will construct the integral, and prove Ito's lemma, which leads to many explict results, including a finer picture of Brownian motion itself.-
**Multi-dimensional Brownian motion and the Dirichlet problem.**The Dirichlet problem asks for harmonic functions on a domain D in Euclidean space with prescribed boundary conditions. The approach to this problem based on Brownian motion has a number of advantages over purely analytic approaches, including (a) more natural treatment of domains that are unbounded and/or do not have smooth boundary, and (b) a probabilistic interpretation for the solution(s). This theory can then be used to deduce properties of Brownian motion in R^d.

**Due April 14.**Problems 1.7, 1.16, 1.19, 1.20(a), 1.22, 1.31.**Due April 28.**Problems 1.53(for tau_3),1.67, 1.81 and 1.82 are to be turned in. Problems 1.55, 1.57,1.58,1.63 and 1.84 are straightforward, You should do them, but not to hand in.**Due May 12.**Problems 1.99,1.109,1.110 and 1.127 are to be turned in. You should do problems 1.100,1.106,1.107 and 1.101, but these are not to be turned in.**Due June 2.**Problems 5.7,5.15,5.25,5.34,5.45 and 5.62 are to be turned in. You should do problems 5.8,5.18,5.19,5.26,5.36 and 5.63, but these are not to be turned in.