Differential-nondifferentiable games and applications in economics

Abstract

The main objective of this thesis is to present the connection between the adjoint variables in the maximum principle (MP) and the value function in the dynamic programming principle (DPP) for two-player nonzero-sum differential games, both in the smooth and nonsmooth cases. This relationship is established in terms of derivatives in the smooth case and through viscosity solutions when the value function is not smooth, with economic interpretations related to the adjoint variables. In the second part, we apply a numerical method based on the Jacobi spectral method (JSM) to solve the nonlinear two-point boundary value problems (TPBVPs) derived from the maximum principle. These problems are then transferred into a system of algebraic equations to determine the open-loop Nash equilibrium (OLNE) for nonzero-sum differ- ential games. Illustrative examples are presented to demonstrate the effectiveness andvalidity of the proposed method.