In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.[-]

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large
repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.[-]

In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. This convergence result is proved by assuming only that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, under these assumptions, we obtain a new proof of existence of solutions for such equations.
We apply our results to several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.[-]

The mathematical framework of variational inequalities is a powerful tool to model problems arising in mechanics such as elasto-plasticity where the physical laws change when some state variables reach a certain threshold [1]. Somehow, it is not surprising that the models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations [2] are equivalent to (finite dimensional) stochastic variational inequalities (SVIs) [3]. This presentation concerns (a) cycle properties of a SVI modeling an elasto-perfectly-plastic oscillator excited by a white noise together with an application to the risk of failure [4,5]. (b) a set of Backward Kolmogorov equations for computing means, moments and correlation [6]. (c) free boundary value problems and HJB equations for the control of SVIs. For engineering applications, it is related to the problem of critical excitation [7]. This point concerns what we are doing during the CEMRACS research project. (d) (if time permits) on-going research on the modeling of a moving plate on turbulent convection [8]. This is a mixture of joint works and / or discussions with, amongst others, A. Bensoussan, L. Borsoi, C. Feau, M. Huang, M. Laurière, G. Stadler, J. Wylie, J. Zhang and J.Q. Zhong.[-]

We provide a general framework to study viability and arbitrage in models for financial markets. Viability is intended as the existence of a preference relation with the following properties: It is consistent with a set of preferences representing all the plausible agents trading in the market; An agent with such a preference is in equilibrium, namely, he or she prefers to stay at the initial endowment respect to trade. We extend the original framework of Kreps ('79) and Harrison-Kreps ('79) to accommodate for Knightian Uncertainty: preferences of plausible agents are not necessarily determined by a single probability measure. The relations between arbitrage, viability, and existence of (non-)linear pricing rules are investigated.
This is a joint work with Frank Riedel and Mete Soner.[-]

We will consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We will provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we will establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. Based on joint work with Francois Delarue and Sergey Nadtochiy.[-]

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.[-]