Deterministic and Stochastic Control Problems with Identical Optimal Cost Functions

We consider two control problems on a finite horizon; one stochastic and the other deterministic. In both problems the running cost and the terminal cost are the same. The controllable input in both problems is of an additive nature with cost proportional to the input (which can be both positive and negatve, depending on the size of the input). The dynamics in the stochastic model is the same as that of the deterministic but with additional "noise" term. We show that the value functions in both problems are the same and the optimal policy for the stochastic problem is to keep the controlled process as close as possible to the optimal trajectory for the deterministic problem