Hamilton-Jacobi-Bellman (HJB) equations are nonlinear second order partial
differential equations (PDEs), which arise in many areas such as stochastic optimal control
and computational finance. If there is only one value function, the HJB equation is scalar.
However, when there are multiple value functions, for instance, regime switching models in
option pricing, or nonzero sum stochastic differential games in dynamic Bertrand
oligopoly, the resulting model will result in a system of coupled HJB PDEs. In this talk, we
will present an efficient multigrid method for solving systems of discrete HJB equations.
We will discuss the discretization of the nonlinear systems and the issues of viscosity
solutions and monotone finite difference schemes. We will derive a relaxation scheme as a
smoother for multigrid, a control preserving restriction and interpolation. Finally, we will
demonstrate the performance of the method by examples of pricing European option under
the regime switching model and the dynamic Bertrand oligopoly problem.|