Tracking a Random Walk First-Passage Time Through Noisy Observations

TL;DR

This paper investigates the limits of estimating the first-passage time of a noisy Gaussian random walk, revealing fundamental bounds and impossibility results, especially in the no-drift case, with implications for stochastic process monitoring.

Contribution

It provides tight bounds on the estimation error of first-passage times in noisy observations and shows the limitations of tracking in the no-drift scenario.

Findings

Bounds on mean absolute deviation become tight as level increases

Estimation error does not decrease with more information in the no-drift case

Impossible to accurately track first-passage time without drift

Abstract

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 $\tau_\ell$ of a given level $\ell$ with a stopping time $\eta$ defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation $\inf_\eta \ex|\eta-\tau_\ell|$ which become tight as $\ell\to\infty$. Interestingly, in this regime the estimation error does not get smaller if we allow $ \eta$ 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 $\tau_\ell$: $\inf_\eta \ex|\eta-\tau_\ell|^p=\infty$ for any $\ell>0$ and $p\geq1/2$.