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 first-passage time τ l of a given level l 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 —τ l which become tight as l →∞. 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 τ l : infη Elη — τ l | p = ∞ for any l > 0 and p ≥ 1/2.