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 $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|$ 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 $\tau_{\ell}$: $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|^{p}=\infty$ for any $\ell>0$ and $p\geq1/2$.