Omega is located both at INRIA Lorraine and INRIA Sophia-Antipolis.

On October 26th Axel Grorud suddenly died. Axel was a member of the Omega team since
its creation in 1994. He brought us his mathematical skills and his thorough knowledge
in finance. He also brought us his warm friendship. We all miss him a lot.
We dedicate this report to his memory.

The Inria Research team Omega is located both at Inria Sophia-Antipolis and Inria Lorraine.
The team develops and analyzes stochastic models and probabilistic numerical methods.
The present fields of applications are
in 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 and numerical analysts.

Concerning the probabilistic resolution of linear and nonlinear partial differential equations,
the Omega team studies Monte Carlo methods, stochastic particle methods and ergodic methods.
For example, we are interested in fluid mechanics equations
(Burgers, Navier-Stokes, etc.), in equations of chemical kinetics and in homogeneization problems for
PDEs with random coefficients.

We develop simulation methods which take into account the boundary conditions. We provide non asymptotic error estimates in order to describe the global numerical error corresponding to each choice of numerical parameters: number of particles, discretization step, integration time, number of simulations, etc. The key argument consists in interpreting the algorithm as a discretized probabilistic representation of the solution of the PDE under consideration. Therefore part of our research consists in constructing probabilistic representations which allow us to derive efficient numerical methods. In addition, we validate our theoretical results by numerical experiments.

In financial mathematics and in actuarial science, Omega
is concerned by market modelling and specific Monte Carlo methods.
In particular we study calibration questions,
financial risks connected with modelling errors, and the dynamical control of such risks.
We also develop numerical methods of simulation to compute prices and sensitivities of various
financial contracts.

In neurobiology we are concerned by stochastic models which describe the neuronal activity. We also develop a stochastic numerical method which will hopefully be useful to the Odyssée project to make more efficient a part of the inverse problem resolution whose aim is to identify magnetic permittivities around brains owing to electro-encephalographic measurements.

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 has 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 of data or the inaccuracy of these 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 PDE 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.

Omega is interested in developing stochastic models and probabilistic numerical methods.
Our present motivations come from Fluid Mechanics, Chemical Kinetics,
Finance and Biology.

In Fluid Mechanics Omega develops probabilistic methods to solve
vanishing vorticity problems and to study complex flows
at the boundary, in particular their interaction with the boundary.
We elaborate and analyze stochastic particle algorithms. Our expertise concerns

The convergence analysis of the stochastic particle methods on theoretical test cases. In particular, we explore speed up methods such as variance reduction techniques and time extrapolation schemes.

The design of original schemes for applicative cases. A first example concerns the micro-macro model of polymeric fluid (the FENE model). A second one concerns the Lagrangian modeling of turbulent flows and its application in combustion for two–phase flows models (joint collaboration with Électricité de France).

The Monte Carlo methods for the simulation of fluid particles in a fissured (and thus discontinuous) porous media.

An important part of the work of the Omega team
concerns the coagulation and fragmentation models.

The areas in which coagulation and fragmentation models appear are numerous : polymerization, aerosols, cement and binding agents industry, copper industry (formation of copper particles), behavior of fuel mixtures in engines, formation of stars and planets, population dynamics, etc.

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.

For a long time now Omega has collaborated with researchers and practitioners
in various financial institutions and insurance companies. We are particularly interested in
calibration problems, 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.

For a couple of years Omega has studied stochastic models in biology, and developed
stochastic methods to analyze stochastic resonance effects and to solve inverse problems.
For example, we are concerned by the identification of an elliptic operator
involved in the calibration of the magnetic permittivity owing to electro-encephalographic
measurements. This elliptic operator has a divergence form and a discontinuous coefficient.
The discontinuities make difficult the construction of a probabilistic interpretation
allowing us to develop an efficient Monte Carlo method for the numerical resolution of the
elliptic problem.

The MEG problem, which consists in estimating the conductivity
coefficients of the different brain layers, is an inverse problem that
the Odyssee project at INRIA solves numerically using an iterative
algorithm. At each step of the algorithm, it is necessary to compute a
few values of the solution of a particular elliptic PDE
defined in the brain

The matrix

However, in the one-dimensional context, the stochastic process
associated to

where

This year, we have continued our work on the modeling of stochastic resonance effects in the neuronal activity. We have to face important technical difficulties due to the huge complexity of the analytical formulae describing the probability densities of particular stopping times related to our model for the electric potential along the neurons. We are trying to deal with approximate formulae which would allow us to quantify the level of random noise which should be added to the internal noise in order to improve the efficiency of the neuronal activity, in the sense that the period of an electrical input signal is better recognized by the neuronal system.

The well known bootstrap method belongs to the family of modern statistical tools exploiting Monte Carlo simulations to obtain precise estimators and powerful tests for complex models. It has been used for over twenty years to refine the performances of estimators in various statistical settings. Yet, as far as we know, very little is known on the theory of the bootstrap in the context of Brownian diffusions. Thus, O. Bardou and D. Talay propose a new methodology to construct bootstrap corrections to the maximum likelihood estimators of diffusion processes.

Let

The process

Exploiting the self normalized martingale structure of this expression, we are able to construct a bootstrap procedure correcting for the bias and the coverage errors of confidence intervals of

Let

The process

The coefficient

O. Bardou and M. Martinez have also extended their results to the case of several discontinuities,
and now aim to introduce a drift term in the dynamics of the process

S. Maire has worked on sequential Monte Carlo methods to compute approximations
and integrals of multivariate real-valued functions. In

In collaboration with Emmanuel Gobet (CMAP, École Polytechnique),
S. Maire has developed an adaptive Monte Carlo method to compute a spectral approximation
of the solution of the Poisson equation. The variance and the
bias due to the simulations of the involved stochastic processes
geometrically decrease when the number of steps increases. The method is compared to deterministic
methods (see

In

In collaboration with Arturo Kohatsu-Higa (Universitat Pompeu Fabra, Barcelona), M. Bossy and D. Talay work on Romberg extrapolation methods for stochastic particle methods for McKean–Vlasov equations. The aim is to accelerate the convergence rate of these methods with respect to the time discretization step.

In the particular case of the Burgers equation with a smooth initial condition, it is shown that the Romberg extrapolation leads to a discretization scheme of order 2. A paper is in preparation.

In collaboration with P. Calka (Université Paris 5)
and A. Mézin (École des Mines, Nancy), P. Vallois
works on a stationary process on the real line which models
the positions of the multiple cracks that are observed in
composite materials submitted to a fixed unidirectional stress

A paper was submitted in June 2003 to ``Stochastic Processes and their
Applications''

A. Diop, M. Bossy and D. Talay have finished to study
the discretization of generalized Cox–Ingersoll–Ross and Hull–White models
for instantaneous interest rates. In these models the
drift coefficient has bounded derivatives whereas the diffusion coefficient
is of the type

According to the Black-Scholes model, stock prices are
exponentials of Brownian motions with drifts. The drifts are the
instantaneous expected rates of return of the stocks.
Thus, it seems possible to construct an estimator of an unknown drift
based upon the successive price amplitudes and
to provide a rigorous mathematical framework for
technical analysis methods used by practitioners in financial institutions.
To this end, in

In collaboration with Rajna Gibson (Zürich University) and Christophette Blanchet (Université de Nice Sophia-Antipolis), A. Diop, É. Tanré, D. Talay, M. Martinez and D. Rovanova elaborate an appropriate mathematical framework to develop the analysis of the financial performances of some financial techniques which are often used by the traders; to study the impact of model risk on such strategies and to study the question: is it possible to improve some techniques used in practice by adding mathematical models? We are finishing to prepare a paper on our results. The involved mathematical techniques are issued from statistics of random processes and stochastic control. This research is funded by NCCR FINRISK (Switzerland) and is a part of its project ``Conceptual Issues in Financial Risk Management''.

In addition, Ouaile El Fetouhi and D. Talay have developed an original and promising model for the VWAP which is a financial index that the traders try to beat. This model may allow one to analyse the performances of the VWAP techniques.

In collaboration with Nadia Maizi (CMA, École des Mines de Paris), Geert Jan Olsder (Delft University, the Netherland) and Odile Pourtallier (Comore and Miaou projects, INRIA Sophia Antipolis), M. Bossy and É. Tanré have applied game theory to model the market of electricity.

The deregulation of the market of electricity in European countries,
initiated in December 1996, has raised lot of modifications,
in particular new spot markets of electricity have emerged.
These markets are close to the pollution right markets
that start to appear as a consequence of the application of the Kyoto protocol
and thus are very peculiar. As, in addition, the electricity cannot be stored,
the classical market analysis methods do not apply,
and new approaches need to be explored.
We have analyzed a simple model with one market
and

In collaboration with A. Volpi (ESSTIN), P. Vallois works on the ruin time of insurance companies.
Let

for all

When

Suppose now that

where

We also study the rate of decay of the ruin probability
and prove that, after a suitable normalization, the triplet

Two papers have been submitted to the ``Annales de l'Institut
Henri Poincaré''

This work is originally motivated by a desire to understand precisely what, among all possible convex quantities, singles out the Shannon entropy. In the framework of distribution functions with a certain number of prescribed moments, J-F. Collet gives an explicit relation between the structure of the minimizer (say, a Gaussian distribution) and the quantity minimized at equilibrium (say, the Shannon entropy). This correspondance being established, properties of the entropy may be read-off from the expression of the minimizer. We then show that the entropy satisfies a natural homogeneity property if and only if the minimizer is Gaussian. If one then relaxes the homogeneity assumption in a natural way, a new distribution may arise, which turns out to be the Tsallis distribution. Besides providing a new characterization of this ditribution, this establishes a link between classical moment systems and non extensive thermodynamics, which we plan to investigate in the future.

In many examples of dissipative systems arising in probability theory (for instance some Kolmogoroff equations associated to stationary Markov processes), the uniqueness of the equilibrium measure together with a dissipation property (e.g. the existence of a Lyapunov functional) may be used to derive trend to equilibrium for large times. We are interested in systems which do not possess any invariant measure. For some of them J-F. Collet has shown that some quantities do decrease as time increases, a fact which may be used to study the large time asymptotics. A typical example is that of linear parabolic PDEs with time-dependent coefficients. In some cases this may be used to yield very quick proofs of the existence of some intermediate asymptotics.

The phycisist Smoluchowski introduced in 1917 a mathematical model which describes coagulation phenomena. It has many applications such as polymerization, formation of stars and planets, behavior of fuel mixtures in engines, etc. This system describes a non linear evolution equation of infinite dimension.

The aim of the probabilistic approach to this system is to give new results or to confirm conjectures formulated by analysts or physicists, with the methods of stochastic analysis.

The model

The equation describes the dynamics of an infinite system of particles in which coagulation phenomena occur.
The particles are characterized by their mass. From a physical point of view, it is natural to suppose that
the rate of coagulation of two particles depends on their masses.

Let us denote by

The Smoluchowski coagulation equation gives the time evolution of

where coagulation kernel. We assume that

Due to the presence of the infinite series, this problem is not a classial initial value problem for a system of non linear ordinary differential equations.

Results

In represents the solution of

Thanks to this representation, we study the asymptotic behaviour of the solution in the particular case
when the kernel is homogeneous

In a more complicated model, we also allow particles to break into two particles (phenomenon of fragmentation). In this context, during the summer school CEMRACS 2003, Francis Filbet (CNRS-Orléans) and É. Tanré studied the large time behaviour of the solution. When the fragmentation and coagulation kernels are well chosen, every solution converges numerically to a stationary solution. We endeavor to prove this convergence.

This methodology was successfully applied to a problem originating in the modelling of industrial crushers. More precisely, a fundamental problem related to the optimization of the crushing process is to estimate the minimum amount of time required to achieve a prescribed degree of crushing. This question was suggested to us by R. Rebolledo (Pontificia Universidad Católica de Chile) during his visit in the framework of the INRIA-CONICYT collaboration programme. By modelling crushing as a pure fragmentation process, we designed an algorithm which yields a method for the computation of residence tumes in crushers.

Another generalization of Smoluchowski's model is obtained when one takes into account the position of particles as a supplementary
variable (spatially non-homogeneous model). Nicolas Fournier (Université Nancy 1), B. Roynette and É. Tanré have proved
the almost sure convergence of the position to 0 and of the mass to infinity with a particular choice
of diffusion for the position process.

In this section we present our results on issues which are more abstract than the preceding ones and, at first glance, might appear decorrelated from our applied studies. However most of them are originally motivated by modelling problems, or technical difficulties to overcome in order to analyse in full generality stochastic numerical methods or properties of stochastic models.

The theory of rough paths allows one
to define stochastic integrals path by path as continuous functionals
of their integrands (see, e.g.,

The aim of this study is the development of a generalized
stochastic calculus applied to processes which are not
semimartingales. We especially focus on extension of Itô's
formula to the fractional Brownian motion

Recall that if the Hurst exponent

where

I. Nourdin studies the one-dimensional stochastic differential equation

where

P. Vallois and P. Salminen (University of Abo) consider first range times (with randomised
range level) of a linear diffusion on

We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is shown that squared Bessel processes with drift are serving hereby as a Markovian element.

Let

If

The knowledge of the rate of decay of

where

A paper is accepted for publication

In collaboration with Gerard Lorang (who has a permanent position in Luxembourg),
Christophe Ackermann and Bernard Roynette
consider a Bernoulli symmetric random walk

We continue our collaboration with the company Électricité de France on the process of formation of electricity prices on local spot markets directed by several producers. We aim to model the spot prices on the different markets resulting from a production equilibrium between the asks and the producers. We have elaborated simple assumptions on the main components of the problem, that is, the ask and bid functions. We fully describe the possible equilibria.

The aim of our collaboration with Gaz de France is to simulate the possible future prices of contracts related to exchange rates and gas and oil indices. We are interested in continuous time stochastic differential models of the indices and the exchange rates involved in the contracts.

The first task was to calibrate the model. We thus have estimated the volatility and the drift coefficient using various non-parametric estimators. In addition, since we are interested in the simultaneous evolution of the indices and the exchange rates, we are also interested in finding the correlations between the different stock prices. We therefore had to estimate the number of Brownian motions driving the stochastic evolution of the indices and the exchange rates. We have finally proposed a model to Gaz de France. Our partners have approved this model.

We now focus on variance reduction techniques which may allow us to improve the efficiency of Monte Carlo simulations to compute the prices and sensitivities of the contracts.

A. Lejay is co-responsible with Iraj Mortazavi (Université Bordeaux 1) for the project ``Probabilistic models and particles methods for nuclear waste disposal'' within the ``Groupe de Recherche MOMAS''.

The team Omega participates to the ``Groupe de Recherche GRIP'' on stochastic interacting particles. D. Talay serves as a member of the scientific committee of this GdR.

D. Talay serves as Associated Editor of:
Stochastic Processes and their Applications,
Annals of Applied Probability,
ESAIM Probability and Statistics,
Stochastics and Dynamics,
SIAM Journal on Numerical Analysis,
Mathematics of Computation,
Monte Carlo Methods and Applications,
Oxford IMA Journal of Numerical Analysis,
Stochastic Environmental Research and Risk Assessment.

M. Bossy is member of the administration committee of the French Society of Applied Mathematics (SMAI). M. Deaconu is member of the scientific committee of the MAS group (Probability and Statistics) within SMAI.

M. Deaconu is member of the ``Comité des Projets'' of LORIA and INRIA Lorraine and of the ``Commission pour les postes d'accueil'' of INRIA Lorraine.

M. Deaconu is member of the ``Conseil du laboratoire'' of the Institut Élie Cartan and of the ``Commission de spécialistes'' of the Mathematics Department of Université Nancy 1.

M. Deaconu and A. Lejay are responsibles for the organization of the conference Journées
MAS 2004 to be held in Nancy in September 2004.

M. Bossy, M. Deaconu and E. Tanré are the organizing committee of the international conference MC2QMC wich will be held in Juan-les-Pins, June 7-10, 2004 (MC2QMC is the Sixth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, twinned with the Second International Conference on Monte Carlo and Probabilistic Methods for Partial Differential Equations). D. Talay and H. Niederreiter (University of Singapore) co–chair the Conference.

M. Deaconu and D. Talay are permanent reviewers for the Mathematical Reviews.

A. Lejay is member of the ``Commission des moyens informatiques'' of INRIA Lorraine.

A. Lejay and Iraj Mortazavi (Université Bordeaux I) are organizing
a workshop Probabilistic models and particles
methods for nuclear waste disposal to be held in Paris in January 2004.

B. Roynette is the head of the Mathematics Department of Université Nancy 1.

P. Vallois is the head of the Probability and Statistics group of Institut Élie Cartan.

P. Vallois is member of the council of the UFR STMIA.

D. Talay has a part time position of Professor at École Polytechnique. He also teaches probabilistic numerical methods at Université Paris 6 (DEA de Probabilités) and within the FAME Ph.D. program (Switzerland).

M. Bossy gives a course on ``Stochastic calculus and financial mathematics'' in the DESS IMAFA (``Informatique et Mathématiques Appliquées à la Finance et à l'Assurance'', Université de Nice Sophia Antipolis), and a course on ``Risk management on energetic financial markets'' in the master ``Ingénierie et Gestion de l'Energie'' (École des Mines de Paris) at Sophia-Antipolis.

M. Bossy and É. Tanré give the course on ``Stochastic modeling and financial applications'' in the DEA de Mathématiques of Université de Nice Sophia Antipolis.

M. Deaconu gives a 30h course in Mathematical Finance at the IUP Sciences Financières of Université Nancy 2.

P. Vallois gives courses in Mathematical Finance in a DESS programme and at the IUP Sciences Financières. He also gives the DEA course ``Introduction to the Brownian motion and continuous martingales'' at Université Nancy 1.

Christophe Ackermann defended his Ph.D. thesis
entitled Processus associés à l'équation de diffusion
rapide ; indépendance du temps et de la position pour un processus
stochastique at Université Nancy 1 in December 2003.

Christophe Berthelot defended his Ph.D. thesis entitled Évaluation d'une architecture
vectorielle pour des méthodes de Monte-Carlo. Analyse probabiliste de conditions au bord
artificielles pour des inéquations variationnelles at the
Université Paris 6 in September 2003.

Awa Diop defended her Ph.D. thesis entitled Sur la discrétisation et le comportement
à petit bruit d'EDS unidimensionnelles dont les coefficients sont à
dérivées singulières at Université de Nice Sophia-Antipolis in May 2003.

Agnès Volpi defended her Ph.D. thesis entitled Étude asymptotique de temps de ruine
et de l'overshoot at Université Nancy 1 in June 2003.

Marian Ciuca defended his Ph.D. thesis entitled Estimation paramétrique sous contraintes.
Applications en finance stochastique at Université de Provence in December 2003.

M. Bossy gave a seminar leacture at the Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, in February 2003.

A. Lejay has given lectures at the workshop on Numerical homogenization in porous media in December 2003; at the IV IMACS Seminar on Monte Carlo methods in Berlin in September 2003; at the Congresso de Matematicas Capriocaron (COMCA 2003) at Antofagasta (Chile) in August 2003; at the probability seminar of the Institut Élie Cartan in April 2003; at the SIAM Conference on Mathematical and Computational Issues in Geosciences at Austin in March 2003.

A. Lejay gave in mini-lecture at the Ist Winter School of Stochastic Analysis
and Statistics at Valparaiso in August 2003.

I. Nourdin has given a lecture at the Journées
de Probabilités at Toulouse in September 2003.

B. Roynette has given lectures at the Journées processus aléatoires et particules at Orléans
in March 2003; at the Journées de Probabilités at Toulouse in September 2003;
at the Colloque international de mathématiques,
analyse et probabilités at Hammamet (Tunisia) in December 2003.

D. Talay has given two plenary lectures: 21th IFIP TC 7 Conference on System Modeling and
Optimization and IVth IMACS Seminar on Monte Carlo Methods MCM-2003.
He also gave a lecture at the MS
Durham Symposium on Markov Chains at the University of Durham (Grande Bretagne),
at the workshop `Stochastic Processes and Applications to Mathematical Finance'
at the Ritsumeikan University (Kyoto, Japan) and at the meeting Journées processus aléatoires
et particules at Université d'Orléans.

P. Vallois has given lectures at the meeting Journées
processus aléatoires et particules in Orléans
in March 2003; at the International Seminar on stability problems for stochastic
models at Pamplona (Spain) in May 2003; at the 11-th Journées Évry-Nancy-Strasbourg
in May 2003; at the Journées de Probabilités at Toulouse in September 2003;
at the Winter school, Analyse stochastique and applications
in Marrakech in December 2003.

A. Lejay spent two weeks in Chile within the INRIA-CONICYT collaboration (July/August). He has also been invited one week by Prof. T. Lyons at the Mathematical Institute of Oxford University.

É. Tanré has participated to the Summer School CEMRACS 2003.

The seminar Théorie et applications numériques des processus
stochastiques organized at Sophia-Antipolis by M. Bossy has received
the following speakers: Larbi Alili (ETH Zürich),
Vlad Bally (Université de Marne la Vallée),
Sylvain Maire (Université de Toulon),
Florent Malrieu (Université de Rennes),
Carlos M. Mora (Universidad de Concepcion, Chile),
Rolando Rebolledo (Universidad Católica de Chile, Chile),
Maria-Soledad Torres (Universidad de Valparaiso, Chile), Pierre-Louis Lions (Collège de France).

The seminar Mathématiques financières organized at
Sophia-Antipolis by M. Bossy has received the following speakers:
Christophette Blanchet (Université de Nice Sophia-Antipolis), Sébastien Chaumont
(Université Nancy 1), Gaël Giraud (Université de Strasbourg),
Benjamin Jourdain (ENPC–CERMICS), Pierre-Louis Lions (Collège de France).