Tracking a random walk first-passage time through noisy observations

Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its firstpassage time ζ of a given level with a stopping time η defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation infη E|η - ζ | which become tight as →∞. Interestingly, in this regime the estimation error does not get smaller if we allow η to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particular case where there is no drift, we show that it is impossible to track ζ: infη E|η -ζp =∞for any >0 and p ≥ 1/2.