Sequential estimation of a threshold crossing time for a Gaussian random walk through correlated observations

Given a Gaussian random walk X with drift, we consider the problem of estimating its first-passage time τA for a given level A from an observation process Y correlated to X. Estimators may be any stopping times η with respect to the observation process Y. Two cases of the process Y are considered: a noisy version of X and a process X with delay d. For a given loss function f(x), in both cases we find exact asymptotics of the minimal possible risk Ef((η − τA)/r) as A, d → ∞, where r is a normalizing coefficient. The results are extended to the corresponding continuous-time setting where X and Y are Brownian motions with drift.