{"paper":{"title":"Suprema of L\\'{e}vy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jacek Ma{\\l}ecki, Mateusz Kwa\\'snicki, Micha{\\l} Ryznar","submitted_at":"2011-03-04T16:04:18Z","abstract_excerpt":"In this paper we study the supremum functional $M_t=\\sup_{0\\le s\\le t}X_s$, where $X_t$, $t\\ge0$, is a one-dimensional L\\'{e}vy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the L\\'{e}vy-Khintchin exponent of the process increases on $(0,\\infty)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0935","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"}