Mathematics 275C - Spring 2010
- 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
- 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.
No late homework will be accepted.
- 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.