Real-time target tracking: Optimal determination of the characteristics, using martingale tools

We study the problem of real-time target tracking: How do we use all the previous observations in order to get, at a given time, the best knowledge of the position of the target? A realistic approach is to take into account the processing time for each observation, and if the precision on the position is low, the processing will take longer. As was shown by C. Olivier (Real-time observability of targets with constrained processing power, to appear), the tracking is efficient when some affine random walk (characterizing the precision of the tracking) is almost surely bounded. Using martingale techniques, we show that this is the case under proper (and realistic) conditions on the parameters.