This work analyzes the necessary and sufficient conditions for the stability of the stochastic dynamic optimization algorithm for the calculus of the water cost for one hydroelectrical generation plant. We present the theory and different simulations for parameters inside and outside the stability region. A particular relationship between the time integration step, the space discretization step and the maximal incoming and outcoming flows. In order to show some advantages and disadvantages of the method, we show a simulation of the Uruguayan generation system with four hydroelectrical generation plants. We show that the stability conditions impose a small time simulation step, i.e., large total simulation time. As a future research direction, we think that this time can be reduced using non linear integration methods.