Tracking a Random Walk First-Passage Time Through Noisy Observations

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 $\inf_η\ex|η-τ_\ell|$ 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 ${\tau }_{\ell }$: $\inf_η\ex|η-τ_\ell|^p=\infty$ for any $\ell >0$ and $p\ge 1/2$.

Reprint of the original article published in the Annals of Applied Probability