UVA Probability Seminar
Mondays, 2:00 - 2:50pm
Kerchof 326
Organizers: Christian Gromoll & Tai Melcher
Mathematics Other Seminars Maps & Directions

Spring 2011

24 Jan Organizational Meeting
31 Jan Ton Dieker (Georgia Tech)
Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions
7 Feb Christian Gromoll
What is ... the dual of a random intersection graph?
18 Feb David Nualart (U Kansas) *
Clark-Ocone formula and central limit theorems for Brownian local time increments
21 Feb Len Scott
What is ... the role of probability in mathematical finance? I
28 Feb Len Scott
What is ... the role of probability in mathematical finance? II
7 Mar No Seminar
Spring Break
14 Mar No Seminar
21 Mar Marty Keutel
An Overview of the Skorohod Topologies
28 Mar Dan Dobbs
What is ... the Levy stochastic area process?
4 Apr Seminar cancelled
speaker rescheduled
11 Apr Tai Melcher
A subelliptic Taylor isomorphism theorem in infinite dimensions, I
18 Apr Tai Melcher
A subelliptic Taylor isomorphism theorem in infinite dimensions, II
26 Apr Larry Leemis (William & Mary)*
Computing in Probability
* please note time, date, and/or location change

Abstracts

Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions

Ton Dieker (Georgia Tech)

We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU) diffusion processes, which arise from many-server queueing systems with phase-type service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to `overload' and `underload'. The resulting switching behavior cause standard techniques for proving positive recurrence to fail. Using and extending the framework of common quadratic Lyapunov functions from the theory of control, we construct Lyapunov functions for the diffusion approximations corresponding to systems with and without abandonment. Using these Lyapunov functions, we prove that piecewise OU processes have a unique stationary distribution.
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Clark-Ocone formula and central limit theorems for Brownian local time increments

David Nualart (U Kansas) *

The purpose of this talk is to discuss some applications of the Clark-Ocone representation formula. This formula provides an explicit expression for the stochastic integral representation of functionals of the Brownian motion in terms of the derivative in the sense of Malliavin calculus. We will compare this formula with the classical Ito formula and we will discuss its application to derive a central limit theorem for the modulus of continuity in the space variable of the Brownian local time increments.
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A subelliptic Taylor isomorphism theorem in infinite dimensions, I

Tai Melcher

We will discuss recent results for subelliptic heat kernel measures on a particular class of infinite dimensional Lie groups. We will begin with the background for the Taylor isometry in the flat case and in finite dimensions, as well as discussing how we define Brownian motion on these infinite dimensional Lie groups. Then we will state the Taylor theorem in this setting, which says that there is a unitary mapping from the space of holomorphic functions which are square integrable with respect to the law of the Brownian motion onto the space of derivatives of those functions at the identity. We will discuss elements of the proof time permitting. This is joint work with M. Gordina.
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Computing in Probability

Larry Leemis (William & Mary)*

Statistical languages such as SAS and R have been used for decades to analyze large data sets. Relatively little work has been done, however, on using symbolic languages for the manipulation of random variables. This talk introduces a prototype probability language named APPL (A Probability Programming Language) developed by the speaker and his colleagues, and highlights some recent applications.

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Mathematics Other Seminars Maps & Directions