On the Hamilton--Jacobi--Bellman Equation for an Optimal Consumption Problem: II. Verification Theorem

Abstract

We consider an optimal consumption problem where an investor tries to maximize the infinite horizon expected discounted hyperbolic absolute risk aversion utility of consumption. We treat a stochastic factor model such that the mean returns of risky assets depend on underlying economic factors formulated as the solution of a stochastic differential equation. Using a dynamic programming principle, we derive the Hamilton--Jacobi--Bellman (HJB) equation and study its solutions. In part I, we prove the existence of a classical solution for HJB equation under suitable conditions. In part II, we consider the verification theorem stating a candidate of optimal strategy derived from a solution of HJB equation is indeed optimal. We prove the result under two sets of conditions. One is a linear model, and one is a nonlinear model. In both cases, the HJB equations do not have analytical solutions.