Automated Target Tracking in Real-Time: Optimal Adjustment of the Parameters

We study a problem in real-time target tracking by means of a one-dimensional "random" Kalman-Bucy filter. The tracking device has constrained processing power and hence has to choose (1) a subregion to be observed out of a larger region in which the target motion takes place and (2) a finite resolution scale. As a consequence the variance of the best prediction of the location of the target becomes a random process. The tracking is satisfactory if the expected time it takes the variance process to enter a finite strip is finite. We determine the optimal choice of subregion and resolution and give conditions on the model that insure a satisfactory tracking procedure for this choice. In addition, using large deviation estimates for Bernoulli random walks, we obtain exponential bounds for the tail probabilities of the expected entrance time of the variance process.