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Hamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem

We consider an investment problem (IC for short) that the goal is to optimize the expected HARA utility of discounted consumption in infinite time horizon. We consider a factor model and study the HamiltonJacobiBellman (HJB) equation. We show that this investment problem is closely related to the problem of optimizing the large time growth rate of the expected HARA utility of wealth (called IE for short). We show that the solution of the HJB equation for (IE) can be used to construct a nice supersolution (or subsolution depending on the parameter of HARA utility). Then a solution of the HJB equation for (IC) with nice growth conditions will follow. We also prove a verification theorem that a strategy constructed from the solution is optimal.