Quickest detection of a hidden target and extremal surfaces
arXiv:1409.1745 · doi:10.1214/13-AAP979
Abstract
Let be a regular diffusion process started at , let be an independent random variable with a strictly increasing and continuous distribution function , and let be the first entry time of at the level . We show that the quickest detection problem \[\inf_τ\bigl[\mathsf{P}(τ<τ_{\ell})+c\mathsf{E}(τ-τ_{\ell})^+\bigr]\] is equivalent to the (three-dimensional) optimal stopping problem \[\sup_τ\mathsf{E}\biggl[R_τ-\int _0^τc(R_t)\,dt\biggr],\] where is the range process of (i.e., the difference between the running maximum and the running minimum of ) and with . Solving the latter problem we find that the following stopping time is optimal: \[τ_*=\inf \bigl\{t\ge0\vert f_*(I_t,S_t)\le X_t\le g_*(I_t,S_t)\bigr\},\] where the surfaces and can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of and . 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.
Published in at http://dx.doi.org/10.1214/13-AAP979 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)