On the Hamilton--Jacobi--Bellman Equation for an Optimal Consumption Problem: I. Existence of Solution

Abstract

We consider a consumption problem with an infinite time horizon to optimize the discounted expected power utility. The returns and volatilities of the assets are random and affected by some economic factors, modeled as diffusion process. The problem becomes a standard control problem. We derive the Hamilton--Jacobi--Bellman (HJB) equation and study its solutions. In Part I, under some weak conditions we prove the existence of a solution for this HJB equation when we assume the existence of an ordered pair of sub/supersolution. To construct an ordered pair of sub/supersolution, we also consider the risk-sensitive portfolio optimization problem. In Part II [Hata and Sheu, SIAM J. Control Optim., 50 (2012), pp. 2401--2430], we consider the uniqueness of the solution and prove the verification theorem.

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