Multigrid Methods for Second Order Hamilton--Jacobi--Bellman and Hamilton--Jacobi--Bellman--Isaacs Equations

We propose multigrid methods for solving the discrete algebraic equations arising from the discretization of the second order Hamilton--Jacobi--Bellman (HJB) and Hamilton--Jacobi--Bellman--Isaacs (HJBI) equations. We propose a damped-relaxation method as a smoother for multigrid. In contrast with the standard policy iteration, the proposed damped-relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems with large jumps in control, we develop restriction and interpolation methods to capture the jumps on the coarse grids as well as during the coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.