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| Speaker |
Jay Rosen, CUNY |
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| Title |
Large Deviations for Riesz Potentials of Additive Processes |
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| Speaker |
Michael B. Marcus, CUNY |
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| Title |
Infinitely divisible squares of non associated Gaussian vectors |
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| Speaker |
Jan Rosinski, University of Tennessee and Cornell University |
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| Title |
Simulation of Levy Processes: an Overview and Current Issues |
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| Abstract |
Levy processes arise in many areas of applied probability and
stochastic finance. They are continuous time random walks comprised of
independent diffusion and jump parts (Brownian motion and a Levy
process of the Poissonian-type). Simulation of a Brownian motion and/or
of a compound Poisson process can be found in many textbooks and will
not be discussed here. We will concentrate on a situation when a Levy
process has infinitely many jumps in each finite interval. Exact
simulation of such processes is obviously impossible and we must use
approximate methods. A choice of the method depends on the Levy measure
of a process and other characteristics.
We will discuss simulation methods based
on (a) random walk approximation; (b) series representations of Levy
processes; (c) Poisson and Gaussian approximation; together with their
ramifications, and pros and cons. Contrary to the one-dimensional case,
closed formulae for simulation of increments of multidimensional Levy
processes are rarely available. This essentially rules out an
approximation by a random walk from a discrete skeleton. If time
permits, we will discuss current issues of simulation in the
multidimensional case.
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| Speaker |
Christian Benes, Brooklyn College, CUNY |
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| Title |
Some Properties of the Complement of Planar Random Walk |
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| Abstract | Consider
simple random walk S in the plane and the continuous curve obtained
from it by linear interpolation between integer times. If one defines
the set of 'holes' to be the set of connected components of C
S[0,2n] and the set of 'lattice holes' of S to be the set of
connected components of Z2 Sj0 < = j < = 2n,
one can assign to each hole an ärea", its Lebesgue measure, and to each
lattice hole a "lattice area", its cardinality. In this talk, we will
show that the number of holes (resp. lattice holes) of area (resp.
lattice area) greater than A(n)*n is, up to a logarithmic correction
term, asymptotic to A(n)-1, if n-d0 < A(n) < f(n), where f(n) goes to 0 more slowly than any power function and d0
> 0. This confirms an observation by Mandelbrot. A consequence is
that the largest hole has an area which is logarithmically asymptotic
to n. We will also mention the different and mysterious exponent of 5/3
observed by Mandelbrot for 'small' lattice holes
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| Speaker |
Kevin O'Bryant, CUNY, CUNY, College of Staten Island |
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| Title |
The Central Limit Theorem without the limit |
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| Abstract |
A B*[g] set is a set S of integers with the property that the set does not have more than g solutions to x=a1 + a2 = b1 + b2
for any x. The foundational problem here is to bound the size of a
B*[g] set contained in {1,2,...,n}. Recent progress on this problem has
been made through proving statements of the form: If X1, X2 are i.i.d. with diameter less than L, then the pdf of X1+X2
has infinity norm at least Q/L. The "trivial" value of Q is 1/2. I will
present recent progress on this problem and its generalization from 2
summands to h summands. |
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| Speaker |
Ross Pinsky, Technion, Israel Institute of Technology |
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| Title |
Increasing subsequences in random permuations |
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| Speaker |
Alan Michael Hammond, Courant Institute, NYU |
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| Title |
The scaling limit of a biased random walk on a supercritical Galton Watson tree |
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| Abstract |
We study the long-term behaviour of a random walk on a super-
critical Galton Watson tree, where the walk is biased away from the
root.
The walk is liable to be trapped in finite trees that are attached to
the backbone of the tree, and this trapping makes the walk sub-
ballistic, if the bias is strong enough.
I will explain how, if the bias is taken to be a constant at each
vertex, a discrete inhomogeneity is present on all time scales, so
that a scaling limit (at least in a conventional sense) does not exist
for the process.
If we randomize the bias at each vertex so that it is independently
sampled according to a smooth law, on the other hand, we show
convergence of the scaled distance from the root to a stable
subordinator.
Joint work with G. Ben Arous. |
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| Speaker |
Olympia Hadjiliadis, Brooklyn College, CUNY |
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| Title |
Optimal quickest detection of two-sided alternatives and connections to drawdown and rally processes. |
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| Abstract |
In this talk we present the problem of quickest detection of two-sided
alternatives in the Brownian motion model. In particular, we consider
this problem in the min-max setting where the change-point is assumed
to be an unknown constant. We formulate a stochastic optimization
problem that arises as the trade-off between minimizing quick detection
and keeping the mean time between false alarma above a certain
threshold. We present properties of the optimal stopping time. We then
proceed to find the best 2-CUSUM stopping time by means of evaluating
an expression for its first moment. We describe a connection between
the 2-CUSUM process and the drawdown and rally processes. We present
the probability that a drawdown of a given level precedes a rally of an
equal or unequal level in a special class time homogeneous diffusion
processes. We provide closed form expressions of this probability in
the case of an Ornstein Uhlenbeck process, a special case of which
includes the Vasicek model for short-term interest rates.
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|
| Speaker |
Davar Khoshnevisan, University of Utah |
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| Title |
The packing dimension of the range of a Levy process |
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| Abstract |
Let {X(t)}t ³ 0 denote a Lévy process in \Rd with
exponent Y. Taylor (1986) proved that the packing dimension
of the range X([0 ,1]) is described by the index
|
g¢ = |
sup
| |
ì
í
î
|
a ³ 0: |
liminf
r ® 0+
|
|
ó
õ
|
1
0
|
\frac¶ {|X(t)| £ r}ra dt = 0 |
ü
ý
þ
|
. |
| (\theequation) |
We provide an alternative formulation of g¢ in terms of the
Lévy exponent Y. Our formulation, as well as methods,
are Fourier-analytic, and rely on the properties of the
Cauchy transform. We show, through examples, some applications
of our formula.
Time permitting, we introduce also the resolution to a question of
J. D. Howroyd (1997) in geometric measure theory.
This is based on joint work with Yimin Xiao.
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