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December 2014 Quickest detection of a hidden target and extremal surfaces
Goran Peskir
Ann. Appl. Probab. 24(6): 2340-2370 (December 2014). DOI: 10.1214/13-AAP979

Abstract

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

\[\inf_{\tau}[\mathsf{P}(\tau<\tau_{\ell})+c\mathsf{E}(\tau -\tau_{\ell})^{+}]\]

is equivalent to the (three-dimensional) optimal stopping problem

\[\sup_{\tau}\mathsf{E}[R_{\tau}-\int_{0}^{\tau}c(R_{t})\,dt],\]

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:

\[\tau_{*}=\inf \{t\ge0\vert f_{*}(I_{t},S_{t})\le X_{t}\le g_{*}(I_{t},S_{t})\},\]

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.

Citation

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Goran Peskir. "Quickest detection of a hidden target and extremal surfaces." Ann. Appl. Probab. 24 (6) 2340 - 2370, December 2014. https://doi.org/10.1214/13-AAP979

Information

Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1338.60115
MathSciNet: MR3262505
Digital Object Identifier: 10.1214/13-AAP979

Subjects:
Primary: 60G35 , 60G40 , 60J60
Secondary: 34A34 , 35R35 , 49J40

Keywords: diffusion process , excursion , extremal surface , hidden target , maximum process , minimum process , nonlinear differential equation , Optimal stopping , quickest detection , range process , the maximality principle , the principle of smooth fit

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 6 • December 2014
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