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Quickest detection of a hidden target and extremal surfaces. (English) Zbl 1338.60115

Summary: Let \(Z=(Z_{t})_{t\geq0}\) 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 \(\tau_{\ell}=\inf\{t\geq0 \mid Z_{t}=\ell\}\) be the first entry time of \(Z\) at the level \(\ell\). We show that the quickest detection problem \[ \inf_{\tau}\big[P(\tau<\tau_{\ell})+cE(\tau-\tau_{\ell})^{+}\big] \] is equivalent to the (three-dimensional) optimal stopping problem \[ \sup_{\tau}E\Bigg[R_{\tau}-\int_{0}^{\tau}c(R_{t})\,dt\Bigg], \] 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: \[ \tau_{*}=\inf \{t\geq0 \mid f_{*}(I_{t},S_{t})\leq X_{t}\leq g_{*}(I_{t},S_{t})\}, \] 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 [G. Peskir, Ann. Probab. 26, No. 4, 1614–1640 (1998; Zbl 0935.60025)] 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.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
60G70 Extreme value theory; extremal stochastic processes
34A34 Nonlinear ordinary differential equations and systems
35R35 Free boundary problems for PDEs

Citations:

Zbl 0935.60025
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Full Text: DOI arXiv Euclid

References:

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