We consider the problem of bringing the controlled motion z( t) into a neighborhood of the random point y( t). The displacements of y( t) represent a stochastic diffusion process [1], and the motion of z( t) is described by linear differential equations involving the control function u. The control function u[ t, y, z] is formed at each instant of time t on the basis of the realized values of y( t) and z( t). It is shown that the problem of bringing the point z( t) into an ϵ-neighborhood of y( T)( T > 0) with a probability p < 1 has a finite solution u[ t, y, z] if the motion of z( t) is completely controlled in a certain sense and the parameters of the process y( t) are held within certain bounds. When the average value M{ y( t)} is described by linear equations, we obtain an explicit form of the control function u which is a linear function of y and z. Several optimal control problems are discussed incidentally. The problem is solved by the Liapunov function method [2,3] modernized for the present problem. This modernization makes use of concepts from the theory of dynamic programming [4].