A discrete-time, scalar, pursuit-evasion game is considered in which an evader, moving according to a stationary stochastic process, is continually being tracked by a pursuer. The pursuer's observations of the evader's positions are subject to observation error and time lag. It is shown that when the pursuer's strategy is linear in his information, the evader is able to infer the values of the observation errors (subject to the same time lag) and it is assumed that he utilises this information linearly in his evasion strategy. The payoff is taken to be the mean-square distance between pursuer and evader, the latter being subject to a quadratic constraint on his movement. The upper value and the corresponding strategies are found for all values of the variance θ of the observation error. The lower value problem is solved completely only for variances θ below a certain critical value and partially for variances above this critical value.