Informal Probability
Seminars: Spring 2009
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
28/1/09: Pavel Gapeev (LSE)
Perpetual American
options in models with default risk and
random dividends
We present closed form
solutions to the problems of pricing of
perpetual American standard options in two
diffusion models of financial markets with
presence of default risk. The method of
proof is based on reducing the initial
discounted optimal stopping problems to
equivalent free-boundary problems and
solving the latter by means of the
smooth-fit conditions. Applying the
recently derived change-of-variable formula
with local time on surfaces, we verify that
the obtained solutions of the free-boundary
problems turn out to be solutions of the
initial optimal stopping problems.
In the first model, it is assumed that a
firm can spontaneously announce a default
and change the value of the dividend rate
at that time. Being inaccessible for usual
investors trading at the market, such a
change in the dividend rate can be actually
hidden into the behaviour of the firm value
under the risk-neutral measure. Aiming at
explicit expressions for the rational
option prices and related exercise
boundaries, we assume exponential
distribution for the default time at which
the dividend rate changes from one constant
value to another, within the framework of a
diffusion model for the firm value
dynamics. We embed the initial problem into
a discounted optimal stopping problem for a
two-dimensional Markov process, having the
firm value and conditional survival
probability as components. It turns out
that the latter process is equivalent to
the posterior probability of the occurrence
of 'disorder' in the corresponding problem
of detecting a change in the drift rate of
an observed Wiener process. By means of
solving the equivalent free-boundary
problem, we show that the exercise boundary
for the firm value is constant and can be
characterized as a unique solution of a
transcendental equation.
In the second model, it is assumed that the
default happens when the firm value falls
to some random barrier, which is not
observable from the market. Aiming at
closed form expressions for ex-dividend
prices and exercise boundaries, we assume
that the barrier has exponential
distribution and it is independent of the
firm value, which is modelled by a
geometric Brownian motion. We embed the
initial problem into an optimal stopping
problem for a two-dimensional Markov
process having the firm value and its
running minimum as components. Note that
the resulting problem turns out to be
necessarily two-dimensional in the sense
that it cannot be reduced to a problem for
a one-dimensional Markov process. We show
that the optimal exercise boundary for the
firm value can be expressed as a function
of the running minimum process. By means of
solving the equivalent free-boundary
problem, where the normal reflection
condition at the diagonal of the
two-dimensional state space breaks down, we
show that the exercise boundary can be
characterized as a unique solution of a
nonlinear ordinary differential
equation.
4/2/09: Parkpoom Phetpradap
Large deviations for the
range of a simple random walk
12/2/09: Wolfgang
König (Leipzig)
A Variational Formula
for the Free Energy of a Many-Boson
System
Note time and day:
11.15am, Thursday
18/2/09: Mladen Savov
(Paris VI)
Right Inverses for Levy
Processes
An integral criterion in terms of the Levy
triplet is proposed which tests whether a
Levy Process has a right inverse,
i.e. there exists a subordinator K(x) such
that X(K(x))=x at least for each x in
(0,a), where a>0 is a random variable. This
completes previous studies by Evans and
Winkel.
25/2/09: Vitali Wachtel (Munich)
Heavy-traffic analysis
of the maximum of an asymptotically
stable random walk
For families of random walks
$\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} =
-ka < 0$ we consider their maxima $M^{(a)}
= \sup_{k \ge 0} S_k^{(a)}$. We investigate
the asymptotic behaviour of $M^{(a)}$ as $a
\to 0$ for asymptotically stable random
walks. This problem appeared first in the
1960's in the analysis of a single-server
queue when the traffic load tends to $1$
and since then is referred to as the
heavy-traffic approximation
problem. Kingman and Prokhorov suggested
two different approaches which were later
followed by many authors. We give two
elementary proofs of our main result, using
each of these approaches. It turns out that
the main technical difficulties in both
proofs are rather similar and may be
resolved via a generalisation of the
Kolmogorov inequality to the case of an
infinite variance.
4/3/09: Stanislav Volkov (Bristol)
Going through a
passport control with wife, or sequential
adsorption at extremes.
In a simple version of a cooperative
sequential adsorption model, particles
consecutively arrive on the set of vertices
{1,2,...,M} uniformly spaced on a
circumference. A particle arriving at time
t=0,1,2,... gets attached to a vertex j
with probability proportional to
beta^{N(t,j)} where N(t,j) is a number of
particles already attached to the vertices
in a certain neighbourhood of vertex
j. Examples of such a neighbourhood
are:
(1) the vertex itself and its
left neighbour (asymmetric case);
(2) the vertex itself and its left and
right neighbours (symmetric case).
We are interested in the long-time
behaviour of this process. Our most recent
results cover the cases when beta=0 and
beta=+infinity, the situations which can be
quite naturally interpreted in a queueing
theory setup
The talk is a sequel to the seminar given
by Vadim Shchervakov on the 29 October
2008, and will highlight the most recent
findings based on our joint work.
11/3/09: Mathew Penrose
Normal approximation
for an allocation model
Consider throwing n balls independently
into m boxes of possibly non-equal size
(probability). Let S be the resulting
number of isolated balls. We discuss error
bounds in the normal approximation for S,
and bounds for the variance of S in terms
of the underlying parameters. Advanced
prerequisites are at a minimum for this
talk.
18/3/09: Matthias Winkel
(Oxford)
A new family of Markov
branching trees: the alpha-gamma model
We introduce a simple tree growth process
that gives rise to a new two-parameter
family of discrete fragmentation trees that
extends Ford's alpha model to
multifurcating trees and includes the trees
obtained by uniform sampling from Duquesne
and Le Gall's stable continuum random
tree. We call these new trees the
alpha-gamma trees. In this paper, we obtain
their splitting rules, dislocation measures
both in ranked order and in sized-biased
order, and we study their limiting
behaviour.
Note time and location:
12.45, 6E 2.1
8/4/09: Günter
Last (Karlsruhe)
Compound Poisson
asymptotics for level crossings of
piecewise deterministic Markov
processes
We consider a piecewise-deterministic
Markov process governed by a jump intensity
function, a rate function that determines
the behaviour between jumps, and a
stochastic kernel describing the
conditional distribution of jump
sizes. First we discuss a version of Rice's
formula relating the stationary density of
the process to level crossing
intensities. Under additional assumptions
we will then show that a suitably
time-scaled point process of level
crossings can be approximated by a
geometrically compound Poisson process.
The talk is based on joint work with Kostya Borovkov.
29/4/09: Manfred
Salmhofer (Heidelberg)
Diffusion of a quantum
particle in a random environment
6/5/09: Pierre Tarrès
(Oxford)
Brownian
polymers
We consider a process $X_t\in\R^d$,
$t\ge0$, introduced by Durrett and Rogers
in 1992 in order to model the shape of a
growing polymer; it undergoes a drift which
depends on its past trajectory, and a
Brownian increment. Our work concerns two
conjectures by these authors (1992),
concerning repulsive interaction functions
$f$ in dimension $1$ ($\forall x\in\R$,
$xf(x)\ge0$).
We showed the first one with T. Mountford
(AIHP, 2008), for certain functions $f$
with heavy tails, leading to transience to
$+\infty$ or $-\infty$ with probability
$1/2$. We partially proved the second one
with B. Tóth (preprint), for rapidly
decreasing functions $f$, through a study
of the local time environment viewed from
the particule: we explicitly display an
associated invariant measure, which enables
us to prove under certain initial
conditions that $X_t/t\to_{t\to\infty}0$,
and that the process is at least diffusive
asymptotically.
13/5/09: Frank Aurzada
(TU Berlin)
Small deviations of
general Lévy processes
The small deviation problem for a
stochastic process $(X_t)_{t\in[0,1]}$
consists in determining the rate of the
quantity $$ P( \sup_{t\in[0,1]} |X_t| \leq
\varepsilon ) $$ when $\varepsilon\to
0$. This problem has several connections to
approximation quantities and the path
regularity of the processes. The talk
starts with a short overview of the field
of small deviation probabilities and its
applications.
Then recent results are presented when $X$
is a real-valued Lévy processes. In this
case, it is possible to obtain tight
estimates for the above probability (on the
exponential scale) only in terms of the
characterizing triplet $(c,\sigma2,\nu)$ of
$X$. Several examples are discussed as well
as ideas of the proofs, which include
exponential change of measure and exit time
estimates. In the multi-dimensional
setting, the problem is open, and I briefly
describe the arising difficulties.
This is joint work with Steffen Dereich
(Berlin).
20/5/09: Tom Rosoman
Critical probabilities
in continuum percolation
27/5/09: Marcel Ortgiese
Ageing in the parabolic
Anderson model
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