A penalty method for the numerical solution of hamilton-jacobi-bellman (HJB) equations in finance

We present a simple and easy-to-implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, classical finite difference discretizations can be shown to converge to the unique viscosity solutions of the considered problems. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ < 0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalized discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results. © 2011 Society for Industrial and Applied Mathematics.