Numerical solution of the Hamilton-Jacobi-Bellman equation for stochastic optimal control problems

Abstract

This paper provides a numerical solution of the Hamilton-Jacobi-Bellman (HJB) equation for stochastic optimal control problems. The computation's difficulty is due to the nature of the HJB equation being a second-order partial differential equation which is coupled with an optimization. By using a successive approximation algorithm, the optimization gets separated from the boundary value problem. This makes the problem solveable by standard numerical methods. For a problem of portfolio optimization where no analytical solution is known, the numerical methods is applied and its usefulness demonstrated.