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Alex Cox
Email: A.M.G.Cox@bath.ac.uk

Informal Probability
Seminars: Winter 2008
Our seminars our usually held at 12.15 p.m. on
Wednesdays in room 1W 3.6 . If you
wish to find out more, please contact one of the
organisers. The speakers are mostly internal (Bath)
unless otherwise stated. Details of previous
semesters can be found
here
8/10/08: Ron Doney (Manchester)
Convergence of local times of Levy
processes The following problem is
discussed: if a sequence of Levy processes (or normed
random walks) converges to a limiting Levy process, does
the corresponding sequence of local times of the
reflected processes converge?
15/10/08: Peter Mörters
Two remarks on a paper by
Kallenberg In a recent discussion
paper, Kallenberg observed a great similarity in the
local hitting, scaling, and conditioning behaviour of
several random measures, including simple point
processes, local times of regenerative sets and the
states of superprocesses. In this talk I would like to
add two further observations: First, the intersection
local times of Brownian paths also share the common
features of these random measures, and second, all these
measures also have strikingly similar multifractal
behaviour. I will also discuss similarities in the
Hausdorff and packing measure properties of their
supports.
29/10/08: Vadim Shcherbakov
Stability of a growth process
generated by cooperative sequential
adsorption
We are interested in a long time behaviour (stability)
of a growth process generated by the cooperative
sequential adsorption (CSA)model .
The CSA model is a probabilistic model of adsorption
processes. In the talk, we consider the CSA model on a
lattice, which in turn can be viewed as a variant of the
Polya urn scheme with interactions: namely, we
sequentially attach particles to lattice sites with
probabilities which depend on the configuration of
already allocated particles.
The growth process is then described by the numbers of
attached (adsorbed) particles. Stability of the growth
process now means, loosely speaking, that its components
grow at approximately the same rate, therefore, for
instance, no extraordinary high peaks are observed. For
different models of attachment configurations, we
rigorously prove the stability (or instability) of the
process.
This is a joint work with Stanislav Volkov (University of Bristol).
5/11/08: Andreas Kyprianou
Refracted Lévy
processes
12/11/08: Nikolai Leonenko (Cardiff)
Multifractality of products of
geometric OrnsteinUhlenbeck type
processes
This is joint work with V.V. Anh (Queensland University
of Technology, Brisbane) and N.R. Shieh (National
Taiwan University, Taipei). We consider multifractal
products of stochastic processes as defined in
Mannersalo et al. (2002), but we provide a new
interpretation of the conditions on the mean, variance
and covariance functions of the resulting cumulative
processes in terms of the moment generating
functions. We show that the logarithms of the
corresponding limiting processes have an infinitely
divisible distribution such as the gamma and variance
gamma distributions (resulting in the loggamma and
logvariance gamma scenarios respectively), the inverse
Gaussian and normal inverse Gaussian distributions
(yielding the loginverse Gaussian and lognormal
inverse Gaussian scenarios respectively). We describe
the behavior of their qth order moments and
Rényi functions, which are nonlinear, hence
displaying their multifractality. A property on the
dependence structure of the limiting processes, leading
to their possible longrange dependence, is also
obtained. We consider very different scenarios such as
the loggamma and loginverse Gaussian scenarios as
typical examples of our approach. We should also note
some related results by BarndorffNielsen and Schmiegel
(2004) who introduced some Lévybased
spatiotemporial models for parametric modelling of
turbulence. Loginfinitely divisible scenarios related
to independently scattered random measures were
introduced in Bacry and Muzy (2003) and others. We
should note that Chris Heyde (1999) proposed to use a
multifractality into risky asset model with fractal
activity time (see also Heyde and Leonenko (2005)).
19/11/08: Hubert Lacoin (Paris)
Directed random polymer on a diamond
lattice. (Joint work with Gregorio
Moreno, supervised by Pr. Francis Comets and Pr.
Giambattista Giacomin)
A great number of model in statistical physics
(e.g. Ising model, Potts Model, first/last passage
percolation, wetting/pinning models) have been studied
on a certain class of hierarchical graphs: the diamond
lattices. The main advantage of these lattices is that
the renormalisation methods that are usually used by
physicists can be performed in a rigourous manner. In
this talk, I will present some results obtained with
Gregorio Moreno for the random polymer model on the
diamond lattice and the different tools that are used
for the proofs (fractional moment method, shift of the
environment, percolation argument and second moment
method) and discuss about the similarities of this model
with the one on $Z^{d+1}$.
3/12/08: Juan Carlos
Pardo Millan
Exact and asymptotic
ntuple laws at first and last
passage
Understanding the spacetime features of
how a Lévy process crosses a
constant barrier for the first time, and
indeed the last time, is a problem which is
central to many models in applied
probability such as queueing theory,
financial and actuarial mathematics,
optimal stopping problems, the theory of
branching processes to name but a few. In
Kyprianou and Doney a new quintuple law was
established for a general Lévy
process at first passage above a fixed
level. In this work, we use the quintuple
law to establish a family of related joint
laws, which we call ntuple laws, for
Lévy processes, Lévy
processes conditioned to stay positive and
positive selfsimilar Markov processes at
both first and last passage over a fixed
level. Here the integer n typically ranges
from three to seven. Moreover, we look at
asymptotic overshoot and undershoot
distributions and relate them to overshoot
and undershoot distributions of positive
selfsimilar Markov processes issued from
the origin. Although the relation between
the ntuple laws for Lévy processes
and positive selfsimilar Markov processes
are straightforward thanks to the Lamperti
transformation, by interplaying the role
of a (conditioned) stable processes as both
a (conditioned) Lévy processes and a
positive selfsimilar Markov processes, we
obtain a suite of completely explicit first
and last passage identities for socalled
Lampertistable Lévy processes. This
leads further to the introduction of a more
general family of Lévy processes
which we call hypergeometric Lévy
processes, for which similar explicit
identities may be considered.
(Joint work with A.E. Kyprianou and
V. Rivero)
10/12/08: Jochen Voss (Warwick)
An SPDEbased Sampling
Method
In many situations it is useful to be able
to simulate paths from an SDE conditioned
on some interesting event occurring. But,
while simulating an unconditioned SDE is
often trivial, simulating paths from a
conditioned SDE can be a challenge. In
this talk we present a new sampling method
to generate paths from the distribution of
a second order SDE under endpoint
constraints.
The proposed method works by constructing a
fourthorder SPDE which is ergodic and has
the distribution of the conditioned SDE as
its invariant measure ("space" of the SPDE
corresponds to "time" of the SDE). Paths
of the conditioned SDE can then be found by
simulating the sampling SPDE until it is
close to stationarity and statistical
properties of the conditioned SDE can be
found by taking ergodic averages of the
solution of the SPDE. In contrast to
earlier works (were we used a second order
SPDE to sample for the distribution of a
conditioned first order SDE) the present
method also allows to consider the case
where the drift in the original SDE does
not have a gradient structure.
We illustrate the method with the help of
numerical simulations of the resulting
SPDE, obtained by using finite elements
discretisation.
(Joint work with Martin Hairer and Andrew
Stuart.)
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