Microscopic and Kinetic Models in Financial Markets

The aim of this PHD thesis is to rewiew some of the more influential models of multi-agent interactive systems in financial markets and to present a new kinetic approach to the description of etherogeneous systems, where different populations of agents are involved and interact each others. In the first chapter, we present the Levy-Levy-Solomon model and The Lux-Marchesi model as mi- crospic models. In the second chapter staring from the microsopic description we derive kinetics model for both Levy-Levy solomon and Lux-Marchesi models, furthermore through the introduction ok Fokker-Plank appoximation models, we are able yo illustrate some analitycal results and numerical simulations. In the third chapter we present a more realistic whic generalize the works of chapter two. For such model, starting from a mesoscopic decription an hydrodynamic model is derived and analytical and numerical results are provided. We leave as appendix A and B full details of some technical proofs of the second chapter, in order to let it more readable. Appendix C contains a pub- blication in the Esaim Proceedings where I’m co-author. It was the results of the CEMRACS summer school held in Marseille in the August 2010. Here a spatial coupling of an asymptotic preserving scheme with the asymptotic limit model, associated to a singularly perturbed, highly anisotropic, elliptic problem is investigated and compared with the numerical discretization of the initial singular perturbation model or the purely asymptotic preserving scheme.