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Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time \\[\\tau_*=\\inf\\{t\\ge0\\vert X_t\\ge f_*(I_t)\\}\\] minimizes $\\mathsf{E}(\\vert\\theta-\\tau\\vert-\\theta)$ over all stopping times $\\tau$ of $X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as the mi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.2891","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-12T14:20:07Z","cross_cats_sorted":[],"title_canon_sha256":"4d3b005238d4b279521cc35dd1383a43d3ce228b40420056b9ae950ea1c6db1f","abstract_canon_sha256":"8545240256a83e5654a9c5f2ef147d012bfab43161cb669bb062a5773f76c983"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:07.687855Z","signature_b64":"hkrUsmWZx/KLLeO4QiuVuM9adQm0m4Yk+4UmfzwbN7Mjs4SY0FY9ef/O6xiHwEpdhNP+O9m1iUOt1FSgc+kcCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2dfbfb29c0222aa644864f5c62ece1d5f1b11596bfdefda7bd5192c1fab76403","last_reissued_at":"2026-05-18T03:31:07.687233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:07.687233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three-dimensional Brownian motion and the golden ratio rule","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir, Hardy Hulley, Kristoffer Glover","submitted_at":"2013-03-12T14:20:07Z","abstract_excerpt":"Let $X=(X_t)_{t\\ge0}$ be a transient diffusion process in $(0,\\infty)$ with the diffusion coefficient $\\sigma>0$ and the scale function $L$ such that $X_t\\rightarrow\\infty$ as $t\\rightarrow \\infty$, let $I_t$ denote its running minimum for $t\\ge0$, and let $\\theta$ denote the time of its ultimate minimum $I_{\\infty}$. 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