March 31 & April 1, 2000

CALL FOR PAPERS:

Southeast Applied Analysis Center, School of Mathematics and School of Industrial and Systems Engineering at Georgia Tech

We are holding our fifth annual stochastics meeting in the Southeast. The keynote speakers this year will be

J. BERTOIN (Paris VI)

and I. KARATZAS (Columbia University),

and each will be giving two one hour talks. These talks are titled:
Clustering statistics for a system of sticky particles (J. Bertoin),
Renewal theory for nested regenerative sets (J. Bertoin), and
Probabilistic aspects of finance, I and II (I. Karatzas);
the abstracts of these talks are included below.

As in previous years, the conference will be loosely structured, with opportunities for contributed talks. The main objective is to make/renew contacts and exchange ideas.

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REGISTRATION FEE:
NONE (both attending and presenting a talk at the conference is free!)

REGISTRATION:
If you are planning to attend, please contact Lorraine Shaw. Lorraine can be reached at:

E-mail: lshaw@isye.gatech.edu
Fax: 404-894-2301.

If you plan to attend please send information in the following format:

Name: Sigrún Andradóttir
Affiliation: Georgia Tech
Address:

School of Industrial and Systems Engineering
Georgia Tech
Atlanta, GA 30332
Phone: 404-894-3933
E-mail: sa@isye.gatech.edu

We encourage attendees to contribute talks on their latest research. If you wish to give a paper please include the following information:

Title: Conditions for finite moments of waiting times in G/G/1 queues

Coauthor(s): Christian Houdré

Abstract: We find conditions for moments to be finite in a G/G/1 queue.

CONFERENCE ORGANIZERS:
Sigrún Andradóttir, School of Industrial and Systems Engineering
Christian Houdré, School of Mathematics

LODGING:
You can make your own hotel arrangements with one of the many hotels in town. Some hotels close by Tech's campus are:

Holiday Inn Express (404-881-0881),
Days Inn, 683 Peachtree Street (404-874-9200),
Renaissance Hotel, W. Peachtree Street (404-881-6000),
Marriott Courtyard, 1132 Techwood Drive (404-607-1112), and
Regency Suites, 975 West Peachtree Street (404-876-5003).

In all cases, ask about the Georgia Tech rate.

LOCATION: The conference will be held in the first floor auditorium of the Manufacturing Research Center, 813 Ferst Drive, on the Georgia Tech Campus in Atlanta, Georgia.

MAP:
A map of campus can be found on the web at http://www.gtalumni.org. The conference is in the Manufacturing Research Center (building 126 on the map), which is the rectangular building directly north of the Groseclose building (#56 on the map) and the Instructional Center (#55 on the map).

DIRECTIONS:

By MARTA: Take the North-South Marta train ($1.50) to the North Avenue exit. The station is on the northeast corner of West Peachtree and North Avenue. Walk west along North Avenue past the Varsity and over the expressway. After the football stadium, take the steps up and enter the campus. Walk diagonally across the campus and ask some students where to find the Manufacturing Research Center.

By car, if you are entering Atlanta from I-20 or while travelling north on I-75 or I-85:
I-75 and I-85 merge in Atlanta to form I-75/85. (If you are on I-20, go north on I-75/85 in the center of Atlanta.) Exit the expressway at Exit 100 which is the Spring Street and West Peachtree Street exit. Turn left at the second light onto West Peachtree Street. Turn left at the first light onto North Avenue. Travel west on North Avenue and follow the signs to the "Center for the Arts". These signs will ask you to turn right onto Tech Parkway which is the second traffic light along the GT campus, then turn right at the first light, and then you are forced to turn either right or left onto Ferst Drive. Now go to the parking directions section below.

By car, if you are entering Atlanta while travelling south on I-75 or I-85:
I-75 and I-85 merge in Atlanta to form I-75/85. Exit the expressway at Exit 100 which is the North Avenue exit. Turn right at the top of the ramp onto North Avenue. Travel west on North Avenue and follow the signs to the "Center for the Arts." These signs will ask you to turn right onto Tech Parkway which is the second traffic light along the GT campus, then turn right at the first light, and then you are forced to turn either right or left onto Ferst Drive. Now go to the parking directions section below.

PARKING DIRECTIONS: Turn right onto Ferst Street, then turn left into the student center driveway which is the second driveway on your left. There is a fee of $4. Walk north past the Instructional Center to the Manufacturing Research Center.

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1. Clustering statistics for a system of sticky particles.

We shall present a connection between two different models of clustering: the deterministic model of sticky particles which describes the evolution of a system of infinitesimal particles governed by the dynamic of completely inelastic shocks (i.e. clustering occurs upon collision with conservation of masses and momenta), and the random model of the so-called additive coalescent in which velocities and distances between clusters are not taken into account. The connection is obtained when at the initial time, the particles are uniformly distributed on a line and their velocities are given by a Brownian motion. An important point in the study is the fact that the evolution of the system of sticky particles can be analysed in terms of the inviscid Burgers equation.

2. Renewal theory for nested regenerative sets.

Consider the age processes $A^{(1)}\geq \cdots \geq A^{(n)}$ associated to a nested sequence $\r^{(1)}\subseteq\cdots\subseteq \r^{(n)}$ of regenerative sets. We present limit theorems in distribution for $\left(A^{(1)}_t, \cdots, A^{(n)}_t\right)$ and for $\left({1\over t}A^{(1)}_t, \cdots, {1\over t}A^{(n)}_t\right)$ which correspond to multivariate versions of the renewal theorem and of the Dynkin-Lamperti theorem, respectively. The latter has interesting applications to the concave minorant of a Cauchy process.

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PROBABILISTIC ASPECTS OF FINANCE, I and II

In the first of these talks, we shall present the standard model of Samuelson-Merton and Black-Scholes for a financial market consisting of one risk-free asset (bond) and several risky assets (stocks). Within its context, we shall introduce and discuss notions of portfolio and consumption strategies, arbitrage and its absence, equivalent martingale measures, contingent claims, complete and incomplete markets. We shall broach the problem of hedging contingent claims, such as options, and show how it can be solved in the context of complete markets using the methodologies of Stochastic Analysis and of linear parabolic Partial Differential Equations. Along with other examples of this methodology, we shall present the famous Black-Scholes (1973) formula for the price of a European call-option. Finally, we shall indicate briefly how the methodology has to be modified when dealing with American options, as well with incomplete/constrained markets, different interest rates for borrowing and lending, transactions costs, etc. All these developments are now well documented in the literature; see, for instance, the recent monograph by Karatzas and Shreve (1998).

In the second lecture we shall try to examine what happens when some of the assumptions of the standard model are not satisfied. In particular, we shall assume that we start with funds insufficient for hedging without risk, and that there is uncertainty about the stock appreciation rates. This uncertainty will be modelled in Bayesian fashion, i.e., by positing that the appreciation rates are unobservable random variables, independent of the primary sources of randomness in the market and with known probability distributions. Based on an observation-flow and past-and-present stock-prices, what is then the portfolio that maximizes the probability of perfect hedge? We shall provide an explicit solution to this adaptive stochastic control problem, using techniques of filtering, the Neyman-Pearson lemma, and dynamic programming methodologies leading to a parabolic version of the famous Monge-Ampere equation. These results are taken from Karatzas (1997) and extend those of Kulldorff (1993) and Heath. Similar techniques can be used to solve more general utility maximization problems in a similar context, as in Karatzas and Zhao (1998).

References

1. BLACK, F. and SCHOLES, M. (1973) The pricing of options and corporate liabilities. J. Political Econ. 81, 637--659.
2. KARATZAS, I. (1997) Adaptive control of a diffusion to a goal, and a parabolic Monge-Ampere-type equation.
3. KARATZAS, I. and SHREVE, S. E. (1998) Methods of Mathematical Finance. Springer-Verlag, New York.
4. KARATZAS, I. and ZHAO, X. (1998) Bayesian adaptive portfolio optimization. Preprint, Columbia University.
5. KULLDORFF, M. (1993) Optimal control of favorable games with a time-limit.SIAM J. Control and Optimization 31, 52--69.