Refined Estimation of the Bellman Function for Stochastic Optimal Control Problems with Probabilistic Performance Criterion

In this paper, the optimal control problem for a discrete-time stochastic system with a general-form probabilistic criterion is considered. Using dynamic programming and the properties of the Bellman function, new two-sided bounds of this function that refine the earlier results are constructed. The derived bounds are then adopted to justify the application of the modified strategy that is optimal in the two-step investment portfolio management problem under risk to the corresponding multistep problem. An example that illustrates the advantages of such a strategy over other well-known strategies is given.