Tosca-NGE aims to significantly contribute to discern and
explore new horizons for stochastic modeling. To this end we need to
better understand the issues of stochastic modeling and the
objectives pursued by practitioners who need them: we thus need to deeply
understand other scientific fields than ours (e.g., Fluid Mechanics,
Ecology, Biophysics) and to take scientific risks.
Indeed, these risks are typified by the facts that often new and complex models do not behave as expected, mathematical and numerical difficulties are harder to overcome than forecast, and the increase of our knowledge in target fields is slower than wished.

In spite of these risks we think that our scientific approach is relevant for the following reasons:

To bring relevant analytical and numerical answers to the preceding problems, we feel necessary to attack in parallel several problems arising from different fields. Each one of these problems contributes to our better understanding of the advantages and limitations of stochastic models and algorithms.

Of course, this strategy allows each researcher in the team to have
her/his own main topic. However
we organize the team in order to maximize internal collaborations.
We consider this point, which justifies the existence of
Inria project-teams, as essential to the success of our programme
of research. It relies on the fact that,
to develop our mathematical and numerical studies, we share
a common interest for collaborations with engineers, practitioners,
physicists, biologists and numerical analysts, and we also share
the following common toolbox:

We finally emphasize that the unifying theme of our research is to develop analytical tools that can be effectively applied to various problems that come from extremely diverse subjects. For example, as described in more detail below, we study: branching processes and their simulation with the view of advancing our understanding of population dynamics, molecular dynamics, and cancer models; the theory and numerical analysis of McKean-Vlasov interacting particle systems in order to develop our models in biology, computational fluid dynamics, coagulation and fragmentation; hitting times of domains by stochastic processes so that we can improve on the current methods and theory used in finance and neuroscience.

Our recent directions of research, combining stochastic modelling and data analysis, will conduct to a new Inria team, named PASTA. PASTA research program is focused on spatio-temporal stochastic processes and their applications.

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 of its complexity and/or its
dimension or because precise measurements are impossible. Therefore, they abandon trying to get
the exact description of the state of the system at future times given its initial
conditions, and try instead to get a statistical description of the evolution of the system.
For example, they desire to compute occurrence probabilities for critical events such as the
overstepping of a given threshold 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 to a very small size of a large proportion of a given population of particles.
By nature such problems lead to complex modelling issues: one has to choose appropriate
stochastic models, which require 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 Tosca 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
Tosca 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 (SDEs), 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 Tosca team develops new probabilistic representations in order
to propose probabilistic numerical methods for equations such as conservation law equations,
kinetic equations, and nonlinear Fokker–Planck equations.

Tosca-NGE is interested in developing stochastic models and probabilistic numerical methods. Our present motivations come from
models with singular coefficients, with applications in Geophysics, Molecular Dynamics and Neurosciences; Population Dynamics, Evolution and Genetics; and Financial Mathematics.

Stochastic differential equations with discontinuous coefficients arise in Geophysics, Chemistry, Molecular Dynamics, Neurosciences, Oceanography, etc. In particular, they model changes of diffusion of fluids, or diffractions of particles, along interfaces.

For practitioners in these fields, Monte Carlo methods are popular as they are easy to interpret — one follows particles — and are in general easy to set up. However, dealing with discontinuities presents many numerical and theoretical challenges. Despite its important applications, ranging from brain imaging to reservoir simulation, very few teams in mathematics worldwide are currently working in this area. The Tosca project-team has tackled related problems for several years providing rigorous approach. Based on stochastic analysis as well as interacting with researchers in other fields, we developed new theoretical and numerical approaches for extreme cases such as Markov processes whose generators are of divergence form with discontinuous diffusion coefficient.

The numerical approximation of singular stochastic processes can be combined with backward stochastic differential equations (BSDEs) or branching diffusions to obtain Monte Carlo methods for quasi-linear PDEs with discontinuous coefficients. The theory of BSDEs has been extensively developed since the 1980s, but the general assumptions for their existence can be quite restrictive. Although the probabilistic interpretation of quasi-linear PDEs with branching diffusions has been known for a long time, there have been only a few works on the related numerical methods.

Another motivation to consider stochastic dynamics in a discontinuous setting came to us from time evolution of fragmentation and coagulation phenomena, with the objective to construct stochastic models for the avalanche formation of soils, snow, granular materials or other geomaterials. Most of the models and numerical methods for avalanches are deterministic and involve a wide variety of physical parameters such as the density of the snow, the yield, the friction coefficient, the pressure, the basal topography, etc. One of these methods consists in studying the safety factor (or limit load) problem, related to the shallow flow of a visco-plastic fluid/solid with heterogeneous thickness over complex basal topography. The resulting nonlinear partial differential equation of this latter theory involves many singularities, which motivates us to develop an alternative stochastic approach based on our past works on coagulation and fragmentation. Our approach consists in studying the evolution of the size of a typical particle in a particle system which fragments in time and to study particular fragmentation kernels.

Diffusion hitting times are of great interest in many domains as in finance (a typical example is the study of barrier options), in Geophysics and Neurosciences. On the one hand, analytic expressions for hitting time densities are well known and studied only in some very particular situations (essentially in Brownian contexts). On the other hand, the study of the approximation of the hitting times for stochastic differential equtions is an active area of research since very few results still are available in the literature. The mainly existing methods are based on Euler and Monte Carlo procedures. We work on alternative methods based on the construction of particular nonlinear boundaries for which we can express explicitly the distribution of the hitting time. We couple this explicit form with the construction of the exit time and exit position for well chosen domains.

The activity of the team on stochastic modeling in population dynamics and genetics mainly concerns application in adaptive dynamics, a branch of evolutionary biology studying the interplay between ecology and evolution, ecological modeling, population genetics in growing populations, and stochastic control of population dynamics, with applications to cancer growth modeling. Stochastic modeling in these areas mainly considers individual-based models, where the birth and death of each individual is described. This class of model is well-developed in Biology, but their mathematical analysis is still fragmentary. Another important topic in population dynamics is the study of populations conditioned to non-extinction, and of the corresponding stationary distributions, called quasi-stationary distributions (QSD). This domain has been the object of a lot of studies since the 1960’s, but we made recently significant progress on the questions of existence, convergence and numerical approximation of QSDs using probabilistic tools rather than the usual spectral tools.

Our activity in population dynamics also involves a fully new research project on cancer modeling at the cellular level by means of branching processes. In 2010 the International Society for proton dynamics in cancer was launched in order to create a critical mass of scientists engaged in research activities on these subjects, leading to the facilitation of international collaboration and translation of research to clinical development. Actually, a new branch of research on cancer evolution is developing intensively; it aims in particular to understand the role of proteins acting on cancerous cells' acidity, their effects on glycolysis and hypoxia, and the benefits one can expect from controlling pH regulators in view of proposing new therapies.

In the financial industry, there are three main approaches to investment: the fundamental approach, where strategies are based on fundamental economic principles; the technical analysis approach, where strategies are based on past price behavior; and the mathematical approach where strategies are based on mathematical models and studies. The main advantage of technical analysis is that it avoids model specification, and thus calibration problems, misspecification risks, etc. On the other hand, technical analysis techniques have limited theoretical justifications, and therefore no one can assert that they are risk-less, or even efficient.

Popular models in financial mathematics usually assume that markets are perfectly liquid. In particular, each trader can buy or sell the amount of assets he/she wants at the same price (the “market price”). They moreover assume that the decision taken by the trader does not affect the price of the asset (the small investor assumption). In practice, the assumption of perfect liquidity is never satisfied but the error due to liquidity is generally negligible with respect to other sources of error such as model error or calibration error, etc.

Derivatives of interest rates are singular for at least two reasons: firstly the underlying (interest rate) is not directly exchangeable, and secondly the liquidity costs usually used to hedge interest rate derivatives have large variation in times.

Due to recurrent crises, the problem of risk estimation is now a crucial issue in finance. Regulations have been enforced (Basel Committee II). Most asset management software products on the markets merely provide basic measures (VaR, Tracking error, volatility) and basic risk explanation features (e.g., “top contributors” to risk, sector analysis, etc).

C. Fritsch organized with Pascal Moyal (Univ. de Lorraine) the weekly Seminar of Probability and Statistics of IECL, Nancy until June 2020.

N. Champagnat was a member of the organizing committee of the conference Mathematical Models in Evolutionary Biology, part
of the Thematic Month on Mathematical Issues in Biology (CIRM, Luminy, 10–14 Feb. 2020).

A. Lejay was a member of the organizing committee of the conference TRAG 2020 (Toulouse, 4-6 Nov. 2020) on rough paths.

A. Lejay is a editor of the project Success Stories (AMIES and FSMP) dedicated to create 2-page sheets to present
successful interactions between industry and academia.