Penalty and penalty-like methods for nonlinear HJB PDEs

Abstract

There are numerous financial problems that can be posed as optimal control problems, leading to Hamilton–Jacobi–Bellman or Hamilton–Jacobi–Bellman–Issacs equations. We reformulate these problems as nonlinear PDEs, involving max and/or min terms of the unknown function, and/or its first and second spatial derivatives. We suggest efficient numerical methods for handling the nonlinearity in the PDE through an adaptation of the discrete penalty method Forsyth and Vetzal(2002)[1] that gives rise to tridiagonal penalty matrices. We formulate a penalty-like method for the use with European exercise rights, and extend this to American exercise rights resulting in a double-penalty method. We also use our findings to improve the policy iteration algorithms described in Forsyth and Labahn(2007)[2]. Numerical results are provided showing clear second-order convergence, and where applicable, we prove the convergence of our algorithms.