Estimating a Random Walk First-Passage Time From Noisy or Delayed Observations

A Gaussian random walk (or a Wiener process), possibly with drift, is observed in a noisy or delayed fashion. The problem considered in this paper is to estimate the first time
$\tau $
the random walk reaches a given level. Specifically, the average
$p$
-moment (
$p\geq 1$
) optimization problem
$\inf _{\eta} {\BBE} \vert \eta -\tau \vert ^{p}$
is investigated where the infimum is taken over the set of stopping times that are defined on the observation process. When there is no drift, optimal stopping rules are characterized for both types of observations. When there is a drift, upper and lower bounds on
$\inf _{\eta} {\BBE} \vert \eta -\tau \vert ^{p}$
are established for both types of observations. The bounds are tight in the large-level regime for noisy observations and in the large-level-large-delay regime for delayed observations. Noteworthy, for noisy observations there exists an asymptotically optimal stopping rule that is a function of a single observation. Simulation results are provided that corroborate the validity of the results for non-asymptotic settings.