Math/Stat 832 Theory of Probability, Spring 2008

Meetings: TR 11-12:15, B211 Van Vleck

Instructor: Timo Seppäläinen

Office: 419 Van Vleck, Office Hours: Tuesdays after class and other times by appointment

Phone: 263-2812

E-mail: seppalai at math dot wisc dot edu

Extra Class

• Wednensday, April 9, 5-6 PM, VV 901.

List of Homework Problems. I will add homework problems to this list. The next homework is due May 1.

Spring 2008 Schedule

The chapters are from Kallenberg: Foundations of Modern Probability, 2nd edition.

1. Week 1. (Chapter 12) Generalities about random measures. Uniqueness for Laplace transforms. Poisson point processes, marking, decomposition.
2. Week 2. (Chapters 8, 12) Generalities about Markov processes. Homogeneous Poisson process on the line as a renewal process with exponential waiting times.
3. Week 3. (Chapter 12) Pure jump Markov processes: construction, Kolmogorov's Backward Equation, semigroup and generator. M/M/1 queue. Markov chain run by a Poisson clock. (Chapter 13) Review of Gaussian distributions, review of basic Hilbert space facts.
4. Week 4. (Chapter 3) Kolmogorov-Centsov theorem. (Chapter 13) Isonormal Gaussian process on a separable Hilbert space. Construction of Brownian motion, its martingale and Markov properties. Transformations that preserve Brownian motion.
5. Week 5. (Chapter 13) Quadratic variation of Brownian motion. Bounded variation functions as integrators. Construction of the integral of a deterministic function with respect to Brownian motion in the L² sense. Strong Markov property of Brownian motion with respect to the augmented filtration. Blumenthal 0-1 law. Brownian motion immediately crosses the axis infinitely often. Definition of Brownian bridge, Ornstein-Uhlenbeck process, Bessel process. (Chapter 7) Continuous-time martingales.
6. Week 6. (Chapter 7) Optional sampling for continuous-time martingales. (Chapter 17) Local martingales.
7. Week 7. (Chapter 17) More about local martingales. Elementary stochastic integral of a predictable step process. Existence of quadratic covariation for continuous local martingales.
8. Week 8. (Chapter 17) Existence and properties of quadratic covariation. Existence of stochastic integral with a progressive integrand and a continuous local martingale integrator.
9. Week 9. Classes rescheduled.
10. Week 10. (Chapter 17) Continuous semimartingales. Properties of the stochastic integral. Itô's formula and applications.
11. Week 11-12. (Chapter 19) Generators and semigroups of Feller processes on locally compact separable metric spaces.
12. Week 13-15 (Chapter 21) Linear SDE's. Strong existence and uniqueness. Weak existence and local martingale problem. Feller diffusions from solutions to SDE's with bounded Lipschitz coefficients.

Overview

The spring term will focus on continuous time stochastic processes. We start with a brief segment on Poisson processes and continuous-time Markov chains from Chapter 12, then move on to Chapter 13 to begin our study of Brownian motion. The subsequent material to be covered appears in Chapters 14-24 in Kallenberg's text. Below is a list of possible topics, not all of which probably can be covered. (Suggestions are also very welcome.)

• Brownian motion: construction, some basic properties, embedding martingales into Brownian motion, local time, connections with partial differential equations, Feynman-Kac formula.
• Stable processes.
• Stochastic integration with continuous integrators (in particular, Brownian motion), Girsanov transformation, basic stochastic differential equations.
• Diffusion processes.
• Weak convergence of stochastic processes.
• Semigroups and generators of processes.

Check out the Probability Seminar for talks on topics that might interest you.