Approximations to Expected Utility Optimization in Continuous Time

Abstract

In this paper, we explore approximate solutions to optimal control problems that cannot be solved analytically with existing techniques. Inspired by the mean-variance analysis of the single period environment, an advanced and a simple method are developed in order to approximate optimal investment strategies in continuous time. In the advanced method, the original problem is approximated by a Taylor series expansion in the conditional mean of terminal wealth. As the point of expansion is thereby continuously changing, the approximation results in a non-standard optimal control problem that can be characterised by an extended HJB equation. In the simple method, the problem is expanded in the initial mean, leading to a problem that can be solved using the classical HJB equation in an unconventional way. The advanced approximated problem inherits more features from the original problem than the simple approximated problem. In a numerical example, we illustrate how the advanced approximate strategy gives a better approximation than the simple approximated strategy. An approximate solution is determined to a prospect theory investment problem, utilising the advanced method of approximation. The solution reflects the same behaviour as the classical life-cycle investment strategy, where the proportion of wealth invested in the risky asset is decreasing over time and independent of the level of wealth.