All Years Seminars
[INMA] 2007-04-19 (16.30pm) : Colloquium CESAME/MAPA "Integral stochastic orderings for comparing actuarial risks and dependence structures."
At Auditoire Ch.-J de La Vallée Poussin (cycl 01)
Speaker :
Michel Denuit (UCL)
Abstract :
This lecture aims to discuss various stochastic inequalities and their applications in risk management.
Broadly speaking, a stochastic ordering is an order relation that allows to compare random variables.
Integral stochastic orderings are defined by reference to classes of measurable functions and inherit
their properties from the defining class. According to the class of functions under consideration, integral
stochastic orderings aim to express intuitive ideas like "being smaller than" or "being less variable than", for instance.
Integral stochastic orderings can also be used to compare random vectors. In this case, it is often convenient to distinguish
the marginal behavior from the underlying dependence structure (described by the copula function in the continuous case).
Dependence orders allow to compare the strength of dependence and to express the fact that the components of some random
vector are « more positively dependent » than those of another random vector. In the management of insurance portfolios, the main
risk is the simultaneous occurrence of large losses. This behavior is appropriately described by various notions of positive dependence,
like co-monotonicity, multivariate total positivity, conditional increasingness, association, positive quadrant dependence, etc. Such dependence
notions are closely related to dependence orders. Several applications of stochastic orders are discussed.
[INMA] 2007-04-17 (14.00) : Clinical data based optimal STI strategies for HIV : a reinforcement learning approach
At Auditoire Euler
Speaker :
Guy-Bart Stan (Cambridge University)
Abstract :
This research addresses the problem of computing optimal structured treatment interruption strategies for HIV infected patients.
We show that reinforcement learning may be useful to extract such strategies
directly from clinical data, without the need of an accurate
mathematical model of HIV infection dynamics. To support our claims,
we report simulation results obtained by running a recently proposed
batch-mode reinforcement learning algorithm, known as fitted Q
iteration, on numerically generated data.
[INMA] 2007-03-27 (14.00) : "The equations of self-similarity and the joint spectral radius"
At Auditoire Euler
Speaker :
Protasov, Vladimir
Abstract :
The joint spectral characteristics of linear operators such as the joint spectral radius, p-radius, the Lyapunov exponent etc.
have found numerous applications.
The computing or estimating these characteristics are known to be hard. We present a new approach to this problem based
on the so-called self-similarity equation. Equations of this type naturally appear in the wavelets theory, approximation theory,
dynamical systems etc. We give the definition of general self-similarity equation and prove that for any family of affine operators this equation
has at most one L_p - solution. This solution exists precisely when the p-radius of this family is less then one, in which case the solution is unique and continuously depend on the
operators. Approximating recursively this solution we can estimate the p-radius, including the joint spectral radius (if p is infinite).
To this end we generalize the notion of Barabanov's norm for p-radii for any p.
As a corollary we derive a criterion for the L_p-contractibility property of linear operators and estimate the asymptotic growth of orbits for any point.
This, in particular, gives a geometric interpretation of the p-radius as a spectral radius of a certain infinite-dimensional operator.
[INMA] 2007-03-22 (16.30pm) : Colloquium CESAME/MAPA "Four examples of how mathematics is useful, even essential, for understanding complex systems."
At Euler building - local a.207
Speaker :
Robert Mackay (Warwick university)
Abstract :
Four examples, from my own and others' work, are presented to demonstrate that mathematics is
useful and essential for understanding complex systems:
1. Adding new road capacity can increase travel time for everyone;
2. Self-sustaining clusters can emerge even when it appears death rate exceeds reproduction;
3. Everlasting self-localised excitations can occur even when linear theory would suggest they would radiate away, and media which are insulating for linear theory can be conducting nonlinearly;
4. Indecomposable spatially extended deterministic systems (as well as stochastic ones) can exhibit non-unique probabilistic behaviour and sensitivity to boundary conditions.
[INMA] 2007-03-20 (14pm) : "A lecture on homogeneity"
At Auditoire Euler
Speaker :
Tuna Emre (ULg)
Abstract :
Being a superclass of linear systems, homogenous systems can serve as better approximations to certain nonlinear systems. Their special
structure lets us study them locally to get a global idea. In this talk we give an overview of homogeneity and homogeneous systems with
emphasis on numerically realizable optimization schemes. Starting from the classical definition of dilations (and homogeneity with respect to
them), we take a natural path to generalization and obtain broader definitions which let us generalize previous results. Following that,
for a homogeneous discrete-time system, we consider an infinite horizon optimization problem (the solution of which is a control
Lyapunov function) and present a numerical algorithm to obtain a global solution. Later, we point our attention to switched systems.
We provide a method to generate a feedback law that stabilizes a homogeneous system under arbitrary switching. To the end, we talk
about the growth rate of a switched homogeneous system which is the analog of joint spectral radius of switched linear systems.
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