Introduction
The Inria Research team Omega is located both at Inria Sophia-Antipolis and
Inria Lorraine. The team aims to develop and analyze stochastic models and probabilistic
numerical methods. The present fields of application are : finance, neurobiology,
chemical kinetics.
Our competences cover the mathematics behind stochastic modeling and stochastic numerical methods. We also benefit from a wide experimental experience on calibration and simulation techniques for stochastic models, and on the numerical resolution of deterministic equations by probabilistic methods. We pay a special attention to collaborations with engineers, practitioners, physicists, biologists, numerical analysts. See http://www-sop.inria.fr/omega/index.html
Scientific foundations
Most often physicists, economists, biologists, engineers need a stochastic model
because they cannot describe the physical, economical, biological, etc., experiment
under consideration with deterministic systems either because of its complexity
and/or its dimension or because precise measurements are impossible. Then, they
renounce to get the description of the state of the system at future times given
its initial conditions and, instead, try to get a statistical description of
the evolution of the system. For example, they desire to compute occurrence
probabilities for critical events such as overstepping of given thresholds by
financial losses or neuronal electrical potentials, or to compute the mean value
of the time of occurrence of interesting events such as the fragmentation up
to a very low size of a large proportion of a given population of particles.
By nature such problems lead to complex modeling issues : one have to choose
appropriate stochastic models, which requires a thorough knowledge of their
qualitative properties, and then one has to calibrate them, which requires specific
statistical methods to face the lack or the inaccuracy of the data. In addition,
having chosen a family of models and computed the desired statistics, one has
to evaluate the sensitivity of the results to the unavoidable model specifications.
The Omega team, in collaboration with specialists of the relevant fields, develops
theoretical studies of stochastic models, calibration procedures, and sensitivity
analysis methods.
In
view of the complexity of the experiments, and thus of the stochastic models,
one cannot expect to use closed form solutions of simple equations in order
to compute the desired statistics.
Often one even has no other representation than the probabilistic definition
(e.g., this is the case when one is interested in the quantiles of the probability
law of the possible losses of financial portfolios). Consequently the practitioners
need Monte Carlo methods combined with simulations of stochastic models. As
the models cannot be simulated exactly, they also need approximation methods
which can be efficiently used on computers. The Omega team develops mathematical
studies and numerical experiments in order to determine the global accuracy
and the global efficiency of such algorithms.
The
simulation of stochastic processes is not motivated by stochastic models only.
The stochastic differential calculus allows one to represent solutions of certain
deterministic partial differential equations in terms of probability distributions
of functionals of appropriate stochastic processes. For example, elliptic and
parabolic linear equations are related to classical stochastic differential
equations, whereas nonlinear equations such as the Burgers and the Navier--Stokes
equations are related to McKean stochastic differential equations describing
the asymptotic behavior of stochastic particle systems. In view of such probabilistic
representations one can get numerical approximations by using discretization
methods of the stochastic differential systems under consideration. These methods
may be more efficient than deterministic methods when the space dimension of
the P.D.E. is large or when the viscosity is small. The Omega team develops
new probabilistic representations in order to propose probabilistic numerical
methods for equations such as conservation law equations, kinetic equations,
nonlinear Fokker--Planck equations.
Industrial
collaborations and application domains
Since its creation the Omega team collaborated with various industrial companies,
some of them, such as Electricité de France, on a long term basis. We
now describe a short list of past success stories and of new application fields
which we aim to develop in the next future.
Finance
For a long time now the Omega team has collaborated with financial institutions
and researchers in finance and insurance. We are particularly interested in
calibration methods, risk analysis (especially model risk analysis), optimal
portfolio management, Monte Carlo methods for option pricing and risk analysis,
asset and liabilities management. We also work on the partial differential equations
related to financial issues, for example the stochastic control Hamilton--Jacobi--Bellman
equations. We study existence, uniqueness, qualitative properties and appropriate
deterministic or probabilistic numerical methods. At the time being we pay a
special attention to the financial consequences induced by modeling errors and
calibration errors on hedging strategies and portfolio management strategies.
We
list here some of our industrial collaborations in this area : Caisse des Dépots,
Fédération Française des Sociétés d'Assurance,
PREDICA, Crédit Agricole IndoSuez, Risklab, Gaz de France etc.
Fluid Mechanics
In Fluid Mechanics the Omega team develops probabilistic methods to solve vanishing
vorticity problems and to study the behavior of complex flows at the boundary,
and their interaction with the boundary. We elaborate and analyze stochastic
particle algorithms. Our expertise concerns
Random media
and changes of scales
A random medium is a material with a lot of heterogeneities which can be described
statistically only because of its complexity or because it is partially unknown.
A typical example is a fissured porous media with rocks of different types.
Its permeability is discontinuous, rapidly varying and generally unknown. Similarly,
it is interesting to consider as random media turbulent fluids and unknown or
deficient materials in which polymers evolve or waves propagate.
Generally, a random medium is described at a small scale. A lot of techniques, such as the techniques issued from the homogenization theory, allow one to approximate some properties of the media by a simpler model at a higher scale by using its statistical properties. Although some of these change of scales techniques can be done analytically, most often a practical description of the media at a higher scale can only be achieved by using intensive numerical computations, including Monte Carlo methods.
The Omega team
has a good expertise in analytical and numerical computations of change of scales
techniques. In addition, the Omega team is now developing a new class of Monte
Carlo methods aiming to simulate diffusion phenomena in discontinuous media
with applications to inverse problems in electro-encephalography and geophysics.
Chemical kinetics
The areas in which coagulation and fragmentation models appear are numerous:
For all these applications we are led to consider kinetic equations using coagulation
and fragmentation kernels (a typical example being the kinetics of polymerization
reactions). The
Omega team aims to analyze and to solve numerically these kinetic equations.
By using a probabilistic approach we describe the behavior of the clusters in
the model and we develop original numerical methods. Our approach allows to
intuitively understand the time evolution of the system and to answer to some
open questions raised by physicists and chemists.
More precisely,
we can compute or estimate characteristic reaction times such as the gelification
time (at which there exists an infinite sized cluster) the time after which
the degree of advancement of a reaction is reached, etc.
Softwares
As explained above, one of the aims of the Omega team is to develop new Monte
Carlo methods from rigorous mathematical analyses of models.
We take benefit from our strong experience on the programming of probabilistic
algorithms on various architectures including intensive computation architectures
(see our joint collaboration <<Amazone>> with Bull company).
We
have developed demonstrators such as the LICS software for financial and insurance
problems and a solver for diphasics fluids within a collaboration with Electricité de France, etc. We also are interested in designing algorithms of resolution
of specific equations in relationship with practitioners.
French
teams similar to Omega
Three other French research groups have interests comparable to those of Omega
: