Investment and Consumption with Forward Criteria and Black's Inverse Investment Problem

We consider the problem of optimal portfolio selection using forward investment and consumption criteria. Such criteria were introduced in and allow the investor to consider utility from both investment and consumption also when investing over unspecified or infinite horizons. Our focus is on criteria which satisfy a specific dynamic property in that their volatility component is identically zero. We provide an explicit characterization of those criteria and explicit formulae for the associated optimal investment and consumption strategies. We further show that an important class of criteria may be decomposed into a combination of pure forward investment and infinite horizon Merton criteria. Our second contribution is that we provide the solution to a specific inverse investment problem proposed by Black. While this result is of independent interest, it also allows us to obtain further results on the adaptedness of non-volatile forward criteria, and provides key insights on the relation between the notion of forward criteria and this inverse investment problem