Optimal stochastic pursuit algorithms are often derived from quadratic loss functions. The smoothness of these functions enables standard variational methods of optimization and, thus, the optimal solutions are easily tractable. This gives algorithms optimized in terms of physical energy. In fact, fuel efficiency is a more natural criterion for these problems. The main difficulty is that this corresponds to absolute loss functions which are not differentiable. In this paper we use dynamic programming to derive an optimal control algorithm for the problem of pursuit with the absolute loss function. In our formulation the location of the target is not known precisely in advance. The localization of the target is gradually improved by Kalman filtering noisy data that becomes available in the process of pursuit. In the beginning of pursuit it is very cheap to make large maneuvers towards the target, but the problem is that the target location estimate is not accurate yet. Our algorithm finds an optimal balance in the tradeoff between saving fuel by making earlier maneuvers and waiting until the the target location estimate becomes more accurate.