{"paper":{"title":"Predicting the ultimate supremum of a stable L\\'{e}vy process with no negative jumps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Goran Peskir, Robert C. Dalang, Violetta Bernyk","submitted_at":"2010-04-13T09:38:29Z","abstract_excerpt":"Given a stable L\\'{e}vy process $X=(X_t)_{0\\le t\\le T}$ of index $\\alpha\\in(1,2)$ with no negative jumps, and letting $S_t=\\sup_{0\\le s\\le t}X_s$ denote its running supremum for $t\\in [0,T]$, we consider the optimal prediction problem \\[V=\\inf_{0\\le\\tau\\le T}\\mathsf{E}(S_T-X_{\\tau})^p,\\] where the infimum is taken over all stopping times $\\tau$ of $X$, and the error parameter $p\\in(1,\\alpha)$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann--Liouville type, and finding an explicit solution to the latter, we show that there exists $\\al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2133","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"}