Section: Overall Objectives

Overall Objectives

The team develops and analyzes stochastic models and probabilistic numerical methods. The present fields of applications are in fluid mechanics, molecular dynamics, chemical kinetics, neuroscience, population dynamics, and financial mathematics.

The problems where stochastic models arise are numerous, and the critical reasons for which stochastic models are used make analysis and simulations difficult.

The Tosca team thus aims to develop calibration and simulation methods for stochastic models in cases where singularities in the coefficients or boundary conditions make them hard to discretize and estimate. For this, we are willing to tackle theoretical and numerical questions which are motivated by real applications.

We are interested in developing stochastic numerical methods and transverse methodologies that cover several fields of applications, instead of having chosen a particular field of application (e.g., Biology, or Fluid Mechanics, or Chemistry). We justify this way to proceed as follows:

• For a couple of years now, we have attacked singular problems to answer questions coming from economists, meteorologists, biologists and engineers with whom we collaborate within industrial contracts or research programs such as ACI, ANR, GDR. To solve their problems which are so complex that stochastic processes are involved in the modelling, these colleagues need to combine expertise and knowledge in many fields: deterministic computing, computer science, vision, algorithm analysis, etc. We are incompetent in these fields, and therefore we could not pretend to fully treat any of these problems. A contrario, we are requested to bring our expertise in stochastic modelling and simulation to extremely various domains of applications.

• In spite of this diversity, whatever the application is, one has to simulate stochastic processes as solutions to equations of the type

  $\left\{\begin{array}{cc}\hfill X_t\left(\omega \right)& =X_0\left(\omega \right)+\left({\int }_0^t{\int }_{{ℝ}^d}b(X_s,y){\mu }_s\left(dy\right)ds\right)\left(\omega \right)\hfill \\ & +\left({\int }_0^t{\int }_{{ℝ}^d}\sigma (X_s,y){\mu }_s\left(dy\right)d{Z}_s\right)\left(\omega \right),\hfill \\ \hfill {\mu }_s& =\text{Law}\text{of}{X}_s\text{for}\text{all}s\ge 0,\hfill \end{array}\right.$

  (1)

  in order to compute statistics of the laws of functionals of these solutions. In addition, several fields often produce very similar “pathologies” of the model (1 ) or of the statistics to compute: for example, Pope's Lagrangian stochastic particles in Fluid Mechanics and models in Molecular Dynamics produce the same degeneracy in (1 ), namely, one has to substitute `conditional law of components of $X_s$ given the other ones' for `law of $X_s$'; as well, when studying chartist strategies in Finance and stochastic resonance in the electrical working of neurons, we encounter close questions on the density functions of the random passage times of processes $\left(X_t\right)$ at given thresholds.

• Theory and numerical experiments show that each `pathology' of the model (1 ) requires specific analysis and numerical methods. However, they require common abstract tools (Malliavin calculus, propagation of chaos theory, nonlinear PDE analysis, etc.) and common numerical methodologies (stochastic particle systems, Monte Carlo simulations, time discretization of stochastic differential equations, etc.). Thus each application takes benefit from the modelling and numerical knowledge developed for all the others.

The Tosca team is currently studying models in relation with Geophysics, Neuroscience, Fluid Mechanics, Chemical Kinetics, Meteorology, Molecular Dynamics, Population Dynamics, Evolutionary Dynamics and Finance. We also construct and study stochastic particle systems for Fluid Mechanics, coagulation–fragmentation, stationary nonlinear PDEs, variance reduction techniques for Monte-Carlo computations and numerical methods combining deterministic and stochastic steps to solve nonlinear PDEs in Finance.