Columbia Probability Seminar
The seminar covers a wide range of topics in
pure and applied probability. The seminar is organized jointly by the Mathematics and Statistics departments and is
run by Julien
Dubedat, Clement
Hongler, Fredrik
Johansson Viklund, Ioannis
Karatzas, and Philip Protter.
E-mail the organizers at probability_seminar -- at --
math.columbia.edu.
In the spring, the seminar usually takes place in Room
903, Statistics Department, School of Social Work Building (1255
Amsterdam Avenue) on Fridays at 12noon-1pm.
Directions
to the Mathematics Department.
Join the Columbia
Probability Seminar Mailing List.
Spring Semester 2012
-
Minerva Lectures, Fridays 3.15pm in Math 520
- Grigori Olshanski (Institute for Information
Transmission Problems of the Russian Academy of Sciences)
Determinantal Processes and Related Topics
More
information
-
Friday, January 13, 2012
- Pietro Siorpaes (Vienna)
A simple proof of Bichteler–Dellacherie–Mokobodzki Theorem
Abstract: We give a simple and quite elementary proof of the celebrated
Bichteler–Dellacherie–Mokobodzki Theorem, which states that a process
is a good integrator if and only if it is a semi-martingale.
This talk is based on a joint work with Mathias Beiglbock
Special Seminar: Friday, January 27, 2012, 11am-12noon, Stat 903.
- Alexander Drewitz (ETH)
- Effective polynomial ballisticity conditions for higher-dimensional RWRE
We consider random walk in random environment (RWRE) in dimension larger
than or equal to four.
A couple of years ago Sznitman introduced a certain class of conditions
denoted $(T)_\gamma,$ $\gamma \in (0,1],$ where the parameter $\gamma$
governs the (stretched)
exponential decay of certain exit probabilities.
The importance of these conditions stems, among others, from the fact that
they imply
ballistic and diffusive behaviour
of the RWRE. They are known equivalent for parameters $\gamma \in (0, 1).$
We do propose here a new class of conditions $(\mathfrak P)_M$ which are
similar in nature to $(T)_\gammma$ but only require
polynomial decay of the corresponding exit probabilities.
Our main result states that these newly introduced conditions already imply
$(T)_\gamma,$ $\gamma \in (0,1),$
and hence all the results deduced from that.
(Work in progress with Noam Berger and Alejandro Ram\'{i}rez)
-
Friday, January 27, 2012
- Pierre Nolin (ETH)
- Monochromatic arm exponents for 2D percolation
Abstract: We investigate the so-called monochromatic arm exponents for
critical
percolation in two dimensions. These exponents describe the probability
to
observe a given number of disjoint paths of the same color across annuli
of large modulus.
After showing that they exist, we investigate how they are related to
the
(now well-understood) polychromatic exponents (describing probabilites
when arms of both colors are present). In particular, an interesting
correlation inequality shows up.
This is a joint work with Vincent Beffara.
-
Friday, February 3, 2012
- Tom LaGatta (Courant)
Geodesics of Random Riemannian Metrics
Abstract: Geodesics are local length-minimizing paths in Riemannian geometry,
but it is an interesting question under what conditions they globally
minimize length. The Cartan-Hadamard theorem, for example, says that
under non-positive curvature assumptions on one's space, geodesics are
globally minimizing. In the context of a random metric, one expects a
presence of positive curvature, and random geodesics should
occasionally run into these positive patches. For perturbations of the
Euclidean plane, we have used the point-of-view of the particle
technique to show that this is indeed the case, and that a geodesic
with randomly selected starting conditions is not minimizing (almost
surely). This is joint work with Janek Wehr.
-
Friday, February 10, 2012
-
Friday, February 17, 2012
- Ramon van Handel (Princeton)
The universal Glivenko-Cantelli property
Abstract:
Uniform laws of large numbers are basic tools in probability and
statistics. It is well understood when the law of large numbers
holds uniformly for a given class of functions and underlying
distribution (the Glivenko-Cantelli property). Uniformity in
other probabilistic limit theorems is much less well understood.
I will discuss a somewhat surprising result which states that if a
class of functions is a Glivenko-Cantelli class for every underlying
distribution (the universal Glivenko-Cantelli property), then uniform
convergence in the law of large numbers, the ergodic theorem, and the
reverse martingale convergence theorem are all equivalent.
Moreover, this property can be characterized in terms of certain
equivalent combinatorial and geometric conditions.
-
Special Seminar: Friday, February 24, 2012, 11am-12noon, Stat 903.
- Alexander G. Tartakovsky (USC)
- Monotone Properties of the First Exit Time of a Markov Process Started at a
Quasistationary Distribution
Quasistationary distributions come up naturally in the context of first exit times of
Markov processes, and of special interest in certain applications is the case of a
nonnegative Markov process, where the first time T(A) that the process exceeds
a fixed level A signals that some action is to be taken. The quasistationary
distribution Q_A(x) is the distribution of the state of the process if a long time
has passed and yet no crossover of A has occurred. Various topics pertaining
to quasistationary distributions are existence, calculation, simulation, etc. In this
talk, I will discuss a monotonicity property of quasistationary distribution Q_A
and apply it to the behavior of the expected time of the first exceedance of A by
a Markov process started at a random point sampled from Q_A, as a function of
A. Specifically, I will provide conditions under which Q_A is nonincreasing and
the corresponding stopping time T(A) is stochastically nondecreasing in A. While
this is of considerable interest on its own merit, our interest stems from certain
aspects in changepoint detection theory where it is of importance to establish
monotonicity properties of the mean time to false alarm (as a function of the
detection threshold A) of (almost optimal) detection schemes that start off at a
random point that has quasistationary distribution.
This is a joint work with Moshe Pollak of the Hebrew University of Jerusalem.
Friday, February 24, 2012
- Chris Burdzy (University of Washington)
Shy couplings, lion and man, and rubber band domains
Abstract: The talk will be an exciting mixture of probability,
metric geometry, differential games and algebraic geometry.
(OK, "rubber band domains", in fact, are not a topic in
algebraic geometry but I thought that "algebraic
geometry" would sound sophisticated.)
Shy couplings are pairs of processes that do not come
close to each other (ever!). Every well educated
person should know whether the lion can catch the man.
The most surprising claims will be: (i) Pursuit problems
for Brownian particles (which have "infinite" velocity)
are related to pursuit problems with bounded velocities; and
(ii) It appears that nobody has thought about "rubber band
domains" so far, although they are a very natural concept.
Joint work with M. Bramson and W. Kendall.
-
Friday, March 2, 2012
- Columbia-Princeton Probability Day 2012 (At Columbia
University)
Please visit the conference website for more information.
-
Friday March 9, 2012
- Spring Recess: No Seminar
Friday March 16, 2012
- Spring Recess: No Seminar
Friday, March 23, 2012
- Per Mykland (Chicago)
Econometrics of High Frequency Data: Background and New Developments
Abstract:
Recent years have seen a rapid growth in high frequency financial data,
This has opened the possibility of accurately determining volatility in
small time periods, such as one day, or even less. We introduce the
types of data, and discuss what quantities can reasonably be estimated
in this setting, such as skewness, and high frequency regression
parameters. The talk provides background for this kind of high
frequency inference, and then discusses challenges and recent
innovations in the area. The talk is particularly focused on issues
involving endogenous times, market microstructure, and local likelihood.
-
Friday, March 30, 2012
- Ed Waymire (Courant)
Dispersion in the Presence of Interfacial Discontinuities
Abstract: This talk will focus on questions arising in the geophysical and
biological sciences concerning dispersion in highly heterogeneous
environments, as characterized by abrupt changes (discontinuities)
in the diffusion coefficient. Some specific laboratory and field experiments
involving breakthrough curves (first passage times), occupation times,
and local times will be addressed within a probabilistic framework based
on variants of the Ito-McKean-Feller classic skew Brownian motion.
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Friday, April 6, 2012
- Kavita Ramanan (Brown)
CANCELLED
Abstract: -
Friday, April 13, 2012 [Note! Two consecutive seminars]
-
-
Johannes Muhle-Karbe (ETH) [12noon SSW 903]
-
On the existence of shadow prices
Abstract: For utility maximization problems under proportional transaction
costs, it has been observed that the original market with transaction costs
can sometimes be replaced by a frictionless "shadow market" that yields the
same optimal strategy and utility. However, the question of whether or not
this indeed holds in generality has remained elusive so far. In this paper
we present a counterexample which shows that shadow prices may fail to
exist. On the other hand, we prove that short selling constraints are a
sufficient condition to warrant their existence, even in very general
multi-currency market models with possibly discontinuous bid-ask-spreads.
This is joint work with Giuseppe Benedetti, Luciano Campi, and Jan Kallsen.
-
Friday, April 20, 2012
- Tom Alberts (CalTech)
Diffusions of Multiplicative Cascades
Abstract: A multiplicative cascade is a randomization of any
measure on the unit interval, constructed from an iid collection of
random variables indexed by the dyadic intervals. Given an arbitrary
initial measure I will describe a method for constructing a continuous
time, measure valued process whose value at each time is a cascade of
the initial one. The process also has the Markov property, namely at
any given time it is a cascade of the process at any earlier time. It
has the further advantage of being a martingale and, under certain
extra conditions, it is also continuous. I will discuss applications of
this process to models of tree polymers and one-dimensional random
geometry.
Joint work with Ben Rifkind (University of Toronto).
-
Friday, April 27, 2012
- Ori Gurel-Gurevich (UBC)
Cancelled
Fall Semester 2011
-
Minerva Lectures, September 7-15, 2011
- Denis Talay (INRIA)
Model Risk: Modeling, Analysis, Control and Numerics
More
information
-
Minerva Lectures, September 22-October 27, 2011
- Vladas Sidoravicius (IMPA)
Phase transitions in stochastic processes with long-range
interactions:
a multiscale analysis approach
More
information
-
Friday, September 30, 2011
- Craig Tracy (UC Davis)
The asymmetric simple exclusion process: Recent progress and
open problems
Abstract:
In the asymmetric simple exclusion process (ASEP), particles are at
integer sites on the line. Each particle waits exponential time, and
then with probability p (q=1-p) moves one step to the right (left) if
the site is unoccupied otherwise it does not move. In this lecture we
review the Bethe Ansatz solution of the transition probability and how
this leads to a formula for the current fluctuations in the case of
step initial condition. The connection with the Kardar-Parisi-Zhang
equation will be briefly described. Some open problems associated
with other initial conditions will be presented.
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Friday, October 7, 2011
- Jianfeng Zhang (USC)
On Viscosity Solutions of Path Dependent PDEs
Abstract:
In this talk we propose a notion of viscosity solutions for semi-linear
parabolic Path-dependent PDEs. This can also be viewed as viscosity
solutions of non-Markovian Backward SDEs, and thus extends the well
known nonlinear Feynman-Kac formula to non-Markovian case. We shall
prove the existence, uniqueness, stability, and comparison principle
for the viscosity solutions. The key ingredient of our approach is a
functional Ito's calculus recently introduced by Dupire and further
developed by Cont and Fournie. The talk is based on a joint work with
Ibrahim Ekren (USC), Christian Keller (USC) and Nizar Touzi (Ecole
Polytechnique).
-
Friday, October 14, 2011
- Amir Dembo (Stanford)
Factor models on locally tree-like graphs
Consider factor (graphical) models on sparse graph sequences that
converge locally to a random tree T. Using a novel interpolation scheme
we prove existence of limiting free energy density under uniqueness of
relevant Gibbs measures for the factor model on T. We demonstrate this
for Potts and independent sets models and further characterize this
limit via large-deviations type minimization problem and provide an
explicit formula for its solution, as the Bethe free energy for a
suitable fixed point of the belief propagation recursions on T (thereby
rigorously generalize heuristic calculations by statistical physicists
using ``replica'' or ``cavity'' methods).
This talk is based on a joint work with Andrea Montanari and Nike Sun.
-
Friday, October 21, 2011
- Alexander Fribergh (Courant)
Phase transition for the speed of a biased random walk on a
supercritical percolation cluster.
Abstract: The effect of a bias on a random walk in
$\mathbb{Z}^d$ is obviously to speed up the walk. In a disordered
medium, in particular on a percolation cluster, the situation is quite
different. Indeed, the bias can turn specific areas of the environment
into traps leading to a seemingly surprising phenomenon: for strong
external fields the random walker asymptotically moves at zero speed.
We will explain this phenomenon and present the key elements that led
to the proof of the following conjecture: as the bias increases there
exists a phase transition from positive to zero speed.
This work is joint with Alan Hammond.
-
Friday, October 28, 2011
- Jonathon Peterson (Purdue)
Weak quenched limiting distributions for one-dimensional
random walks in random environments.
Abstract:
Random walks in random environments (RWRE) are a very simple model for
random motion in a non-homogeneous environment. Despite the
similarities between RWRE and classical random walks, RWRE can exhibit
behavior that is very different from that of classical random walks.
For instance, for certain distributions on the environment, the
averaged limiting distribution of the random walk is non-Gaussian, and
there does not exist an almost sure quenched limiting distribution. In
this talk I will discuss recent work with Gennady Samorodnitsky which
helps explain both the averaged limiting distribution and the
non-existence of quenched limiting distributions. The main result is a
``weak'' quenched limiting distribution in the sense that the quenched
distribution of the random walk, viewed as a random probability
measure, converges weakly to a probability measure that is a random
linear combination of exponential random variables.
-
Special Seminar: Friday, October
28, 2011 ( Math 528 3pm--4pm)
- Yuri Kifer (Hebrew University)
A Zoo of Nonconventional Limit Problems
Abstract (pdf).
-
Special Seminar: Monday, October
31, 2011 ( Stat SSW 1025 12noon--1pm)
- Walter Schachermayer (University of Vienna)
Doob's maximal inequality, the Bichteler-Dellacherie
theorem, and Arbitrage
Abstract: Some ideas commonly used in Mathematical Finance,
in particular the arguments of ``no arbitrage'' and ``super-hedging''
type, allow for a fresh look on some classical theorems of stochastic
analysis.
-
Friday, November 4, 2011
- Marcel Nutz (Columbia)
A Quasi-Sure Approach to the Control of Non-Markovian
Stochastic
Differential Equations
Abstract: We study stochastic differential equations (SDEs) whose drift
and diffusion
coefficients are path-dependent and controlled. We construct a value
process
on the canonical path space, considered simultaneously under a family
of
singular measures, rather than the usual family of processes indexed by
the
controls. This value process is characterized by a second order
backward
SDE, which can be seen as a non-Markovian analogue of the
Hamilton-Jacobi-Bellman partial differential equation. Moreover, our
value
process yields a generalization of the G-expectation to the context of
SDEs.
[arXiv:1106.3273]
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Friday, November 11, 2011
- Soumik Pal (Washington-Seattle)
Random matrix theory for sparse random regular graphs
Abstract:
A random regular graph is a probability distribution on the set of
labeled undirected graphs where every vertex has the same degree. A
sparse regular graph is one for which the degree is far smaller than
the number of vertices (order). The adjacency matrix for such a graph
is an interesting model of random matrices whose eigenvalues have been
the focus of several recent mathematical and simulation studies. From
the point of view of Random Matrix Theory (RMT) these matrices test the
limit of results universal for the classical RMT models. We show that
when the degree is kept fixed and the order grows these matrices
produce results not shared by the classical models. The "Wignerian"
proofs based on the concentration of measure arguments no longer work
due to sparsity. However, the two models seem to converge when the
degree grows (ever so slowly) with the order. These recent results are
obtained via a combinatorial approach. The broad idea is that of local
tree approximation formalized under the rubric of local weak
convergence independently by Aldous and Benjamini-Schramm. We discuss
how (i) the tree determines the limiting global spectral distribution;
(ii) the rate of convergence to the tree determines spectral
convergence in "short scales"; (iii) the distribution of small cycles
that break the tree determines (possibly non-Gaussian) fluctuation of
linear eigenvalue statistics; and (iv) vertices not near these cycles
cannot have an eigenvector localized at them. A bigger picture for
other sparse random graph models will emerge naturally. This talk is
based on joint works with Ioana Dumitriu, Tobias Johnson, and Elliot
Paquette.
-
Thursday--Friday, November 17--18, 2011
- Northeast Probability Seminar (Courant Institute)
See here
for more information.
-
Special Seminar: Monday, November
21, 2011 (Math 622 2pm-3pm)
- Hugo Duminil-Copin (Geneva)
Near-critical random-cluster model: beyond the pivotal sites
phenomenon
Abstract: In 1982, Kesten proved that the critical value of bond
percolation on
the square lattice equals 1/2. The proof harnessed a clever estimation
of the number of pivotal sites for crossing events. It enabled the
study of the probability of these crossing events with respect to the
edge-parameter p. Five years later, Kesten rigorously proved a
relation between the probability of being pivotal and the so-called
correlation length of percolation. We shall discuss similar questions
in random-cluster models on the square lattice.
joint works with V. Beffara, and with C. Garban and G. Pete
-
Special Seminar: Monday, November
21, 2011 (Stat 903 4pm-5pm)
- Jean Jacod (Universite Paris 6)
Statistical treatment of microstructure noise
Abstract: The presence of microstructure noise is a
considerable nuisance
in high-frequency statistics, Several methods allow us to de-noise the
data
in order to estimate the integrated volatility and other quantities of
interest, but only when the noise is an additive white noise, or at
least
presents some kind of (rather strong) independence between successive
observation times. In this talk we will present some (very preliminary)
ideas for dealing with more general forms of noise.
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Friday, November 25, 2011
- Thanksgiving -- No Seminar
-
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Friday, December 2, 2011
- Jaksa Cvitanic (CalTech)
Equilibrium in Financial Markets with Heterogeneous Agents
In this talk we present results on equilibrium in a complete,
continuous time financial market when investors have three possible
sources of heterogeneity. Investors may differ in their beliefs, in
their level of risk aversion and in their time preference rate. We
study the impact of investors heterogeneity on the consumption shares,
the market
price of risk, the risk free rate, the bond prices at different
maturities,
the stock price and volatility, as well as the stock's cumulative
returns, and
optimal portfolio strategies. We relate the heterogeneous economy with
the
family of associated homogeneous economies with only one class of
investors,
and we study asymptotic survival of various agents in the economy.
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Friday, December 9, 2011
- Tom Kurtz (Wisconsin)
Particle representations for SPDEs and a model of asset
price determination
Abstract: Stochastic partial differential equations arise naturally as
limits of
finite systems of interacting particles. For a variety of purposes, it
is
useful to keep the particles in the limit obtaining an infinite
exchangeable
system of stochastic differential equations. The corresponding de
Finetti
measure then gives the solution of the SPDE. These representations
frequently simplify existence, uniqueness and convergence results. The
ideas will be illustrated by a model of asset prices set by an infinite
system of competing traders.
Spring Semester 2011
-
Friday, January 28, 2011
- Gregory F. Lawler (University of Chicago)
Defining the Schramm-Loewner evolution in multiply
connected domains with the Brownian loop measure
We define the Schramm-Loewner evolution in multiply
connected domains for kappa \leq 4 using the Brownian loop
measure. We show that in the
case of the annulus, this is the same as the measure obtained recently
by
Dapeng Zhan. We use the loop formation to give a different derivation
of the partial differential equation for the partition function in the
annulus.
-
February 1--8, 2011
- Johannes Muhle-Karbe (ETH Zurich)
Minerva Lectures: Shadow Prices in Portfolio Optimization
with Transaction Costs
Time and place:
Lecture I: 4--6 pm February 1 in Room 312 Math
Lecture II: 4--6 pm February 3 in Room 312 Math
Lecture III: 4--6 pm February 7 in Room 622 Math
Lecture IV: 4--6 pm February 8 in Room 312 Math.
Abstract and more information on the
Minerva Lectures webpage.
Friday, February 4, 2011
- Clément Hongler (Columbia University)
Ising model interfaces with free boundary conditions
We show conformal invariance of the Ising interfaces in
presence of free boundary conditions. In particular we prove the
conjecture of Bauer, Bernard and Houdayer about the scaling limit of
interfaces arising in a so-called dipolar setup and generalize a result
of Chelkak and Smirnov.
The limiting process is a Loewner chain guided by a drifted Brownian
motion, known as dipolar SLE or SLE(3,-3/2) in the literature. This
case is made harder by the absence of natural discrete holomorphic
martingales, requiring us to introduce "exotic" martingale observables.
The study of these observables is performed by Kramers-Wannier duality
and Edwards-Sokal coupling, and the computation of the scaling limit is
made by appealing to discrete complex analysis methods, to three
existing convergence results about discrete fermions, to the scaling
limit of critical Fortuin-Kasteleyn model interfaces and to the
introduction of Coulomb gas integrals.
Our result allows also to show conjectures by Langlands, Lewis and
Saint-Aubin about conformal invariance of crossing probabilities for
the Ising model.
Joint work with Kalle Kytölä.
-
Friday, February 11, 2011
- Philip Protter (Columbia University)
Recent Results on Filtration Shrinkage
The expansion of filtrations was a popular subject in the 1980s, as it
(among other things) allowed one to increase the space of possible
integrands for stochastic integration. However filtration shrinkage,
its mirror image sister, went largely unstudied, possibly due to the
lack of potential applications. Current progress in the theory of
credit risk, however, has made filtration shrinkage seem much more
important. We study the issue of the behavior of local martingales
under filtration shrinkage (this is based on joint work with Hans
Föllmer), and also the behavior of compensators of stopping
times (based on joint work with with Svante Janson and Sokhne M'Baye).
-
Friday, February 18, 2011
- Johannes Ruf (Columbia University)
Hedging under arbitrage
Explicit formulas for optimal trading strategies in terms of minimal
required initial capital are
derived to replicate a given terminal wealth in a continuous-time
Markovian context. To achieve
this goal this talk does not assume the existence of an equivalent
local martingale measure.
Instead a new measure is constructed under which the dynamics of the
stock price processes
simplify. It is shown that delta hedging does not depend on the ``no
free lunch with vanishing
risk'' assumption. However, in the case of arbitrage the problem of
finding an optimal strategy is
directly linked to the non-uniqueness of the partial differential
equation corresponding to the
Black-Scholes equation. The recently often discussed phenomenon of
``bubbles'' is a special case
of the setting in this talk.
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Friday, February 25, 2011
- Partha Dey (Courant Institute)
Central Limit Theorem for First-Passage Percolation along
thin cylinders
We
consider first-passage percolation on the graph Z x [-h_n, h_n]^{d-1}
where each edge has an i.i.d. non-negative weight. The passage time for
a path is defined as the sum of weights of all the edges in that path
and the first-passage time between two vertices is defined as the
minimum passage time over all paths joining the two vertices. We show
that the first-passage time between the origin and the vertex
(n,0,...,0) satisfies a Gaussian CLT as long as h_n <<
n^{1/(d+1)}. The proof is based on moment estimates, decomposition of
the first passage time as an approximate sum of independent random
variables and a renormalization argument. We conjecture that the CLT
holds upto $h_n=o(n^{2/3})$ for $d=2$ and provide some numerical and
heuristic support for that. Based on joint work with Sourav Chatterjee.
-
Friday, March 4, 2011
- Murad Taqqu (Boston University and Columbia University)
The Rosenblatt distribution
The Rosenblatt process is the next simplest self-similar
process which belongs to the Wiener Chaos after fractional
Brownian motion. While fractional Brownian motion is Gaussian,
the Rosenblatt process is not Gaussian. Thus obtaining the
Rosenblatt distribution which is the distribution of the
Rosenblatt process at time 1 is not easy. Our goal is to derive
various properties of the Rosenblatt distribution and to
describe a technique for computing it numerically with a high
degree of precision. This is joint work with Mark Veillette.
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Friday, March 11, 2011
- Spring Recess -- No Seminar
-
-
Friday, March 18, 2011
- Spring Recess -- No Seminar
-
-
Friday, March 25, 2011
- Nicolai Krylov (University of Minnesota)
Filtering partially observable diffusions up to the
exit time from a domain
We consider a two component diffusion
process with the second component treated as
the observations of the first one. The observations
are available only until the first exit time of the first
component from a fixed domain. We derive filtering
equations for the conditional distribution
of the first component before it hits the boundary
and give a formula for the conditional distribution
of the first component at the first time it hits the boundary.
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March 29--April 1
- Darrell Duffie (Stanford University)
Minerva Lectures: Dark Markets
Tuesday, March 29
10:30am - 12:00pm, SSW 1025
Wednesday, March 30
10:30am - 12:00pm, SSW 1025
Friday, April 1
12:00pm - 1:00pm, SSW 903
Abstracts and more information on the
Minerva Lectures webpage
-
Friday, April 1, 2011
Note: two seminars: D. Duffie 12noon-1pm,
M. Freidlin 230pm-330pm, both in Rm SSW 903.
- Darrell Duffie (Stanford University) [12noon-1pm in Rm SSW 903.]
Dark Markets
The financial crisis of 2007-2009
brought significant concerns and regulatory action regarding the role
of over-the-counter markets, particularly from the viewpoint of
financial instability. Over-the-counter markets for derivatives,
collateralized debt obligations, and repurchase agreements played
particularly important roles in the crisis and in subsequent
legislation in U.S. and Europe. This legislation has also focused on
increasing competition and transparency. The modeling of OTC markets,
however, is still relatively undeveloped in comparison to the available
research on central market mechanisms.
Rather than trading a centralized mechanism such as an
auction, specialist, or limit-order book, over-the-counter market
participants negotiate terms privately with other market participants,
often pair-wise. Over-the-counter investors may be largely unaware of
prices that are currently available elsewhere in the market, or of
recent transactions prices. In this sense, OTC markets are relatively
opaque; investors are somewhat in the dark about the most attractive
available terms and about who might offer them. These lectures
addresses how prices, asset allocations, and information transmission
in OTC markets are influenced by this form of opaqueness. The objective
is to provide a brief introduction to OTC markets, including some of
the key conceptual issues and modeling techniques, and to provide a
foundation for reading more advanced research in this topic area. The
lectures assume a graduate-level background in probability theory.
- Mark Freidlin (University of Maryland at College Park)
[2.30pm-3.30pm in Rm SSW 903.]
Stochastic and deterministic perturbations of systems with
multiple invariant measures: long-time evolution
I will consider perturbations of dynamical systems and stochastic
processes with conservation laws. Long-time behavior of such systems
is defined by slow evolution of first integrals due to the
perturbations.
In an appropriate time scale, this slow evolution, even in the case of
pure deterministic perturbations of deterministic systems, should be
approximated, in general, by a stochastic process. The averaging
principle and the large deviation theory are the main
tools in this class of problems.
-
Friday, April 8, 2011
- Jean Jacod (Universite Pierre et Marie Curie)
The quadratic variation of an It\^o semimartingale without
Brownian part
In the context of high frequency data, one of the main objects of
interest is the quadratic variation. When the underlying process is an
Itˆo semimartingale one knows the rate of convergence of the
``approximated'' quadratic variation when it is computed on the basis
of
a regular sampling with a mesh going to 0. This rate is the square
root of the number of observations.
When the process has no Brownian part, the above limit vanishes,
meaning that the rate is not appropriate. We show that, under some
(unfortunately rather strong) assumptions, there is a faster rate, and
we describe the corresponding limiting process. (Joint work with Viktor
Todorov.)
Abstract in pdf.
-
Friday, April 15, 2011
- Columbia-Princeton
Probability Day (At Princeton University)
Main Speakers:
* Alexei Borodin (Caltech)
* Shige Peng (Shandong U.)
* Yakov Sinai (Princeton)
* Horng-Tzer Yau (Harvard)
Student Speakers:
* Antonio Auffinger (Courant)
* Hana Kogan (CUNY)
* Ilya Vinogradov (Princeton)
Registration is free and will be open until April 1st, 2011.
The registration form can be found on the Probability Day website.
Continental breakfast
and lunch will be provided for all registered participants.
Directions and a schedule can be found on the website. Titles
and abstracts
of the talks will be posted there when they are available.
-
April 16--May 3
- Josef Teichmann (ETH Zurich)
Minerva Lectures
-
Friday, April 22, 2011
- Gabor Pete (University of Toronto)
The scaling limits of dynamical and near-critical
percolation, and
the Minimal Spanning Tree
Let each site of the triangular lattice, with small mesh $\eta$,
have an independent Poisson clock with a certain rate $r(\eta) =
\eta^{3/4+o(1)}$ switching between open and closed such that, at any
given
moment, the configuration is just critical percolation. In particular,
the
probability of a left-right open crossing in the unit square is close
to
1/2. Furthermore, because of the scaling, the expected number of
switches
in unit time between having a crossing or not is of unit order.
We prove that the limit (as $\eta \to 0$) of the above process exists
as a
Markov process, and it is conformally covariant: if we change the
domain
with a conformal map $\phi(z)$, then time has to be scaled locally by
$|\phi'(z)|^{3/4}$. A key step in the proof is that the counting
measure
on the set of pivotal points in a percolation configuration has a
conformally covariant scaling limit. Similar limit measures can be
constructed for other special points: length measures on interfaces and
area measures on clusters.
The same proof yields a similar result for near-critical percolation,
and
it also shows that the scaling limit of (a version of) the Minimal
Spanning Tree exists, it is invariant under translations, rotations and
scaling, but *probably* not under general conformal maps. I will
explain
what is missing for a proof.
Joint works with Christophe Garban and Oded Schramm.
Fall Semester 2010
-
Friday, September 24, 2010
- Fredrik Johansson Viklund (Columbia University)
Convergence rates for loop-erased random walk
Loop-erased random walk (LERW) is a self-avoiding random walk obtained
by
chronologically erasing the loops of a simple random walk.
In the plane, the lattice size scaling limit of LERW is known to be
SLE(2),
a random fractal curve constructed by solving the Loewner
differential equation with a Brownian motion input.
In
the talk, we will discuss our recent work on obtaining a rate for the
convergence of LERW to SLE(2).
More precisely, we will outline our derivation of a rate for the
convergence of the Loewner driving function for loop-erased random walk
to Brownian motion with speed 2 on the unit circle, the Loewner driving
function for radial SLE(2).
We will also, time permitting, discuss how to obtain a rate for the
convergence with respect to Hausdorff distance.
This is joint work with C. Benes (CUNY) and M. Kozdron (U. of Regina).
- Friday, October 1, 2010
- Misha Sodin (Tel Aviv University)
Nodal lines of random waves
In the talk, I will introduce random spherical harmonics
and random plane waves, and will describe recent attempts
to understand the mysterious and beautiful structure of
their nodal lines. The talk is based on a joint
work with Fedor Nazarov.
- Friday, October 8, 2010
- John Palmer (Arizona)
I will survey some of what is known about the correlations
of the solvable Ising model in two dimensions. This includes
formulas for the lattice correlations, the scaling functions
which exhibit the asymptotics of the lattice correlations at the
correlation length scale, and the short distance behavior
of the scaling functions summarized in correlations predicted
by conformal field theory. Beyond the accident of being "solvable"
there are some surprises in the sort of mathematics
that has applications to understanding the correlations. I
will say something about the role that elliptic curves and
uniformization play in understanding the lattice correlations.
- Friday, October 15, 2010
- Henrik Hult (Royal Institute of Technology)
Sanov's theorem under importance sampling
We present a version of Sanov's theorem for the weighted empirical
measure based on importance sampling. The theorem provides a way to
quantify the probability of large errors of importance sampling
estimates. It is, in addition, potentially useful for designing
efficient importance sampling algorithms.
- Friday, October 22, 2010
- Vadim Gorin (Moscow State University)
From random tilings to representation theory
Lozenge tilings of planar domains provide a simple, yet sophisticated
model of random surfaces. Asymptotic behavior of such models has been
extensively studied in recent years.
We will start from recent results about q-distributions on
tilings of a hexagon or, equivalently, on boxed plane partitions. (This
part is based on the joint work with A.Borodin and E.Rains).
In the second part of the talk we will explain how
representation theory of the infinite-dimensional unitary group is
related to random lozenge tilings with a certain Gibbs property. We
will discuss applications of this correspondence and results on the
classification of Gibbs measures on tilings of the half-plane.
- Friday, October 29, 2010
- Richard Bass (University of Connecticut)
Convergence of symmetric Markov chains
Suppose one has a symmetric Markov chain on the
integer lattice in one or more dimensions. One might hope that,
properly
normalized, in the limit one would obtain a diffusion that corresponds
to a
second order elliptic operator in divergence form. I will give a
sufficient condition for such a result to hold, even when the
symmetric Markov chain is not a nearest neighbor chain or one
with bounded range. Then I will discuss generalizations where the limit
process has jumps as well as a continuous part. This is joint work
with Takashi Kumagai and Toshihiro Uemura.
- Friday, November 5, 2010
- Ivan Corwin (Courant Institute)
The KPZ universality class and equation
The Gaussian central limit theorem says that for a wide
class of stochastic systems, the bell curve (Gaussian distribution)
describes the statistics for random fluctuations of important
observables. In this talk I will look beyond this class of systems to
a collection of probabilistic models which include random growth
models, polymers, particle systems, matrices and stochastic PDEs, as
well as certain asymptotic problems in combinatorics and
representation theory. I will explain in what ways these different
examples all fall into a single new universality class with a much
richer mathematical structure than that of the Gaussian.
- Wednesday, November 10, 2010, 5--6 PM, Math 520. Special joint Probability/Geometry/Analysis Seminar.
- Horng-Tzer Yau (Harvard University)
Universality of random matrices and Dyson Brownian motion
Random matrices were introduced by E. Wigner to model the excitation
spectra of large nuclei.
The central idea is based on the hypothesis that the local statistics
of the excitation spectrum for a large complicated system is
universal
in the sense that it depends only on the symmetry class of the
physical system but not on other detailed structures.
Dyson Brownian motion is the flow of eigenvalues of random matrices
when each matrix element performs independent Brownian motions.
In this lecture, we will explain the connection between the
universality of random matrices and
the approach to local equilibrium of Dyson Brownian motion. This
connection has led to a complete solution
of the universality conjecture by Wigner, Dyson and Mehta.
The main tools in our approach are an estimate on the flow of entropy
in Dyson Brownian motion and a local semicircle law.
One key feature of the entropy estimate is an extension of the
logarithmic Sobolev inequality
to cases not covered by the convexity criterion of Bakry and Emery.
- Friday, November 12, 2010
- Jason Miller (Stanford University)
CLE(4) and the Gaussian Free Field
The discrete Gaussian free field (DGFF) is the Gaussian measure on
functions $h \colon D \to \R$, $D \subseteq \Z^2$ bounded, with
covariance given by the Green's function for simple random walk. The
graph of $h$ is a random surface which serves as a physical model for
an effective interface. We show that the collection of random loops
given by the level sets of the DGFF for any height $\mu \in \R$
converges in the fine-mesh scaling limit to a family of loops which is
invariant under conformal transformations when $D$ is a lattice
approximation of a non-trivial simply connected domain. In
particular, there exists $\lambda > 0$ such that the level sets
whose
height is an odd integer multiple of $\lambda$ converges to a nested
conformal loop ensemble with parameter $\kappa=4$ [so-called
$\CLE(4)]$, a conformally invariant measure on loops which locally
look like $\SLE(4)$. Using this result, we give a coupling of the
continuum Gaussian free field (GFF), the fine-mesh scaling limit of
the DGFF, and $\CLE(4)$ such that the GFF can be realized as a
functional of $\CLE(4)$ and conversely $\CLE(4)$ can be made sense as
a functional of the GFF. This is joint work with Scott Sheffield.
- Thursday--Friday, November 18--19, 2010
- Ninth Northeast Probability Seminar (At CUNY
Graduate Center)
Speakers:
Nathanael Berestycki (Cambridge University)
Persi Diaconis (Stanford University)
Yves Le Jan (Universite Paris Sud)
Edwin Perkins (University of British Columbia)
Further
information
- Friday, November 26, 2010
- Friday, December 3, 2010
- Boris Rozovsky (Brown University)
Stochastic Quantization and Navier-Stokes Equation
Minerva
Lectures
- Friday, December 10, 2010
- Grigori Olshanski (Dobrushin Math. Lab.,
Kharkevich Institute, Moscow;
Independent University of Moscow)
A model of infinite-dimensional Markov dynamics
The talk is based on recent joint work with Alexei Borodin. The problem
of harmonic analysis on the infinite-dimensional unitary group
U(infinity) leads to a family of probability measures living on an
infinite-dimensional space Omega, a kind of dual object to
U(infinity). The measures in question can also be interpreted as
determinantal point processes; earlier we explicitly computed their
correlation functions. Now we construct continuous-time Markov
processes
on Omega, preserving those measures. In contrast to many works on
infinite-dimensional Markov dynamics, our approach does not use
Dirichlet forms or other advanced analytic techniques, it relies on a
mix of elementary tools from Markov chains and some ideas of random
matrix theory.
Spring Semester 2010
Please see Spring
Semester 2010 and older.