CONSUMPTION AND INVESTMENT UNDER CONSTRAINTS

We study the problem of an investor who is endowed with a deterministic level of initial capital and must use this endowment to consume and invest in a financial market. The investor is constrained to consume at a minimum rate throughout the period and has to maintain his wealth over a lower bound at all times. Both these minimal requirements may be stochastic. An example of such an investor would be an investment firm with cash flow commitments and subject to regulatory capital constraints. In addition, the investor's portfolio choices are limited to a convex constraint set. This accounts for prohibitions on short-selling and incompleteness of the market. Using the space of associated risk-neutral measures, we construct portfolio policies to support target consumption-investment plans. We also consider a utility maximization problem for the investor and provide a related dual minimization problem over the space of measures. The solution of this dual problem provides a price system under which the investor can solve for his optimal consumption investment policy. In general, however, we are not assured of existence due to the fact that the investor's minimal requirements may themselves be beyond the span of the constrained portfolios.