Optimal sure portfolio plans

This paper is a sequel to [2], 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. non-random) portfolio plan can be defined in such a way that solutions of this problem correspond to the distant future is sufficiently discounted. This has been proved in [2], land 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 non-existence are identified. Programming conditions characterising an optimal sure portfolio plan are also obtained.