Optimal sure portfolio plans

This paper is a sequel to the author's "Certainty Equivalence in the Continuous-Time Portfolio-cum-Saving Model" in Applied Stochastic Analysis (eds. M. H. A. Davis and R. J. Elliot), where a model of optimal accumulation of capital and portfolio choice over an infinite horizon in continuous time was considered in which the vector process representing returns to investment is a general semimartingale with independent increments and the welfare functional has the discounted constant relative risk aversion (CRRA) form. A problem of optimal choice of a sure (i.e., nonrandom portfolio plan can be defined in such a way that solutions of this problem correspond to solutions of optimal choice of a portfolio-cum-saving plan, provided that the distant future is sufficiently discounted. This has been proved in the earlier paper, and is in part proved again here by different methods. Using the canonical representation of a PII-semimartingale, a formula of Lévy-Khinchin type is derived for the bilateral Laplace transform of the compound interest process generated by a sure portfolio plan. With its aid. the existence of an optimal sure portfolio plan is proved under suitable conditions, and various causes of nonexistence are identified. Programming conditions characterizing an optimal sure portfolio plan are also obtained