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 $\tau_A$ for a given level $A$ from an observation process $Y$ correlated to $X$. Estimators may be any stopping times $\eta$ 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 $\mathbf E f((\eta-\tau_A)/r)$ as $A,d\to\infty$, 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.