Prob-L@b
Seminars: Winter 2011
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
17/10/11: Mohammud
Foondun (Loughborough)
Stochastic heat
equation with spatially coloured random
forcing
The aim of this talk
is to present some results concerning the
long term behaviour of a class of
stochastic heat equations. We will showcase
some relationship between Lévy
processes and the solution to these
SPDEs. We will talk about both the additive
noise case as well as the multiplicative
case. We will also cover the case of white
noise and various other coloured noise.
24/10/11: Sergey
Nadtochiy (Oxford)
Static hedging of
barrier options: exact solutions and
semi-robust extensions
We solve the problem of static hedging of
standard (call, put and digital) barrier
options in models where the underlying is
given by a time-homogeneous diffusion
process with, possibly, independent
stochastic time-change. The main result of
the paper includes analytic expression for
the payoff of a (single) European-type
contingent claim (which pays a certain
function of the underlying value at
maturity, without any path-dependence),
such that it has the same price as the
barrier option up until hitting the
barrier. We then address the issues of
numerical approximation of the static hedge
payoff, and, in particular, investigate the
performance of the approximate static hedge
consisting of vanilla (call and put)
options of two strikes only. Finally, we
show how the above results allow to
construct static sub- and super-replicating
strategies which are semi-robust with
respect to implied volatility. More
precisely, for each range of the implied
volatility values, we construct the static
sub- and super-replicating strategies which
work in any continuous model for the
underlying, as long as the corresponding
implied volatility stays within the
prescribed range.
(Joint work with Peter Carr)
31/10/11: Jozsef
Lorinczi (Loughborough)
Spectral Properties of
some Non-Local Operators through Stochastic
Methods
Fractional Schrödinger
and jump-diffusion operators provide useful
tools in modelling quantum and anomalous
kinetic phenomena. Driven by such
applications I will formulate some problems
involving spectral and analytic properties
of semigroups generated by these
operators. Then I will discuss some of
these properties by using a probabilistic
representation of the semigroups.
7/11/11: Alex
Mijatović (Warwick)
Limit Distributions of
Continuous-State Branching Processes with
Immigration
In this talk we
describe the characterisation of the limit
distributions (as time tends to infinity)
of the class of continuous-state branching
processes with immigration
(CBI-processes). The Levy-Khintchine
triplet of the limit distribution L of
the process X will be given explicitly in
terms of the characteristic triplets of the
Levy subordinator XF and the spectrally
positive Levy process XR, which describe
the immigration and branching mechanism of
the CBI-process X respectively. The Levy
density of L is essentially given by the
generator of XF applied to the scale
function of the spectrally positive Levy
process XR. We will also show that the
class of limit distributions of
CBI-processes is strictly larger
(resp. smaller) than the class of
self-decomposable (resp. infinitely
divisible) distributions. This is joint
work with Martin Keller-Ressel from TU
Berlin.
14/11/11: Peter Mörters
Shifting Brownian motion
Let \(\{B_t : t\in\mathbb{R}\}\) be a standard linear
Brownian motion. A stopping time T is
called an unbiased shift if \(\{B_{T+t}-B_T
: t\in\mathbb{R}\}\) is a Brownian motion
independent of \(B_T\). We solve the
Skorokhod embedding problem for unbiased
shifts and discuss optimality of our
solution. The talk is based on joint work
with Günter Last
(Karlsruhe) and Hermann Thorisson
(Reykjavik).
Note venue: 6W
1.1
18/11/11: Louigi
Addario-Berry (McGill University)
Cutting down trees with
a Markov chainsaw
Let \(T_n\) be a
uniformly random rooted tree on labels
\(1,\ldots,n\). Removing an edge from \(T_n\)
separates \(T_n\) into two components. Throw
away the component not containing the
root. Repeat this process until all that is
left is the root, and call the number of
edge removals (cuts) \(C_n\). In 2003,
Panholzer showed that \(C_n/\sqrt{n}\)
converges in distribution to a Rayleigh
random variable; Janson (2005) generalized
the result to all critical conditioned
Galton-Watson trees with finite variance
offspring distribution.
We present a new, coupling-based proof of
Janson's result, based on a modification of
the Aldous-Broder algorithm for generating
spanning trees. Our approach also yields a
family of novel transformations from
Brownian excursion to Brownian bridge, and
uncovers how to reconstruct a CRT from the
sawdust created by the Aldous--Pitman
fragmentation.
Based on joint work with Nicolas Broutin,
Cecilia Holmgren, and Gregory Miermont.
Note time, day and
venue: 16.15, Friday, 6W 1.1
28/11/11: Marion Hesse
Branching Brownian
motion in a strip: Survival near
criticality
We consider a
branching Brownian motion in which
particles are killed on exiting a strip and
study the evolution of the process as the
width of the strip shrinks to the critical
value at which survival is no longer
possible. We obtain asymptotics for the
survival probability near criticality and a
quasi-stationary limit result for the
process conditioned on survival.
5/12/11: Alex
Watson
Censored stable processes
We describe a method
for exploiting the self-similarity of
stable processes in order to obtain some
new hitting identities.
Our technique is to construct a positive,
self-similar Markov process which we call
the censored stable process, and from this
obtain a new Levy process via the Lamperti
transformation. It happens that we can find
the Wiener-Hopf factorisation of this Levy
process explicitly, which allows us to
compute a number of key quantities for
it. These translate to identities of the
original stable process.
This is joint work with Andreas Kyprianou
and Juan-Carlos Pardo (CIMAT).
12/12/11: Perla Sousi
(Cambridge)
The effect of drift on
the volume of the Wiener sausage and the
dimension of the Brownian path
The Wiener sausage at
time t is the algebraic sum of a Brownian
path on [0,t] and a ball. Does the
expected volume of the Wiener sausage
increase when we add drift? How do you
compare the expected volume of the usual
Wiener sausage to one defined as the
algebraic sum of the Brownian path and a
square (in 2D) or a cube (in higher
dimensions)? We will answer these
questions using their relation to the
detection problem for Poisson Brownian
motions, and rearrangement inequalities on
the sphere.
14/12/11: Amandine Veber
(CMAP, École Polytechnique)
Large-scale behaviour
of the spatial Lambda-Fleming-Viot process
The SLFV process is a
population model in which individuals live
in a continuous space. Each of them also
carries some heritable genetic type or
allele. We shall describe the long-term
behaviour of this measure-valued process
and that of the corresponding genealogical
process of a sample of individuals in two
cases : one that mimics the evolution of
nearest-neighbour voter model (but in a
spatial continuum), and one that allows
some individuals to send offspring at very
large distances. This is a joint work with
Nathanaël Berestycki and Alison Etheridge.
Note time, day and
venue: 15.15, Wednesday, 4E 2.61
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