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INSTITUTE OF MATHEMATICAL STATISTICS
DEPARTMENT OF MATHEMATICS (UW)
UNIVERSITY OF WASHINGTON




Abstracts of invited talks
 Ioannis Karatzas (Columbia University)
Some Stochastic Optimization Problems in
Mathematical Finance
Methods from Stochastic Analysis, coupled with
convex duality techniques, have been very successful in
tackling optimization problems that arise in mathematical
economics and finance. We survey some 'classical' problems
of this type, then formulate a model of preferences with
nonaddictive habits. Here consumption is required to be
nonnegative at all times but is allowed to fall below a
standardofliving index that aggregates past consumption.
We describe the optimal strategies for this problem, and
show that the consumption constraint is binding up to a
suitable stopping time, after which it becomes slack.
Backwards stochastic equations play a crucial role in
establishing this result. We describe variants of this problem
that remain open; these suggest some very interesting questions
in the analysis of forwardbackward stochastic equations
of a novel type.
 Wenbo Li (University of Delaware)
Large Deviations for Intersection Local Times
There are various important reasons for studying large deviations
of intersection local times.
In this talk, we will present new results on
intersection local times for independent Brownian motions
and symmetric random walks. We will also discuss
selfintersection local times.
Our approach relies on a FeymannKactype large deviation
result for Brownian occupation time,
certain localization techniques from DonskerVaradhan (1975) and
Mansmann (1991), and some general methods developed along the lines
of probability in Banach space. Our treatment of the discrete case
involves rescaling, spectral representation and
invariance principle.
This talk is based on a joint work with Xia Chen.
 Russ Lyons (Indiana University and Georgia Institute of Technology)
Stationary Determinantal Processes (Fermionic Lattice Gases)
Eigenvalues of random matrices arise in various areas of physics and
mathematics. The moststudied such probability measures have a
determinantal form. Several people have studied other specific
determinantal processes, as well as a general theory. We shall discuss the
general theory of stationary random fields on integer lattices that are
defined via minors of multidimensional Toeplitz matrices. Explicit
examples include combinatorial models, finitely dependent processes, and
renewal processes in one dimension. Among the interesting properties of
these processes, we focus mainly on whether they have a phase transition
analogous to that which occurs in statistical mechanics. We describe
necessary and sufficient conditions for the existence of such a phase
transition and give several examples to illustrate the theorem. This is
joint work with Jeff Steif.
 Carl Mueller (University of Rochester)
Some Wave Equations with Noise
We use a certain path representation to get information
about stochastic wave equations with noise. The noise is
colored in space, but white in time. Our first result is
about a process for which the solution is a Schwartz
distribution which has no density. This is one of the few
known examples of an equation with this kind of solution,
other than linear equations with additive noise. We regard
the latter equations as trivial. Our second equation has
smoother solutions. We use the path representation to get
information about the moment Lyapunov exponents of the
solutions, in the sense of Molchanov.
 Balint Toth (Budapest University of Technology and Economics)
Between Equilibrium Fluctuations and Eulerian Scaling
We derive a special class of twocomponent systems of PDEs (hyperbolic
conservation laws), as hydrodynamic limit for interacting particle
systems. The scaling regime interpolates between the Eulerian scaling
and the scaling of equilibrium fluctuations. The PDEs are derived as
"universal laws" driving propagation of small perturbations of
equilibria.
Joint work with Benedek Valko.
