Hedging in Incomplete Markets with HARA Utility

In the context of Merton’s original problem of optimal consumption and portfolio choice in continuous time, this paper treats an extension in which the investor is endowed with a stochastic income that cannot be replicated by trading the available securities. In other words, markets are incomplete in an essential way. The problem is solved by demonstrating, using analytic and in particular “viscosity solutions’ techniques, that the value function of the stochastic control problem is a smooth solution of the associated Hamilton-Jacobi-Bellman (HJB) equation. The optimal policy is then given from the first order conditions of the HJB equation.