Since a pursuer pursuing a maneuvering target does not know what maneuvers an evading target will make, the maneuvers (the target’s control law) appear as a random process to the pursuer. However, he has opinions about what the evader will do. From these, he can assign a prior probability distribution to the evader’s maneuvers. For a linear pursuit evasion problem in which the evader’s control law is modeled as a random process, in which the pursuer has partial noisy linear measurements of his own and the evader’s relative position, and a quadratic optimality criterion is used, past results of the authors imply that the optimal control is a linear function of the "predicted miss." Determining the predicted miss involves estimating the evader’s terminal position from past system measurements, Nonlinear filtering techniques are used to give expressions for computing the conditional expectation of the evader’s terminal position even in the presence of the random unknown maneuvers of the evader.