Measurable Selection and Dynamic Programming

A general multistage problem of stochastic optimization is studied. It is proved under some simple assumptions that the extremum in the problem is attained. The “Bellman functions” are constructed and a criterion of optimality in terms of these functions is given. The main tools used are measurable selection theorems. The paper generalizes the previous work of R. T. Rockafellar and R. J.-B. Wets devoted to the convex case.