CONSUMPTION-PORTFOLIO POLICIES - AN INVERSE OPTIMAL PROBLEM

We study a problem that is the ''inverse'' of Merton [J. Econ. Theory 3 (1971), 373-413]. For a given consumption-portfolio policy, we provide necessary and sufficient conditions for it to be optimal for ''some'' agent with an increasing, strictly concave, time-additive, and state-independent utility function when the risky asset price follows a general difussion process. These conditions involve a set of consistency and state independency conditions and a partial differential equation satisfied by the consumption-portfolio policy. We also provide an integral formula which recovers the utility function that supports a given optimal policy. The inverse optimal problem studied here should be viewed as a dynamic recoverability problem in financial markets with continuous trading.