Power penalty method for solving HJB equations arising from finance

Abstract

Nonlinear Hamilton–Jacobi–Bellman (HJB) equation commonly occurs in financial modeling. Implicit numerical scheme is usually applied to the discretization of the continuous HJB so as to find its numerical solution, since it is generally difficult to obtain its analytic viscosity solution. This type of discretization results in a nonlinear discrete HJB equation. We propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term. Under some mild conditions, we give the unique solvability of the penalized equation and show its convergence to the original discrete HJB equation. Moreover, we establish a sharp convergence rate of the power penalty method, which is of an exponential order with respect to the power of the penalty term. We further develop a damped Newton algorithm to iteratively solve the lower order penalized equation. Finally, we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method. We also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method.