An analysis of a stochastic resource allocation model with various utility functions

This paper deals with stochastic models of an optimal sequential allocation of resources between consumption and production. The following results are obtained by means of the theory of martingales. For a model of resource allocation the dynamic programming equation is established and it is shown that a supermartingale characterizes the composition of the model and clear the composition of the optimality, representing the sufficient condition for an allocation to be optimal via a martingale. Further, the following three results are shown as the appliction of the above result. (1) For the Kennedy model another proof is given for the fact on an optimal allocation. (2) For a model with a convex utility function an optimal allocation is represented via a supermartingale. (3) For a model with a logarithmic utility function, an explicit optimal allocation is obtained.