{"paper":{"title":"Killed Brownian motion with a prescribed lifetime distribution and models of default","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"q-fin.RM","authors_text":"Alexandru Hening, Boris Ettinger, Steven N. Evans","submitted_at":"2011-11-13T00:54:12Z","abstract_excerpt":"The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $\\zeta$, there exists a time-varying barrier $b$ such that $\\mathbb{P}\\{B_s>b(s),0\\leq s\\leq t\\}=\\mathbb{P}\\{\\zeta>t\\}$. We study a \"smoothed\" version of this problem and ask whether there is a \"barrier\" $b$ such that $ \\mathbb{E}[\\exp(-\\lambda\\int_0^t\\psi(B_s-b(s))\\,ds)]=\\mathbb{P}\\{\\zeta >t\\}$, where $\\lambda$ is a killing rate parameter, and $\\psi:\\mathbb{R}\\to[0,1]$ is a nonincreasing function. We prove that if $\\psi$ is suitably smooth, the function $t\\mapsto \\mathbb{P}\\{\\zeta>"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2976","kind":"arxiv","version":3},"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"}