Numerical methods for two person games arising from transboundary pollution with emission permit trading

摘要

We propose finite difference methods for solving two dimensional Hamilton–Jacobi–Bellman (HJB) equations and systems arising from the modelling of transboundary pollution with emission permits trading. We prove that our numerical scheme for the HJB equations from cooperative game is consistent, stable, and monotone, therefore guarantees convergence to viscosity solutions. To ensure our scheme is fully implicit and unconditionally monotone, we combine wide stencil with narrow stencil in the discretization. We address coupling between unknown variables and emission controls using policy iteration. We solve the coupled systems of HJB equations from non-cooperative game efficiently through policy-like iteration. We give a theoretical analysis of our scheme for solving the coupled system and find that it converges under certain conditions. Finally, we show that our numerical scheme achieves higher order of convergence than the finite volume method proposed originally.