{"paper":{"title":"Optional splitting formula in a progressively enlarged filtration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shiqi Song","submitted_at":"2012-08-21T00:41:06Z","abstract_excerpt":"Let $\\mathbb{F}$ be a filtration and $\\tau$ be a random time. Let $\\mathbb{G}$ be the progressive enlargement of $\\mathbb{F}$ with $\\tau$. We study the validity of the following formula, called optional splitting formula : For any $\\mathbb{G}$-optional process $Y$, there exist a $\\mathbb{F}$-optional process $Y'$ and a function $Y\"$ defined on $[0,\\infty]\\times(\\mathbb{R}_+\\times\\Omega)$ being $\\mathcal{B}[0,\\infty]\\otimes\\mathcal{O}(\\mathbb{F})$ measurable, such that $$ Y=Y'\\ind_{[0,\\tau)}+Y\"(\\tau)\\ind_{[\\tau,\\infty)} $$ We are interested in this formula, because it has been taken for granted"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4149","kind":"arxiv","version":2},"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"}