{"paper":{"title":"Quickest detection of a hidden target and extremal surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir","submitted_at":"2014-09-05T11:42:51Z","abstract_excerpt":"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=2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1745","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}