Location: Doheny Beach Room B (next to the West Courtyard, room A134B on map) at the UCI Student Center (building 113 on the map).
Title: On the Positive Recurrence of Semimartingale Reflecting Brownian Motions in Three (and Higher) Dimensions*
Let Z be an n-dimensional Brownian motion confined to the non-negative orthant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for multi-station stochastic processing networks. For dimension n = 2, a simple condition is known to be necessary and sufficient for positive recurrence of Z. The obvious analog of that condition is necessary but not sufficient in three and higher dimensions, where fundamentally new phenomena arise.
Building on prior work by Bernard and El Kharroubi (1991) and El Kharroubi et al. (2000, 2002), we provide necessary and sufficient conditions for positive recurrence in dimension n = 3. In this context we find that the fluid-stability criterion of Dupuis and Williams (1994) is not only necessary for positive recurrence but also sufficient; that is, in three dimensions Z is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. A recently discovered example strongly suggests that this equivalence fails in four and higher dimensions.
* Joint work with Maury Bramson, Dick Cottle and Jim Dai
Title: How to match two pictures
This problem can be transferred into the distance of two 2-dimensional random vectors in the following way. Let us just consider the black-white photo for simplicity. The photo is composed of dots. One can associate to each dot a probability mass according to its darkness. Thus one photo gives a two-dimensional probability distribution. The best match means the shortest distance between two distributions.
Title: Probabilistic Techniques in Mathematical Phylogenetics
I will discuss recent results on connections between the reconstruction problem on spin systems and two important problems in evolutionary biology: the inference of ancestral molecular sequences and the reconstruction of large phylogenies.
Title: Detection of an Abnormal Cluster in a Network
We consider the model problem of detecting whether or not in a given sensor network, there is a cluster of sensors which exhibit an ``unusual behavior''. Formally, suppose we are given a graph and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses. Under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster (connect component) of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. The cluster is unknown but restricted to belong to a class of interest.
We also address surveillance settings where each sensor in the network transmits information over time. The resulting model is similar, now with a time series is attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal; and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The class of growth models we use to represent
emerging cluster is quite general, and includes threshold growth automata used to model epidemics.
In both settings, we study minimax detection rates for a variety of cluster classes and show that the scan statistic, by far the most popular method in practice, is near-optimal in a wide array of situations.
Joint work with Emmanuel Candes (Stanford) and Arnaud Durand (Universite Paris XI).
Title: Parameter Estimation in Stochastic Equations That are Second-Order in Time
Consider the following problems: 1. Given an undamped harmonic oscillator driven by additive Gaussian white noise, estimate oscillator's frequency from the observations of the oscillations;
2. Given an undamped wave equation driven by additive space-time white noise, estimate the propagation speed from the observations of the solution.
It turns out that the the first (one-dimensional) problem is harder to analyze than the second (infinite-dimensional) problem. The objective of the talk is to study the asymptotic properties of the maximum likelihood estimator in both problems and to discuss various generalizations of both problems.
This is based on joint work with Ning Lin and Wei Liu.
Title: A Simulation Approach to Optimal Stopping under Partial Information
I will discuss the numerical solution of nonlinear partially observed optimal stopping problems. The system state is taken to be a multi-dimensional diffusion and drives the drift of the observation process, which is another multi-dimensional diffusion with correlated noise. Such models where the controller is not fully aware of her environment are of interest in applied probability and financial mathematics. We propose a new approximate numerical algorithm based on the particle filtering and regression Monte Carlo methods. Our method is entirely simulation-based, maintains a continuous state-space and yields an integrated approach to the filtering and control sub-problems. We carry out the error analysis of our scheme and illustrate with several computational examples. An extension to discretely observed stochastic volatility models will also be considered.
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This year's organizers are R. Feldman (feldman[at]pstat.ucsb.edu), J-P. Fouque (fouque[at]pstat.ucsb.edu) and M. Ludkovski (ludkovski[at]pstat.ucsb.edu).