{"paper":{"title":"On the density of exponential functionals of L\\'evy processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Juan Carlos Pardo, Kees van Schaik, Victor Rivero","submitted_at":"2011-07-19T16:20:00Z","abstract_excerpt":"In this paper, we study the existence of the density associated to the exponential functional of the L\\'evy process $\\xi$, \\[ I_{\\ee_q}:=\\int_0^{\\ee_q} e^{\\xi_s} \\, \\mathrm{d}s, \\] where $\\ee_q$ is an independent exponential r.v. with parameter $q\\geq 0$. In the case when $\\xi$ is the negative of a subordinator, we prove that the density of $I_{\\ee_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \\cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $\\xi$ is the negative of a s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3760","kind":"arxiv","version":1},"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"}