ON AN $ \varepsilon $-OPTIMAL POLICY OF DISCRETE TIME STOCHASTIC CONTROL PROCESSES

In this paper, stochastic control processes have been investigated as dynamic programming models with an infinite horizon. In many cases, it is our main purpose to seek for an optimal policy under the various conditions. However, optimal policy may not exist under weak conditions. In such situation, it will be necessary to seek for $ \varepsilon $-optimal policy. Thus, after an $ \varepsilon $-optimal policy has been sought, we show that there exists at least a better policy than the given $ \varepsilon $-optimal one and that the better policy is near to the given $ \varepsilon $-optimal one. Then, the principle of Ekeland\u27s theorem will play an important role. Moreover, we introduce a modified control model with the perturbed one-stage cost functions and show that the better policy is an optimal one of the modified model