Prob-L@b Seminars: Spring 2013
Our seminars our usually held at 12.15 p.m. on Mondays
in room
4W 1.7
. If you wish to find out more, please contact one
of the organisers. The speakers are internal (Bath)
unless otherwise stated. Details of previous semesters
can be found
here
4/2/13: Martin Forde (King's College,
London)
Model-independent bounds for general
path dependent options - a convex duality approach
CANCELLED
- NEW DATE TBC
11/2/13: Tiziano De
Angelis (Manchester)
Optimal stopping
of a Hilbert space valued diffusion and applications to
finance
Pricing American Bond
options in a market model with forward interest rates
corre- sponds, in mathematical terms, to an optimal
stopping problem of an infinite dimensional
diffusion. Motivated by this financial application we
analyse a finite horizon optimal stopping problem for an
infinite dimensional diffusion \(X\) by means of
variational techniques. The diffusion is driven by a SDE
on a Hilbert space \(H\) with a non-linear diffusion
coefficient \(\sigma(X)\) and a generic unbounded
operator \(A\) in the drift term. We show that when the
gain function \(\Theta\) is time-dependent and fulfils mild
regularity assumptions, the value function \(U\) of the
optimal stopping problem solves an infinite-dimensional,
parabolic, degenerate variational inequality on an
unbounded domain. The solution of the variational
problem is found in a suitable Banach space \(V\) fully
characterized in terms of a Gaussian measure \(\mu\)
associated to the coefficient \(\sigma(X)\).
18/2/13: Stefan Grosskinsky (Warwick)
Dynamics of condensation in inclusion processes
The inclusion process is an interacting particle system
where particles on connected sites attract each other in
addition to performing independent random walks. The
system has stationary product measures and exhibits
condensation in the limit of strong interactions, where
all particles concentrate on a single lattice site. We
study the equilibration dynamics on finite lattices in
the limit of infinitely many particles, which, in
addition to jumps of whole clusters, contains an
interesting continuous mass exchange between clusters
given by Wright-Fisher diffusions. During equilibration
the number of clusters decreases monotonically, and the
stationary dynamics consist of jumps of a single
remaining cluster (the condensate). This is joint work
with Frank Redig and Kiamars Vafayi.
25/2/13: Dafydd Evans (Cardiff)
Fast anomaly detection in spatial
point processes
CANCELLED
- NEW DATE TBC
4/3/12: Johannes Ruf (Oxford)
Nonnegative local martingales,
Novikov's and Kazamaki's criteria, and the
distribution of explosion times
I will give a new proof for the famous
criteria by Novikov and Kazamaki, which provide
sufficient conditions for the martingale property of a
nonnegative local martingale. The proof is based on an
extension theorem for probability measures that can be
considered as a generalization of a Girsanov-type change
of measure. In the second part of my talk I will
illustrate how a generalized Girsanov formula can be
used to compute the distribution of the explosion time
of a weak solution to a stochastic differential
equation.
Parts of this talk are based on joined
working papers with Martin Larsson and Ioannis Karatzas.
11/3/13: Antal Jarai
Electrical resistance of the low-dimensional critical
branching random walk
We consider the trace of a critical branching random
walk in \(d+1\) dimensions conditioned to survive
forever. We show that the electrical resistance between
the origin and generation \(n\) grows sublinearly in \(n\) when
\(d<6\). In particular, it follows that in \(d=5\) the spectral
dimension of simple random walk on the trace is strictly
larger than 4/3, answering a question of Barlow, Jarai,
Kumagai and Slade. (Joint work with Asaf Nachmias.)
20/3/13: Charles Bordenave (Toulouse)
Spectrum of Markov generators on sparse random graphs
In this talk, we will consider various
probability distributions on the set of stochastic
matrices with n states and on the set of
Laplacian/Kirchhoff matrices on n states. They will
arise naturally from the conductance model on n states
with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these
processes.
NOTE: Unusual time and date. Wednesday, 14.15 in 1E2.4
8/4/13: Tim Rogers
Demographic noise leads to the spontaneous formation of species
In this talk I will discuss an evolutionary model of
competition, which is a microscopic stochastic analogue
of a famous population-level model in ecology. I will
show how the effects of demographic noise in the
stochastic model give rise to radically different
macro-scale behaviour. The (non-rigorous) analysis uses
an expansion in system size, coupled with a time-scale
separation argument.
15/4/13: David Applebaum (Sheffield)
Brownian motion, martingale transforms and Fourier
multipliers on Lie groups.
We associate a space-time martingale to Brownian motion
on a Lie group \(G\) and transform it to obtain a family
of "differentially subordinate" martingales. Using
powerful inequalities dues to Burkholder, Banuelos and
Wang we construct a family of linear operators which are
bounded on \(L^{p}(G, \tau)\) (where \(\tau\) is a Haar
measure) for all \(1 < p < \infty\). When \(G\) is compact, we
can utilise non-commutative Fourier analysis to
represent these operators as Fourier
multipliers. Examples include second order Riesz
transforms and operators of Laplace transform type.
Talk based on joint work
with Rodrigo Banuelos
22/4/13: Dafydd Evans (Cardiff)
Fast anomaly detection in spatial
point processes
29/4/13: Yan Fyodorov (Queen Mary, London)
Fluctuations and extreme values in multifractal patterns
The goal is to understand sample-to-sample
fluctuations in disorder-generated multifractal
intensity patterns. Arguably the simplest model of that
sort is the exponential of an ideal periodic 1/f
Gaussian noise. The latter process can be looked at as a
one-dimensional "projection" of 2D Gaussian Free Field
and inherits from it the logarithmic covariance
structure. It most naturally emerges in the random
matrix theory context, but attracted also an independent
interest in statistical mechanics of disordered
systems. I will determine the threshold of extreme
values of 1/f noise and provide a rather compelling
explanation for the mechanism behind its
universality. Revealed mechanisms are conjectured to
retain their qualitative validity for a broad class of
disorder-generated multifractal fields.
The presentation will be mainly based on the joint work
with Pierre Le Doussal and Alberto Rosso, J Stat Phys:
149 (2012), 898-920 as well as on some related earlier
works by the speaker.
13/5/13: Denis Denisov (Manchester)
Tail behaviour of stationary distribution for Markov
chains with asymptotically zero drift
We consider a one-dimensional Markov chain
with asymptotically zero drift and finite second moments
of jumps which is positive recurrent. A power-like
asymptotic behaviour of the invariant tail distribution
is proven; such a heavy-tailed invariant measure happens
even if the jumps of the chain are bounded. Our analysis
is based on test functions technique and on construction
of a harmonic function. This is a joint work
with Korshunov and Wachtel.
20/5/13: Peter Mörters
Emergence of condensation in models of selection and
mutation
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