Speakers

Title: A Parcimonious Long Term Mean Field Model for Mining Industries (joint work with P-N. Giraud, J-M. Lasry and P-L. Lions)

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Title: “An adverse selection approach to power tarification” (joint work with Ivar Ekeland, Romuald Elie, Nicola Hernández Santibáñez, Dylan Possomai)

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Title: Stylized model for a grid with distributed generation and storage

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Title: Diffusion versus superdiusion in a stochastic Hamiltonian lattice field model

Abstract: In this talk I will review several recent works obtained in collaboration with M. Jara, P. Gonalves, M. Simon and M. Sasada about the energy diffusion in a Hamiltonian lattice field model with two conserved quantities perturbed by a conservative noise. I will discuss simple mechanisms that provide a crossover regime between the Edwards-Wilkinson universality class, the Zero-Pressure (3/4-fractional superdifusion) universality class and the KPZ universality class.

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Title: PDE strategies for the existence of McKean Nonlinear diffusion models (joint work with Jean Francois Jabir, University of Valpareiso)

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Title: Mean-field SDE driven by a fractional Brownian motion and related stochastic control problem

Abstract: We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient. The talk is baed on a joined work with Shuai Jing (Central University of Finance and Economy, Beijing, PRC)

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Title: Distributed demand control in power grids and ODEs for Markov decision processes (Joint work with Sean Meyn)

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Title: Long time behavior of Mean Field Games (joint work with A. Porretta)

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Title: Mean Field Games with Major and Minor Players: Theory and Numerics

Abstract: We present a (possibly) new formulation of the mean field game problem in the presence of major and minor players, and give new existence results for linear quadratic models and models with finite state spaces. We shall also provide numerical results illustrating the theory and raising new challenges.

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Title: Obliquely Reflected Backward Stochastic Differential Equations (joint work with A. Richou)

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Title: Two-dimensional pseudo-gravity model: particles motion in a non-potential singular force field

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Title: Rough mean field differential equations

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Title: Describing the thermodynamic limit of networks of interacting neurons

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Title: Systemic risk and stochastic games with delay

Abstract: We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of N banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a linear-quadratic stochastic game with delay between N players. A unique open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a close-loop Nash equilibrium using an HJB approach to this stochastic game with delay and we analyze its mean field limit.
Joint work with R. Carmona, M. Mousavi and L.H. Sun.

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Title: Evolution of the Wasserstein distance between the marginals of two Markov processes (joint work with Aurélien Alfonsi and Jacopo Corbetta)

Abstract: The Kantorovich duality leads to a natural candidate for the time derivative of the Wasserstein distance between the marginals of two Markov processes. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding infinitesimal generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate.

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Title: Estimation of the parameters of a discontinuous diffusion (Joint work with Paolo Pigato IECL / Inria Nancy Grand-Est)

Abstract: In this talk, we address the problem of the statistical estimation of the coefficients of a diffusion with piecewise constant coefficients (diffusivity and drift), called the oscillating Brownian motion. This problem of estimation shows a variety of behavior according to the respective signs of the drift, as the convergence is driven by the asymptotic behavior of the occupation time itself dependent of the regime of the diffusion (recurrent, null recurrent, transient). The oscillating Brownian motion could be seen as a continuous time version of the self-exciting threshold autoregressive (SETAR) model. Application to financial data shows estimations which are coherent with the one based on the SETAR model. It also empirically demonstrates the presence of leverage effect and/or mean-reverting properties on some stock prices.
Potential applications to other domains are also considered.

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Title: Weak solutions of mean-field stochastic differential equations

Abstract:In this talk we discuss weak solutions of mean-field stochastic differential equations (SDEs), also known as McKean-Vlasov equations, whose drift $b(s, X_s,Q_{X_s})$, and diffusion coefficient $\sigma(s, X_s,Q_{X_s})$ depend not only on the state process $X_s$ but also on its law. We suppose that $b$ and $\sigma$ are bounded and continuous in the state as well as the probability law; the continuity with respect to the probability law is understood in the sense of the 2-Wasserstein metric. Using the approach through a local martingale problem, we prove the existence and the uniqueness in law of the weak solution of mean-field SDEs. The uniqueness in law is obtained if the associated Cauchy problem possesses for all initial condition $f\in C_0^\infty({\mathbb R}^d)$ a classical solution. However, unlike the classical case, the Cauchy problem is a mean-field PDE as recently studied by Buckdahn, Li, Peng and Rainer (2014). In our approach, we also extend the It\^o formula associated with mean-field problems given by Buckdahn, Li, Peng and Rainer (2014) to a more general case of coefficients.
The talk is based on joint works with Hui Min (SDU, Weihai).

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Title: Beyond stochastic control (and MFG)

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Title: Time scales and spectral gaps for quasi stationary distributions in large populations birth and death processes

Abstract: We study a general class of birth-and-death processes that describe the size of populations going to extinction with probability one. The scale of the population is measured in terms of a ‘carrying capacity’ K. When K is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the time for the process to reach the quasi stationary regime and the mean time to extinction in the quasi stationary distribution as a function of K, for large K. In dimension one, we also give a quantitative description of this quasi-stationary distribution.

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Title: Exponential Ergodicity in a Sobolev Space (coauthors: Adithya Devraj and Ioannis Kontoyiannis)

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Title: Supply and Shorting in Speculative Markets

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Title: Non-asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion

Abstract: We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure of an ergodic Brownian diffusion process and the empirical distribution $\nu$ of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions $f$ such that $f-\nu(f)$ is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr\”obenius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.

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Title: Mean-field contract theory and electricity demand management

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Title: Martingales and Strict Local Martingales

Abstract: Whether a nonnegative solution of an SDE is a martingale or is a strict local martingale can, at times, have profound implications. Works of Delbaen, Shirakawa, Mijatovic, Urusov, Lions, Musiela, Andersen, Piterbarg, Bernard, Cui, and finally McLeish have studied the case of a one dimensional SDE, possibly with stochastic volatility. We will discuss two situations: (1) How a solution of an SDE which is a martingale can morph into a strict local martingale within a financial context by the addition of new information to the underlying filtration, and (2) how various components of a system of SDEs can be strict local martingales for some components of the system, and martingales for others. Our talk is based on joint work with Aditi Dandapani, Columbia PhD (2016), currently at ETH-Zurich.

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Title: Stochastic Nerve Axon Equations

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Title: Network of interacting neurons

Abstract:Some simple neuronal models evolve as follow: in the sub threshold regime, their membrane potential are solution of diffusion (each neuron is governed by independent noise).
Each time at which one of these diffusions hit a fixed threshold define a spiking time of the network. Then, the spiking neuron interacts with the other neurons of the network: connected neurons receive a ‘kick’, that is the membrane potentials of connected neurons have a jump. In addition, the membrane potential of the spiking neurons are reset to a deterministic value (the rest potential). We discuss the asymptotic behavior of such simplified networks of neurons in interaction, the existence of smooth solutions to the limit Mc-Kean Vlasov equation and the selection of particular solutions by two numerical schemes in a case where uniqueness fails.
(common work with F. Delarue, J. Inglis and S. Rubenthaler)

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Title: PDE arising from principal-agent problems

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