Mathematics – Statistics Theory
Scientific paper
2010-05-04
Mathematics
Statistics Theory
18 pages, typo corrected
Scientific paper
Given two dependent stochastic processes X and Y, and a stopping time S on X, the tracking stopping time problem consists in finding a stopping time T on Y that best tracks S, e.g., so as to minimize the mean absolute deviation E|T-S|. This problem formulation applies in several areas including control, communication, and finance. However, the problem is in general hard to solve analytically as it generalizes the well-known (Bayesian) change-point detection problem for which solutions have been reported only for specific settings. In this paper we provide an analytical solution to a tracking stopping time problem that cannot be formulated as a change-point problem. For the setting where X and Y are correlated Gaussian random walks, and where S is the crossing time of some given threshold, we provide upper and lower bounds on the minimum value of E|T-S| whose main asymptotic terms coincide as the threshold tends to infinity. The results immediately extend to the continuous time setting where X and Y are correlated standard Brownian motions with drift.
Burnashev Marat
Tchamkerten Aslan
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