{"paper":{"title":"Exponential functionals of L\\'evy processes with jumps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anita Behme","submitted_at":"2015-04-14T19:02:31Z","abstract_excerpt":"We study the exponential functional $\\int_0^\\infty e^{-\\xi_{s-}} \\, d\\eta_s$ of two one-dimensional independent L\\'evy processes $\\xi$ and $\\eta$, where $\\eta$ is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping $\\Phi_\\xi$ for a fixed L\\'evy process $\\xi$, which maps the law of $\\eta_1$ to the law of the corresponding exponential functional $\\int_0^\\infty e^{-\\xi_{s-}} \\, d\\eta_s$, and study the behaviour of the range of $\\Phi_\\xi$ for varying characteristics of $\\xi$. Mo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03660","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"}