Columbia Probability Seminar

The Probability Seminar takes place Fridays from 12:00 to 1:00 pm. Unless indicated otherwise, talks in Fall 2008 are held in Room 520, in the math Department. an update on the seminar schedule: The seminar is run by Julien Dubedat, Ioannis Karatzas, and Emmanuel Schertzer.

Fall Semester 2009

Minerva Research Foundation Lectures-
    Sept 9- Sept 30.
Lectures every Wednesdays 10am-noon (Math 622) and Fridays 10am-noon (Math 507)

    Jean Bertoin (Universite Paris VI)

    Title: Exchangeable Coalescents.

Exchangeable coalescents form an important class of stochastic processes
with values in spaces of partitions that were introduced by Pitman,
Möhle and Sagitov. They appear in the study of the genealogy of
certain random population models. The best known example is the
celebrated coalescent of
Kingman which is related to the model of Wright-Fisher.

Our purpose in this series of lectures is to present some of their fundamental
aspects (Poissonian construction, look-down construction,
characterization, relation with stochastic flows,
duality with generalized Fleming-Viot processes, ...) and to discuss
properties of key examples (Kingman coalescent,
Bolthausen-Sznitman coalescent, Beta-coalescents).

Plan:

1. Random partitions

2. Kingman's coalescent

3. Exchangeable coalescents

4. Flows of bridges

5. Bolthausen-Sznitman coalescent

6. Beta-coalescents

Lecture Notes:
   ../Desktop/Minerva.pdf

Friday, 18th September
    Xue-Mei Li (Courant/Warwick)

Title: Examples of intertwinned diffusion operators and filtering

Abstract: Consider a map f from N to M and a pair of diffusion operators intertwined by f. They induce a splitting of the operator on N and this is used to interpret the geometry of filtering. We explain this by examples. In particular we look at a family of diffusion operators on the Heisenberg group.

Friday, 25th September.
    Ramon Van Handel (Princeton ORFE)

Title: An asymptotic theory for nonlinear filters

Abstract: The theory of nonlinear filtering is concerned with the conditional law of a Markov process (the signal) given a sequence of noisy observations. A collection of recent results give a rather general characterization of the long time asymptotic properties of nonlinear filters. On the one hand, ergodicity of the filter is inherited from ergodicity of the signal, largely resolving a long-standing gap in a classic paper by H. Kunita (1971). On the other hand, certain structural properties of filtering models, reminiscent of fundamental ideas in systems theory, ensure stability of the filter in the absence of ergodicity. Key to the proofs are various martingale arguments and a new ergodic theorem for Markov chains in random environments. If time permits, I will briefly outline applications to the convergence of particle filtering algorithms and to the asymptotics of the stationary estimation error.

Friday, Oct. 2
    no seminar

Monday Oct. 5
4:00-5:00pm, Math 520
Gordan Zitkovic, University of Texas at Austin

Title : Incomplete-market equilibria with exponential utilities

Abstract: In addition to existence, the excess-demand approach allows us to establish uniqueness and provide efficient computational algorithms for incomplete-market stochastic financial equilibria when agents exhibit constant absolute risk aversion. An overview of recent results (including those jointly obtained with M. Anthropelos and with Y. Zhao) will be given.

5:00-6:00pm, Math 520
Peter Bank, TU Berlin

Title: A large investor trading at market indifference prices

We consider a financial market where a finite number of market makers quote prices for a given security. The market makers rehedge the acquired positions among themselves so as to keep the allocation of risk in a Pareto-optimum. We show how this amounts to a dynamic model for trades with permanent price impact and discuss how the implied continuous-time strategies relate to the Black-Scholes hedges. (Joint work with Dmitry Kramkov).

Friday, Oct. 9
    Rama Cont (Columbia IEOR)

Title: Mimicking the marginal distributions of a semimartingale

Abstract: We give conditions under which the flow of marginal distributions of
a discontinuous semimartingale X can be matched by a Markov
process whose infinitesimal generator can be expressed in terms of
the local characteristics of X, generalizing a result of Gyongy (1986) to the
discontinuous case. This result allows to derive a partial
integro-differential equation for the one-dimensional distribution of
discontinuous semimartingales, extending the Kolmogorov forward
equation to a non-Markovian setting.

Joint work with Amel BENTATA (Universite de Paris VI).

Friday 9 Oct, 2pm
room 1025 in SSW
Aleksandar Mijatovic (Imperial College)

Title: On the martingale property of certain local martingales

Abstract: The stochastic exponential $Z$ of a continuous local
martingale $M$ is itself a continuous local martingale. In this
talk we describe a necessary and sufficient condition for the
process $Z$ to be a true martingale in the case where
$M_t=\int_0^t b(Y_u)dW_u$ and $Y$ is a one-dimensional diffusion
driven by a Brownian motion $W$. Furthermore, we give a necessary
and sufficient condition for $Z$ to be a uniformly integrable
martingale in the same setting. These conditions are deterministic
and expressed only in terms of the function $b$ and the drift and
diffusion coefficients of $Y$. As an application we provide a
deterministic criterion for the absence of bubbles in a one-dimensional
setting. This is joint work with Misha Urusov.

Friday, Oct. 16.
Gennady Samorodnitsky (Cornell ORIE)

Title: Excursion sets over high levels of non-Gaussian infinitely divisible random fields: extreme values, algebra, and geometry

Abstract: we discuss new directions in extreme value theory of random fields, and possible connections with algebra and geometry.

Friday, Oct. 23
Iosif Pinelis (Michigan Technological University)

Title: On the Bennett-Hoeffding inequality

Abstract: The well-known Bennett-Hoeffding bound for sums of
independent random variables is refined, by taking into account
truncated third moments, and at that also improved by using, instead
of the class of all increasing exponential functions, the much larger
class of all generalized moment functions f such that f and f" are
increasing and convex. It is shown that the resulting bounds have
certain optimality properties. Comparisons with related known bounds
are given. The results can be extended in a standard manner to (the
maximal functions of) (super)martingales.

Friday, Oct. 30
Mihyun Kang, Technische University Berlin

Title: Critical behaviour of random planar graphs

Abstract: Since the seminal work of Erdos and Renyi the phase transition of the
largest components in random graphs became one of the central topics in
random graph theory and discrete probability theory. In this talk we
discuss the phase transition of the largest components in a uniform random
planar graph. (Joint work with Tomasz Luczak.)

Friday, Nov. 6.
Sourav Chatterjee (Courant/Berkeley Stats)

Title: Superconcentration

Abstract: We introduce the term `superconcentration' to describe the phenomenon
when a function of a Gaussian random field exhibits a far stronger
concentration than predicted by classical concentration of measure. We
show that when superconcentration happens, the field becomes chaotic
under small perturbations and a `multiple valley picture' emerges.
Conversely, chaos implies superconcentration. While a few notable
examples of superconcentrated functions already exist, e.g. the
largest eigenvalue of a GUE matrix, we show that the phenomenon is
widespread in physical models; for example, superconcentration is
present in the Sherrington-Kirkpatrick model of spin glasses, directed
polymers in random environment, the Gaussian free field and the
Kauffman-Levin model of evolutionary biology. As a consequence we
resolve the long-standing physics conjectures of disorder-chaos and
multiple valleys in the Sherrington-Kirkpatrick model, which is one of
the focal points of this talk.

Friday, Nov. 13.
Yuri Bakhtin (Georgia Tech)

Title: SPDE approximation for random trees.

Abstract: I will present a point of view at large random trees.
I will start with a biological motivation and consider random trees
under Gibbs distribution with nearest neighbour interaction. I will describe
their asymptotics as the tree size grows to infinity. The "thermodynamic limit"
is a limiting infinite random tree that can be viewed as branching process
conditioned on nonextinction. I will introduce a natural way of
embedding the infinite
trees in the plane and study the details of the tree geometry by
tracing progenies of subpopulations. Under an appropriate scaling the
limiting continuum random tree can be described as a solution of an
SPDE w.r.t. a Brownian sheet.

Nov 19th and 20th.

Northeast Probability seminar at Columbia University

Rick Kenyon, Brown University "The vector-bundle Laplacian on a graph"

Claudia Neuhauser University of Minnesota "When Genealogy Meets Ecology"

Giovanni Peccati Université Paris Ouest "Universality of the Gaussian Wiener chaos"

Craig Tracy University of California, Davis "Asymmetric Simple Exclusion Process: Integrable Structure and Limit Theorems"

For more information:

http://www.math.csi.cuny.edu/probability/NortheastProbabilitySeminar/

Friday, Dec. 4
Jan Hannig (UNC Stats)

Title: Continuum modeling of large networks

Abstract: This paper is concerned with approximation of extremely large networks
using time-dependent partial differential equations. In particular, we
present conditions under which a sequence of parameterized Markov
chains converges to its continuum limit, which is the solution to a
PDE. We then apply this result to a communication model of network
traffic.
The practical thesis is that global characteristics of sufficiently
large networks can be captured by continuum modeling. Continuum
modeling provides a powerful way to deal with the number of components in large networks,
and opens up the use of highly sophisticated mathematical tools such as adaptive finite element methods.
This, in turn, makes it possible to carry out—with reasonable computational burden even for very large
systems—network performance eval- uation and prototyping, network design, systematic parameter studies,
and optimization of network characteristics. This is a joint work with E. Chong and D. Estep, both at Colorado State Univeristy.