Differentiability properties of Rank-Linear Utilities.

We study the differentiability properties of concave functionals defined as integrals of the quantile. These functionals generalize the rank dependent expected utility and are called rank-linear utilities in decision theory. Their superdifferential is described as well as the set of random variables where they are Gâteaux-differentiable. Our results generalize those obtained for the rank dependent expected utility in Ref. [Carlier, G., Dana, R.-A., 2003. Core of a convex distortion of a probability. Journal of Economic Theory 113, 199–222.].Optimization and control;

On systems of continuity equations with nonlinear diffusion and nonlocal drifts

This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument

Remarks on existence and uniqueness of Cournot-Nash equilibria in the non-potential case

This article is devoted to various methods (optimal transport, fixed-point, ordinary differential equations) to obtain existence and/or uniqueness of Cournot-Nash equilibria for games with a continuum of players with both attractive and repulsive effects. We mainly address separable situations but for which the game does not have a potential. We also present several numerical simulations which illustrate the applicability of our approach to compute Cournot-Nash equilibria

Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization

We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of $N$ player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article we emphasize the role of optimal transport theory in: 1) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity, 2) the analysis of Cournot-Nash equilibria

Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds), t>0, with initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and: $$v(x,z):=\inf_{u\in V} \{\int_0^{+\infty} e^{-\lambda s} L(y_{x,z,u}(s), u(s))ds \}$$ where $V$ is a class of admissible controls, we prove that $v$ is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: $$\lambda v(x,z)+H(x,z,\nabla_x v(x,z))+D_z v(x,z), \dot{z} >=0$$ in the sense of the theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions

Existence and monotonicity of optimal debt contracts in costly state verification models

This note gives a simple proof of the existence and monotonicity of optimal debt contracts in simple models of borrowing and lending with ex-post asymmetric information, risk-averse agents and heterogeneous beliefs. Our argument is based on the concept of nondecreasing rearrangement and on a supermodular version of Hardy-Littlewood inequality.

Debt Contracts with ex-ante and ex-post Asymmetric Information: An Example

We consider a simple model of lending and borrowing combining two informational problems: adverse selection and costly state verification. Our analysis highlights the interaction between these two informational problems. We notably show that the higher the monitoring cost, the less discriminating the optimal menu of contracts is.debt contracts, diversity of opinions, screening, costly monitoring, pooling