**TOSCA** is a joint project-team between Inria Nancy Grand Est and the University of Lorraine, located in the Institut Elie Cartan de Lorraine (IECL) in Nancy and in the Research Center Inria of Sophia Antipolis.

The TOSCA Project-Team is a follow-up of the **OMEGA** Research-Team that ended on December 2006.

# Team presentation

Most often physicists, economists, biologists and engineers need a stochastic model

because they cannot describe the physical, economical, biological, etc., experiment under

consideration with deterministic systems, either because the experiment has a huge

complexity, or because accurate calibrations of the parameters of the models would

be impossible. Then one abandons attempts to get the description of the state of

the experiment at future times given its initial conditions; instead, one aims to get

a statistical description of the evolution of the system. For example, one desires to

compute occurrence probabilities of critical events such as losses in finance, the overstepping of given thresholds by neuronal electrical potentials, or the fragmentation

up to a very low size of mineral particles in a crusher. By their very nature such problems lead to complex modelling 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 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 model specifications.

We thus develop calibration and simulation methods for general Stochastic Differential Equations whose coefficients and boundary conditions have the singularities which are imposed by Physics or Finance. These singularities make the stochastic equations hard to discretize and estimate.

Using our knowledge on the stochastic integration theory we work on models of interest for the physicists, biologists, engineers, etc., with whom we collaborate.

# Research themes

- Transverse problems
- Long time simulations for nonlinear PDEs.
- Simulations of multivalued models.
- Variance reduction techniques.
- Stochastic partial differential equations.
- Stochastic models in Neurosciences and Biology
- Stochastic models in Finance
- Stochastic models in diffusions in random media
- Stochastic models in Fluid Mechanics and Meteorology
- Stochastic models in Chemical Kinetics
- Softwares, numerical experiments

# International and industrial relations

We have a long term collaboration with

- P. Protter (Cornell University)
- A. Kohatsu-Higa (Osaka University)
- L. Tubaro (Trento University)
- R. Gibson (university of Zurich)
- Chilean probabilists (R. Rebolledo, S. Torres, J. Fontbona and C. Mora)
- A. Veretennikov (Leeds University).

We recently had collaborations with industrial partners:

- Credit Agricole CIB (ex Calyon)
- Natixis