We consider an optimal investment problem proposed by Bielecki and Pliska. The goal of the investment problem is to optimize the long-term growth of expected utility of wealth. We consider HARA utility functions with exponent $-\infty< \gamma< 1$. The problem can be reformulated as an infinite time horizon risk-sensitive control problem. Some useful ideas and results from the theory of risk-sensitive control can be used in the analysis. Especially, we analyze the associated dynamical programming equation. Then an optimal (or approximately optimal) Markovian investment policy can be derived.
"Risk-sensitive control and an optimal investment model II." Ann. Appl. Probab. 12 (2) 730 - 767, May 2002. https://doi.org/10.1214/aoap/1026915623