Summary: The problem of optimally tracking a stochastic process based on noisy measurements through a window is being considered. Such a problem arises in a variety of applications where the field of view of the measuring device is limited (e.g., the ``aiming'' problem). With the objective to minimize the probability to lose track, the problem is formulated as an optimal control problem with partial observations. The existence of an optimal control is proven for both cases of discrete and continuous time (diffusion) signal process observed in white noise. The low observation noise asymptotics are then considered: for a one-dimensional problem, a proposed suboptimal tracker is shown to be asymptotically logarithmically optimal.