Semiclassical Approximation in Stochastic Optimal Control: I. Portfolio Construction Problem

Abstract

This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order HJB partial differential equation to a hierarchy of first order PDEs, followed by a numerical algorithm to solve the first few of the resulting first order PDEs. This method is applicable to stochastic systems with a relatively large number of degrees of freedom, and does not seem to suffer from the curse of dimensionality. Computer code implementation of the method using modest computational resources runs essentially in real time. We apply the method to solve a general portfolio construction problem.