Informal Probability
Seminars: Winter 2007
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
10/10/07: Zaeem Burq
(Melbourne)
Onedimensional Brownian motion
crossing nonsmooth curves
17/10/07: Victor Rivero
On the asymptotic behaviour of
increasing selfsimilar Markov
processes.
It has been proved by Bertoin and Caballero in 2002
that an increasing selfsimilar Markov process
(issMp) has polynomial growth, in the sense of
convergence in distribution, if and only if the
increasing Levy process associated to it via
Lamperti's transformation has finite mean. The
purpose of this talk is to explain the asymptotic
behaviour of an issMp in the case where the
underlying increasing Levy process has infinite mean.
To that end I will start by providing a short
introduction to the theory of positive selfsimilar
Markov processes (pssMp), then I will explain the
transformation introduced by Lamperti relating the
class of Levy processes with that of pssMp. To
finish, I will present some recent results of the
type central limit theorem and law of iterated
logarithm about the rate of growth at infinity of the
logarithm of an increasing selfsimilar Markov
processes.
24/10/07: Steffen Dereich
Random networks with concave
preferentia attachment rule: Degree
evolutions
31/10/07: Peter
Mörters
A tale of two cities: the
parabolic Anderson model with heavytailed
potential
7/11/07: Tobias Mueller
(EURANDOM/Technische Universiteit Eindhoven)
Twopoint concentration in random
geometric graphs
21/11/07: Alex Cox
Optimal Skorokhod Embeddings:
extremal range probabilities
The Skorokhod embedding problem is to find a stopping
time of a Brownian motion such that the stopped
process has a specified distribution. Many solutions
are known, including the solution of Azéma and
Yor, which additionally maximises the probability of
the maximium being greater than a given level among
all Skorokhod embeddings of a given distribution.
Motivated by applications to finance, we construct
embeddings which maximise and minimise the
probability of both passing above a given level, and
passing below a given level. (Joint work with Jan
Obłój, Imperial College.)
5/12/07: Ronnie Loeffen
On optimality of the barrier
strategy in de Finetti's dividend problem for
spectrally negative Levy processes
The classical optimal dividends control problem
introduced by de Finetti has been extensively studied
in the CramerLundberg (CL) risk model. Recently
Avram, Palmowski & Pistorius considered the case
were the risk process is modelled by a general
spectrally negative Levy process. Up till now, an
explicit solution to this control problem has only
been found in two concrete examples: the CL model
with exponentially distributed claims and the case
where the risk process is modelled by a Brownian
motion with drift. In both cases the optimal strategy
is formed by a barrier strategy. In this talk, we
show that if an easytocheck analytical condition on
the Levy measure is satisfied, then the optimal
strategy is always of the barrier type. In the
analysis the socalled scale functions of spectrally
negative Levy processes play a central role.
12/12/07: Robert Knobloch
Uniform conditional ergodicity and
intrinsic ultracontractivity
19/12/07: Renming
Song(Illinois)
Boundary Harnack principle for
subordinate Brownian motions
In this talk I will present some resent progress in
the potential theory of subordinate Brownian motions.
In particular I will show that the boundary Harnack
principle holds for a large class of subordinate
Brownian motions.
This class of subordinate Brownian motion can not be
regarded as perturbations of symmetric stable
processes.
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