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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 non-addictive habits. Here consumption is required to be non-negative at all times but is allowed to fall below a standard-of-living 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 forward-backward 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 self-intersection local times. Our approach relies on a Feymann-Kac-type large deviation result for Brownian occupation time, certain localization techniques from Donsker-Varadhan (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 most-studied 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 multi-dimensional 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 two-component 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 PDE-s are derived as "universal laws" driving propagation of small perturbations of equilibria. Joint work with Benedek Valko.