COLLOQUIUM IN STOCHASTICS
Friday 13 January 2006
International Centre for Mathematical Sciences
14 India Street, Edinburgh EH3 6EZ, UK
Organisers:
Takis Konstantopoulos
and
Andreas Kyprianou
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In celebration of the creation of the new Maxwell institute, a collaborative
research incentive between the School of Mathematics at Edinburgh University
and the School of Mathematical and Computer Sciences at Heriot Watt
University, members of the stochastic community in Edinburgh are hosting
a day of specially invited talks by leading European
experts in stochastics.
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Programme:
09:00-10:00 Welcome talks and coffee
Presentation of the International Centre for Mathematical Sciences (5-10 mins)
Introduction to the Maxwell Institute for Mathematical Sciences (5-10 mins)
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10:00-10:45
Nathalie EISENBAUM
(Laboratoire de Probabilités,
Université Paris VI)
On the continuity of local times of Markov processes
The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process, is still open. Barlow and Hawkes have completely treated the case of the Lévy processes, Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We will present an answer to that problem in the general case of the Borel right Markov processes. This answer unifies the results of Barlow and Hawkes, and Marcus and Rosen, by using a Gaussian process that naturally appears in a Central Limit Theorem involving the local time process. Joint work with Haya Kaspi (Technion).
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11:00-11:45
Martijn PISTORIUS
(Financial Maths, Mathematics Dept, King's C
ollege London)
On the optimal dividend problem for
a spectrally negative Lévy process
In this talk we consider the optimal dividend
problem for an insurance company whose risk process evolves as a
spectrally negative Lévy process in the absence of dividend
payments. The classical dividend problem for an insurance company
consists in finding a dividend payment policy that maximizes the
total expected discounted dividends. Related is the problem where
we impose the restriction that ruin be prevented: the
beneficiaries of the dividends must then keep the insurance
company solvent by bail-out loans. Drawing on the fluctuation
theory of spectrally negative Lévy processes we give an
explicit analytic description of the optimal dividend policies and
corresponding value functions for either of the problems.
Joint work with Florin Avram and Zbigniew Palmowski.
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12:00-14:00 Lunch break
There will be no lunch provided, however the venue is very central
with lots of nice places to eat in the neighbourhood:
Guide to Edinburgh's Restaurants
Henderson's
Edinburgh's Cafes
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14:00-14:45
Zoran VONDRAČEK
(Department of Mathematics, University of Zagreb)
On potential theory of subordinate Brownian motion
Subordinating multidimensional Brownian motion
by means of a subordinator with jumps leads to a discontinuous
Lévy process called subordinate Brownian motion. By use of
specific properties of the subordinator one can deduce various
results about potential theory of the subordinate process.
In this talk I will discuss (1) special subordinators and
consequences on potential theory of subordinate killed Brownian
motion, (2) subordinators with a drift and discontinuous part,
(3) geometric subordinators and corresponding potential theory
of subordinate Brownian motion.
Joint work with Renming Song.
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15:00-15:45
Alexander GNEDIN
(Mathematical Institute, Utrecht University)
Asymptotics of
regenerative composition structures in the case of slow variation
For S
a subordinator and Πn
an independent Poisson process of intensity
n exp(-x), x>0,
we are interested in the number
Kn of gaps in the range of
S that are hit by at least one point of
Πn.
This corresponds to the number of blocks
of a poissonised regenerative composition structure.
Extending previous studies
we focus on the case when the tail of the
Lévy measure of S is
slowly varying.
We view Kn
as the terminal value of a random process
Kn, and
provide an asymptotic analysis of the fluctuations of
Kn,
as n
→∞,
for a wide spectrum of situations.
Joint work with Andrew Barbour.
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16:00-16:30 Coffee break
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16:30-17:15
Jean BERTOIN
(Laboratoire de Probabilités,
Université Paris VI)
Stochastic flows, coalescents, and Fleming-Viot processes
I will present a survey of recent works
in collaboration with J.-F. Le Gall (ENS Paris) in which
important relations between certain stochastic flows,
coalescents with multiple coagulations, and the genealogy
of generalized Fleming-Viot processes will have a key part.
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