{"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 $I_{\\infty}$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2891","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"}