{"paper":{"title":"Embedding laws in diffusions by functions of time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. M. G. Cox, G. Peskir","submitted_at":"2012-01-25T16:26:01Z","abstract_excerpt":"We present a constructive probabilistic proof of the fact that if $B=(B_t)_{t\\ge0}$ is standard Brownian motion started at $0$, and $\\mu$ is a given probability measure on $\\mathbb{R}$ such that $\\mu(\\{0\\})=0$, then there exists a unique left-continuous increasing function $b:(0,\\infty)\\rightarrow\\mathbb{R}\\cup\\{+\\infty\\}$ and a unique left-continuous decreasing function $c:(0,\\infty)\\rightarrow\\mathbb{R}\\cup\\{-\\infty\\}$ such that $B$ stopped at $\\tau_{b,c}=\\inf\\{t>0\\vert B_t\\ge b(t)$ or $B_t\\le c(t)\\}$ has the law $\\mu$. The method of proof relies upon weak convergence arguments arising from "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5321","kind":"arxiv","version":4},"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"}