The Annals of Applied Probability

Quickest detection of a hidden target and extremal surfaces

Goran Peskir

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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


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.

Article information

Ann. Appl. Probab., Volume 24, Number 6 (2014), 2340-2370.

First available in Project Euclid: 26 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65]
Secondary: 34A34: Nonlinear equations and systems, general 35R35: Free boundary problems 49J40: Variational methods including variational inequalities [See also 47J20]

Quickest detection hidden target optimal stopping diffusion process maximum process minimum process range process excursion the maximality principle extremal surface the principle of smooth fit nonlinear differential equation


Peskir, Goran. Quickest detection of a hidden target and extremal surfaces. Ann. Appl. Probab. 24 (2014), no. 6, 2340--2370. doi:10.1214/13-AAP979.

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