We consider the problem of controlling the diffusion coefficient of a diffusion with constant negative drift rate such that the probability of hitting a given lower barrier up to some finite time horizon is minimized. We assume that the diffusion rate can be chosen in a progressively measurable way with values in the interval [0,1]. We prove that the value function is regular, concave in the space variable, and that it solves the associated HJB equation. To do so, we show that the heat equation on a right triangle, with a boundary condition that is discontinuous in the corner, possesses a smooth solution.
Work in Collaboration with Stefan Ankirchner, Nabil Kazi-Tani, Chao Zhou.[-]

We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values on a grid of [0, 1], with mutation and selection. We study of the dynamics of the population in logarithm size and time scales, under a large population assumption. In the first part of the talk, individual mutations are rare but the global mutation rate tends to infinity. Then negligible sub-populations may have a strong contribution to evolution. The traits can also be horizontally transferred, leading to a trade-off between natural evolution to higher birth rates and transfer which drives the population towards lower birth rates. We prove that the stochastic discrete exponent process converges to a piecewise affine continuous function, which can be described along successive phases determined by dominant traits. In the second part of the talk, the individual mutations are small but not rare, we don't have any transfer and we assume the grid mesh for the trait values becoming smaller and smaller. We establish that under our rescaling, the stochastic discrete exponent process converges to the viscosity solution of a Hamilton-Jacobi equation, filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations.
Joint works with N. Champagnat and V.C. Tran, and S. Mirrahimi for the second part.[-]

We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the 'flow solution' is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
Joint work with Aurélien Alfonsi.[-]

In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_{t}^{q}-L_{x}^{p}$ space. Contrary to the large deviation principle approach recently proposed in the litterature (Hoeksama et al, 2020), the main ingredient of the proof here are the Partial Girsanov transformations introduced by (Jabir-Talay-Tomašević.,2018) and developed in a general setting here.[-]

In a factor model for a large panel of N asset prices, a random time $S$ is called a 'systematic jump time' if it is not a jump time of any of the factors, but nevertheless is a jump time for a significant number of prices: one might for example think that those $S$ 's are jump times of some hidden or unspecified factors. Our aim is to test whether such systematic jumps exist and, if they do, to estimate a suitably defined 'aggregated measure' of their sizes. The setting is the usual high frequency setting with a finite time horizon $T$ and observations of all prices and factors at the times $iT /n$ for $i = 0, . . . , n$. We suppose that both $n$ and $N$ are large, and the asymptotic results (including feasible estimation of the above aggregate measure) are given when both go to $\infty$, without imposing restrictions on their relative size.
(joint work with Huidi Lin and Viktor Todorov)[-]

We are interested in monitoring patients in remission from cancer. Our aim is to detect their relapses as soon as possible, as well as detect the type of relapse, to decide on the appropriate treatment to be given. Available data are some marker level of the rate of cancerous cells in the blood which evolves continuously but is measured at discrete (large) intervals and through noise. The patient's state of health is modeled by a piecewise deterministic Markov process (PDMP). Several decisions must be taken from these incomplete observations: what treatment to give, and when to schedule the next medical visit. After presenting a suitable class of controlled PDMPs to model this situation, I will describe the corresponding stochastic control problem and will present the resolution strategy that we adopted. The objective is to obtain an approximation of the value function (optimal performance) as well as build an explicit policy applicable in practice and as close to optimality as possible. The results will be illustrated by simulations calibrated on a cohort of a clinical trial on multiple myeloma provided by the Center of Cancer Research in Toulouse.[-]

Concave and convex functions are basic functions in economy and finance. In derivatives market, options pay-offs as Call and Put are in general convex functions of their underlying $((x-K)^{+}, or (K-x)^{+})$ and their Black-Scholes Prices are also convex. This property can be maintain in a random universe, (without reference to finance). Here, we are looking for the pricing point of view. The data is an underlying random field, $\left\{X_{t}(x) \right\}$, non negative with $X_{t}(0)=0$, $X_{t}(+\infty )=\infty$, and a pricing (strictly) convex function $\Phi (0,z)$ whose the right-derivative is denoted $\phi$, given the price today of convex European derivative. The problem is to characterize a convex pricing rule $\left\{\Phi (t,z) \right\}$ in the future, optimal in the sense that $\left\{\Phi (t,X^{t}(x)) \right\}$ is a martingale. Obviously, without additional constraint, the problem has many solutions. So, thanks to convexity assumptions, it is natural to introduce the convex conjugate random field $\Psi (t,y)$. By the Fenchel theory, the Gap function $G_{\Phi }(t,z,y)=\Phi (t,z)+\Psi (t,y)-zy\geq 0$, $= 0$ if $\phi (t,z)=y$.

Put $Y_{t}(\phi (z)):=\Phi _{z}(t,X_{t}(z))$. The problem is to solve a be revealed problem find a par of conjugate convex random fields $(\Phi (t,z), \Psi (t,y))$ such that $\Phi (t,X_{t}(x))$ and $\Psi (t,Y_{t}(y))$ are martingales. The Legendre formula implies that $X_{t}(z)Y_{t}(\phi (z))$ is a martingale. As for revealed utility, the problem at least a solution if and only if their exists an equivalent intrinsic framework, where necessary the processes ‘$\left\{X_{t}(x) \right\},\left\{Y_{t}(y) \right\},\left\{\Phi (t,z) \right\}$' are supermartingales, and $\left\{X_{t}(x)Y_{t}(\phi (x)) \right\}$ is a martingale. The family $\left\{Y_{t}(\phi (x)) \right\}$ is a family of pricing kernel for $X_{t}(x)$. The relation $Y_{t}(\phi (z)):=\Phi _{z}(t,X_{t}(z))$, and the monotony of $X_{t}(z)$ gives the way to obtained $\Phi _{z}(t,z)=Y_{t}(\phi (X_{t}^{-1}(z)))$ by a pathwise procedure. The convexity of the pricing kernel reduced the arbitrage problems. Itô's semimartingale framework is used to illustrate this characterization. The revealed pricing kernel y is solution of a non-linear SPDE. Many properties can be deduced of this pathwise construction.
Joint work Mohamed Mrad.[-]

In 1927, two Scottish epidemiologists, Kermack and McKendrick, published a paper on a SIR epidemic model, where each infectious individual has an age of infection dependent infectivity, and a random infectious period whose law is very general. This paper was quoted a huge number of times, but almost all authors who quoted it considered the simple case of a constant infectivity, and a duration of infection following the exponential distribution, in which case the integral equation model of Kermack and McKendrick reduces to an ODE.
It is classical that an ODE epidemic model is the Law of Large Numbers limits, as the size of the population tends to infinity, of finite population stochastic Markovian epidemic models.
One of our main contributions in recent years has been to show that the integral equation epidemic model of Kermack and McKendrick is the law of large numbers limit of stochastic non Markovian epidemic models. It is not surprising that the model of Kermack and Mc Kendrick, unlike ODE models, has a memory, like non Markovian stochastic processes. One can also write the model as a PDE, where the additional variable is the age of infection of each infected individual.
Similar PDE models have been introduced by Kermack and Mc Kendrick in their 1932 and 1933 papers, where they add a progressive loss of immunity. We have also shown that this 1932-33 model is the Law of Large Numbers limit of appropriate finite population non Markovian models.
Joint work with R. Forien (INRAE Avignon, France), G. Pang (Rice Univ., Houston, Texas, USA) and A.B. Zotsa-Ngoufack (AMU and Univ. Yaoundé 1)[-]