Some problem in the theory of dynamic programming

The theory of dynamic programming treats problems involving multi-stage processes by means of a transformation of the problem from the space of decisions to the space of functions. This is accomplished by deriving a functional equation whose solution is equivalent to the solution of the original problem. To illustrate this approach most clearly, free of extraneous analytic details, we consider a simple but nontrivial multi-stage investment problem. We show how exact solutions may be obtained in some cases, approximate solutions in others, and how these approximate solutions may be used to obtain more accurate solutions in the general case. Of particular importance is the decrease in the number of independent variables made possible by this approach. This is not only important from the theoretical standpoint, but is also of great value in reducing the cost in time and effort of numerical computation.