Nash equilibrium approximation of some class of stochastic differential games: A combined Chebyshev spectral collocation method with policy iteration

Abstract

This study relates to nonzero-sum stochastic games with perfect information. It proposes an efficient combined Chebyshev spectral collocation method (CSCM) with the policy iteration (PI) algorithm for solving nonlinear coupled Hamilton–Jacobi (HJ) equations. The proposed approach is comprised of two steps. First, the PI algorithm is used to reduce the nonlinear coupled HJ equations to a sequence of linear uncoupled PDEs. Then, these equations are approximated by the CSCM. The CSCM＋PI is especially useful when the CSCM fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations. The main advantage of the resulting method is that it converts nonlinear coupled HJ equations to the systems of linear algebraic equations, which can be solved readily. Convergence analysis of this method is also provided in detail. To confirm the accuracy and validity of the proposed computational algorithm, several illustrative examples are presented.