On Hölder calmness of solution mappings in parametric equilibrium problems

We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Hölder calmness of solutions are established. We also study the Hölder well-posedness for equilibrium problems in metric spaces

First order Mean Field Games on networks

We study deterministic mean field games in which the state space is a network. Each agent controls its velocity; in particular, when it occupies a vertex, it can enter in any edge incident to the vertex. The cost is continuous in each closed edge but not necessarily globally in the network. We shall follow the Lagrangian approach studying relaxed equilibria which describe the game in terms of a probability measure on admissible trajectories. The first main result of this paper establishes the existence of a relaxed equilibrium. The proof requires the existence of optimal trajectories and a closed graph property for the map which associates to each point of the network the set of optimal trajectories starting from that point. Each relaxed equilibrium gives rise to a cost for the agents and consequently to a value function. The second main result of this paper is to prove that such a value function solves an Hamilton-Jacobi problem on the network

Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming

AbstractSample average approximation (SAA) is one of the most popular methods for solving stochastic optimization and equilibrium problems. Research on SAA has been mostly focused on the case when sampling is independent and identically distributed (iid) with exceptions (Dai et al. (2000) [9], Homem-de-Mello (2008) [16]). In this paper we study SAA with general sampling (including iid sampling and non-iid sampling) for solving nonsmooth stochastic optimization problems, stochastic Nash equilibrium problems and stochastic generalized equations. To this end, we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem. Exponential convergence of estimators of both optimal solutions and M-stationary points (characterized by Mordukhovich limiting subgradients (Mordukhovich (2006) [23], Rockafellar and Wets (1998) [32])) are established under mild conditions. We also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic Nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

Stochastics of Environmental and Financial Economics

Systems Theory, Contro

IST Austria Thesis

This thesis is based on three main topics: In the first part, we study convergence of discrete gradient flow structures associated with regular finite-volume discretisations of Fokker-Planck equations. We show evolutionary I convergence of the discrete gradient flows to the L2-Wasserstein gradient flow corresponding to the solution of a Fokker-Planck equation in arbitrary dimension d >= 1. Along the argument, we prove Mosco- and I-convergence results for discrete energy functionals, which are of independent interest for convergence of equivalent gradient flow structures in Hilbert spaces. The second part investigates L2-Wasserstein flows on metric graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein distance, which is proved via a regularisation scheme for solutions of the continuity equation, adapted to the peculiar geometric structure of metric graphs. Based on those results, we show that the L2-Wasserstein space over a metric graph admits a gradient flow which may be identified as a solution of a Fokker-Planck equation. In the third part, we focus again on the discrete gradient flows, already encountered in the first part. We propose a variational structure which extends the gradient flow structure to Markov chains violating the detailed-balance conditions. Using this structure, we characterise contraction estimates for the discrete heat flow in terms of convexity of corresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals

A note on mean field games of controls with state constraints: existence of mild solutions

We show the existence of "mild solutions" for a first-order mean field game of controls under the state constraint that trajectories be confined in a closed and bounded set in euclidean space. This extends the results of Cannarsa and Capuani to the case of a mean field game of controls. Our controls are velocities and we find that the existence of an equilibrium is complicated by the requirement that they should have enough regularity. We solve this by imposing a small Lipschitz constant on the dependence of the Lagrangian on the joint measure of states and controls, and showing that regular paths can be approximated within the same class of functions despite the constraint

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

[Book of abstracts]

USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos