Estimating a Gaussian random walk first-passage time from noisy or delayed observations

Given a Gaussian random walk X with drift, we consider estimating its first-passage time τ, of a given level ℓ, with a stopping time η defined over an observation process Y that is either a noisy version of X, or a delayed version of X. For both cases, we provide lower bounds on average moments E|η - τ|^p, p ≥ 1, for any stopping rule η, and exhibit simple stopping rules that achieve these bounds in the large threshold regime and in the large threshold large delay regime, respectively. The results immediately extend to the corresponding continuous time settings where X and Y are standard Wiener processes with drift.