In this paper sequential tests of one sided hypotheses H1: μ ≤ μ0and H2: μ > μ0for the velocity parameter of a stationary Ornstein-Uhlenbeck process(OUP) are discussed. When the variance parameter of the OUP is known the results obtained here complement a recent work of Roters (1991) by proving the existence of minimax sequential tests under concave and symmetric (about μ0) loss l(μ)with l(μ0) > 0 and linear costs of observation. For unkown variance parameter minimax sequential tests are derived for all loss functions discussed here and in the above mentioned paper.