Quickest detection of a hidden target and extremal surfaces

Let $Z=(Z_t)_{t\ge0}$ be a regular diffusion process started at $0$, let $\ell$ be an independent random variable with a strictly increasing and continuous distribution function $F$, and let $\tau_{\ell}=\inf\{t\ge0\vert Z_t=\ell\}$ be the first entry time of $Z$ at the level $\ell$. We show that the quickest detection problem $$\inf_{\tau}\bigl[\mathsf{P}(\tau<\tau_{\ell})+c\mathsf{E}(\tau -\tau_{\ell})^+\bigr]$$ is equivalent to the (three-dimensional) optimal stopping problem $$\sup_{\tau}\mathsf{E}\biggl[R_{\tau}-\int _0^{\tau}c(R_t)\,dt\biggr],$$ where $R=S-I$ is the range process of $X=2F(Z)-1$ (i.e., the difference between the running maximum and the running minimum of $X$ ) and $c(r)=cr$ with $c>0$. Solving the latter problem we find that the following stopping time is optimal: $$\tau_*=\inf \bigl\{t\ge0\vert f_*(I_t,S_t)\le X_t\le g_*(I_t,S_t)\bigr\},$$ where the surfaces $f_*$ and $g_*$ can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of $X$ and $c$. This is done by extending the arguments associated with the maximality principle [Ann. Probab. 26 (1998) 1614-1640] to the three-dimensional setting of the present problem and disclosing the general structure of the solution that is valid in all particular cases. The key arguments developed in the proof should be applicable in similar multi-dimensional settings.