Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. Let $\tau_l:=\inf\{t>0:X_t\geq l\}$ and $\eta$ an arbitrary estimator of $\tau$ defined over $Y$. I want to show that for a given $h$ there exists a constant $C$ such that
$$\inf_{\eta(Y)}\mathbb{E}f(|\eta-\tau_l|)\leq C\mathbb{E}f(\tau_h/2)$$
Assume that $X$ and $Y$ are $\mathbb{R}$ valued and that $X$ can only be accessed through $Y$.
Here is roughly my progress: if $Z$ is a standard Brownian motion then $X\overset{\mathcal{D}}{=}Z$ and so
$$\inf_{\eta(Y)}\mathbb{E}f(|\eta-\tau_l|)\geq\inf_{\eta(Z)}\mathbb{E}f(|\eta-\tau_l|)$$ since we have removed the noise from $Y$. I was thinking about doing something like
$$\inf_{\eta(Z)}\mathbb{E}f(|\eta-\tau_l|)\geq\inf_{\eta(Z)}\{\mathbb{E}f(|\eta-\tau_l|)\mathbf{1}_{\{X_t\leq h\}}+\mathbb{E}f(|\eta-\tau_l|)\mathbf{1}_{\{X_t> h\}}\}$$ but I'm at a loss as to what to do next.