报告题目：Some Problems Related to Game Theory
The talk consists of two parts, one of which is about the differential game, and the other one is the mean field game. In the first part, we present a stochastic differential game of transboundary industrial pollution with emission permits trading. More generally, the emission permits price is assumed to follow a geometric Brownian motion (GBM). We make use of stochastic optimal control theory to derive the Hamilton-Jacobi-Bellman (HJB) equations satisfied by the value functions for cooperative and noncooperative game respectively, and then propose a so-called fitted finite volume method to solve it. In the second part, we present a mean field game to model the production behaviors of a very large number of producers, whose carbon emissions are regulated by government. By means of the mean field equilibrium, we obtain a Hamilton-Jacobi-Bellman (HJB) equation coupled with a Kolmogorov equation, which are satisfied by the adjoint state and the density of producers (agents), respectively. Then, we propose a so-called fitted finite volume method to solve the HJB equation and the Kolmogorov equation.