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Forthcoming events in this series
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Small time behaviour of double stochastic integrals and hedging under gamma constraints
Abstract
We formulate a problem of super-hedging under gamma constraint by
taking the portfolio process as a controlled state variable. This
leads to a non-standard stochastic control problem. An intuitive
guess of the associated Bellman equation leads to a non-parabolic
PDE! A careful analysis of this problem leads to the study of the
small time behaviour of double stochastic integrals. The main result
is a characterization of the value function of the super-replication
problem as the unique viscosity solution of the associated Bellman
equation, which turns out to be the parabolic envelope of the above
intuitive guess, i.e. its smallest parabolic majorant. When the
underlying stock price has constant volatility, we obtain an
explicit solution by face-lifting the pay-off of the option.
15:45
The Brownian snake and random trees
Abstract
The Brownian snake (with lifetime given by a normalized
Brownian excursion) arises as a natural limit when studying random trees. This
may be used in both directions, i.e. to obtain asymptotic results for random
trees in terms of the Brownian snake, or, conversely, to deduce properties of
the Brownian snake from asymptotic properties of random trees. The arguments
are based on Aldous' theory of the continuum random tree.
I will discuss two such situations:
1. The Wiener index of random trees converges, after
suitable scaling, to the integral (=mean position) of the head of the Brownian
snake. This enables us to calculate the moments of this integral.
2. A branching random walk on a random tree converges, after
suitable scaling, to the Brownian snake, provided the distribution of the
increments does not have too large tails. For i.i.d increments Y with mean 0,
a necessary and sufficient condition is that the tails are o(y^{-4}); in
particular, a finite fourth moment is enough, but weaker moment conditions are
not.
14:15
An extension of Levy-Khinchine formula in semi-Dirichlet forms setting
Abstract
The celebrated Levy-Khintchine formula provides us an explicit
structure of Levy processes on $R^d$. In this talk I shall present a
structure result for quasi-regular semi-Dirichlet forms, i.e., for
those semi-Dirichlet forms which are associated with right processes
on general state spaces. The result is regarded as an extension of
Levy-Khintchine formula in semi-Dirichlet forms setting. It can also
be regarded as an extension of Beurling-Deny formula which is up to
now available only for symmetric Dirichlet forms.
15:45
Weak interaction limits for one-dimensional random polymers
Abstract
Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every
self-intersection to the simple random walk path. The Edwards model (EM) is
obtained by giving a penalty proportional to the square integral of the local
times to the Brownian motion path. Both measures significantly reduce the
amount of time the motion spends in self-intersections.
The above models serve as caricature models for polymers, and we will give
an introduction polymers and probabilistic polymer models. We study the WSAW
and EM in dimension one.
We prove that as the self-repellence penalty tends to zero, the large
deviation rate function of the weakly self-avoiding walk converges to the rate
function of the Edwards model. This shows that the speeds of one-dimensional
weakly self-avoiding walk (if it exists) converges to the speed of the Edwards
model. The results generalize results earlier proved only for nearest-neighbor
simple random walks via an entirely different, and significantly more
complicated, method. The proof only uses weak convergence together with
properties of the Edwards model, avoiding the rather heavy functional analysis
that was used previously.
The method of proof is quite flexible, and also applies to various related
settings, such as the strictly self-avoiding case with diverging variance.
This result proves a conjecture by Aldous from 1986. This is joint work with
Frank den Hollander and Wolfgang Koenig.
14:15
Brownian motion in a Weyl chamber
Abstract
We give a construction of Brownian motion in a Weyl chamber, by a
multidimensional generalisation of Pitman's theorem relating one
dimensional Brownian motion with the three dimensional Bessel
process. There are connections representation theory, especially to
Littelmann path model.
14:15
Brownian motion in tubular neighborhoods around closed Riemannian submanifolds
Abstract
We consider Brownian motion on a manifold conditioned not to leave
the tubular neighborhood of a closed riemannian submanifold up
to some fixed finite time. For small tube radii, it behaves like the
intrinsic Brownian motion on the submanifold coupled to some
effective potential that depends on geometrical properties of
the submanifold and of the embedding. This characterization
can be applied to compute the effect of constraining the motion of a
quantum particle on the ambient manifold to the submanifold.
15:45
A polling system with 3 queues and 1 server
is a.s. periodic when transient:
dynamical and stochastic systems, and a chaos
Abstract
We consider a queuing system with three queues (nodes) and one server.
The arrival and service rates at each node are such that the system overall
is overloaded, while no individual node is. The service discipline is the
following: once the server is at node j, it stays there until it serves all
customers in the queue.
After this, the server moves to the "more expensive" of the two
queues.
We will show that a.s. there will be a periodicity in the order of
services, as suggested by the behavior of the corresponding
dynamical systems; we also study the cases (of measure 0) when the
dynamical system is chaotic, and prove that then the stochastic one
cannot be periodic either.
15:45
Exponents of Growth for SPDEs
Abstract
We discuss estimating the growth exponents for positive solutions to the
random parabolic Anderson's model with small parameter k. We show that
behaviour for the case where the spatial variable is continuous differs
markedly from that for the discrete case.
14:15
Degenerate periodic homogenization
Abstract
The probabilistic approach to homogenization can be adapted to fully
degenerate situations, where irreducibility is insured from a Doeblin type
condition. Using recent results on weak sense Poisson equations in a
similar framework, obtained jointly with A. Veretennikov, together with a
regularization procedure, we prove the homogenization result. A similar
approach can also handle degenerate random homogenization.
15:45
On the exit and ergodicity of reflected Levy processes
Abstract
Consider a spectrally one-sided Levy process X and reflect it at
its past infimum I. Call this process Y. We determine the law of the
first crossing time of Y of a positive level a in terms of its
'scale' functions. Next we study the exponential decay of the
transition probabilities of Y killed upon leaving [0,a]. Restricting
ourselves to the case where X has absolutely continuous transition
probabilities, we also find the quasi-stationary distribution of
this killed process. We construct then the process Y confined in
[0,a] and prove some properties of this process.
14:15
Spectral analysis of stochastic lattice and continuous systems
Abstract
A reveiw of results about spectral analysis of generators of
some stochastic lattice models (a stochastic planar rotators model, a
stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new
results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral
analysis of the generator of stochastic continuous particle system. The
construction of one-particle subspaces of the generators and the spectral
analysis of the generator restricted on these subspaces will be the focus of
the talk.
15:45
15:45
Non-central limit theorems in geometric probability
Abstract
Consider a graph with n vertices placed randomly in the unit
square, each connected by an edge to its nearest neighbour in a
south-westerly direction. For many graphs of this type, the centred
total length is asymptotically normal for n large, but in the
present case the limit distribution is not normal, being defined in
terms of fixed-point distributions of a type seen more commonly in
the analysis of algorithms. We discuss related results. This is
joint work with Andrew Wade.
14:15
A particle representation for historical interacting Fisher-Wright diffusions and its applications
Abstract
We consider a system of interacting Fisher-Wright diffusions
which arise in population genetics as the diffusion limit of a spatial
particle model in which frequencies of genetic types are changing due to
migration and reproduction.
For both models the historical processes are constructed,
which record the family structure and the paths of descent through space.
For any fixed time, particle representations for the
historical process of a collection of Moran models with increasing particle
intensity and of the limiting interacting Fisher-Wright diffusions are
provided on one and the same probability space by means of Donnelly and
Kurtz's look-down construction.
It will be discussed how this can be used to obtain new
results on the long term behaviour. In particular, we give representations for
the equilibrium historical processes. Based on the latter the behaviour of
large finite systems in comparison with the infinite system is described on
the level of the historical processes.
The talk is based on joint work with Andreas Greven and Vlada
Limic.
15:45
Front Fluctuations for the one dimensional Stochastic Cahn Hilliard Equation
Abstract
We consider the Cahn Hilliard Equation in the line, perturbed by
the space derivative of a space--time white noise. We study the
solution of the equation when the initial condition is the
interface, in the limit as the intensity of the noise goes to zero
and the time goes to infinity conveniently, and show that in a scale
that is still infinitesimal, the solution remains close to the
interface, and the fluctuations are described by a non Markovian
self similar Gaussian process whose covariance is computed.
14:15
Rough Paths and applications to support theorems
Abstract
After a brief introduction to the basics of Rough Paths I'll
explain recent work by Peter Friz, Dan Stroock and myself proving that a
Brownian path conditioned to be uniformly close to a given smooth path
converges in distribution to that path in the Rough Path metric. The Stroock
Varadhan support theorem is an immediate consequence.
The novel part of the argument is to
obtain the estimate in a way that is independent of the particular norm used
in the Euclidean space when one defines the uniform norm on path space.
14:15
The solutions to a class of non-linear stochastic partial
differential equations
Abstract
In this talk, we consider a class of non-linear stochastic partial
differential equations. We represent its solutions as the weighted
empirical measures of interacting particle systems. As a consequence,
a simulation scheme for this class of SPDEs is proposed. There are two
sources of error in the scheme, one due to finite sampling of the
infinite collection of particles and the other due to the Euler scheme
used in the simulation of the individual particle motions. The error
bound, taking into account both sources of error, is derived. A
functional limit theorem is also derived. The results are applied to
nonlinear filtering problems.
This talk is based on joint research with Kurtz.
15:45
Surface measures on paths in an embedded Riemannian manifold
Abstract
We construct and study different surface measures on the space of
paths in a compact Riemannian manifold embedded into the Euclidean
space. The idea of the constructions is to force a Brownian particle
in the ambient space to stay in a small neighbourhood of the manifold
and then to pass to the limit. Finally, we compare these surface
measures with the Wiener measure on the space of paths in the
manifold.
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