Optimal long term growth rate of expected utility of wealth

An optimal investment policy model for the long term growth of expected utility of wealth is considered. The utility function is HARA with exponent ..<.<1. The problem can be reformulated as an infinite time horizon, risk sensitive control problem. Then the dynamic programming equations for different HARA exponents and different policy constraints are studied. We obtain some estimates for the solution of each equation. This can be used to derive an optimal policy with some interesting properties.