In Part I of the paper the mean cost for a unit of time arising from a non-terminating diffusion process, denoted by $\Theta$ , is defined. One part of the cost originates from the motion inside the interval between two boundaries, the other part originates in the jumps from these boundaries. $\Theta$ is characterised by Theorem I. In Part II it is supposed that the diffusion coefficient and the coefficient of the local shift of the process depend on a control variable. The optimum $\hat\Theta$ of realizable mean costs may be determined by means of Theorem 2.