An optimal feedback control is developed for fixed-speed, fixed-altitude Unmanned Aerial Vehicle (UAV) to maintain a nominal distance from a ground target in a way that anticipates its unknown future trajectory. Stochasticity is introduced in the problem by assuming that the target motion can be modeled as Brownian motion, which accounts for possible realizations of the unknown target kinematics. Moreover, the possibility for the interruption of observations is included by assuming that the duration of observation times of the target is exponentially distributed, giving rise to two discrete states of operation. A Bellman equation based on an approximating Markov chain that is consistent with the stochastic kinematics is used to compute an optimal control policy that minimizes the expected value of a cost function based on a nominal UAV-target distance. Results indicate how the uncertainty in the target motion, the tracker capabilities, and the time since the last observation can affect the control law, and simulations illustrate that the control can further be applied to other continuous, smooth trajectories with no need for additional computation.