Modeling of socio-economic phenomena by Fokker-Planck Equations.

This thesis is centered on Fokker-Planck Equation and its appplication in Econophysics. Part I is devoted to the modeling of wealth distribution and it is based on this two works that were developed under the supervision of Professor G.Toscani: M. Torregrossa and G. Toscani. On a Fokker-Planck equation for wealth distribution. Kinet. Relat. Models, 11(2):337–355, 2018. M. Torregrossa and G. Toscani. Wealth distribution in presence of debts. A Fokker-Planck description. Commun. Math. Sci, 16(2):537–560, 2018. Within this Part, in Chapter 4, we study a Fokker- Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society. In particular, we analyze the large-time behavior of the solution, by showing that convergence to the steady state can be obtained in various norms at different rates. In Chapter 5, we consider the same Fokker-Planck equation with variable coefficient of diffusion which appears in Chapter 4. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker-Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached. Part II shows an application of Fokker-Planck equation in the formulation of rating model. It is based on a joint work with Professor B.Düring and with Dr.Marie-Therese Wolfram: B. Düring, M. Torregrossa, and M.-T. Wolfram. On a kinetic Elo rating model for players with dynamical strength. ArXiv e-prints, June 2018. In this paper we propose and study a new kinetic rating model for a large number of players, which is motivated by the well-known Elo rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated after each game. We state and analyse the respective Boltzmann type equation and derive the corresponding nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the Boltzmann and Fokker-Planck equation with various numerical experiments. Supervisor: Prof. G. Toscani. PhD candidate: Marco Torregrossa