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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1107.3760","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-19T16:20:00Z","cross_cats_sorted":[],"title_canon_sha256":"7ed7c1f7c68cc668ef30d34f4729ba093c7b8eb83b2bfef0c4009560bc3e122b","abstract_canon_sha256":"cdb74bd732d7b0f8bea755a17e65b13beaa96948570bfa038b46a67a5068ca90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:16.042152Z","signature_b64":"7CIwXPrmpYbVgX9K17Ox51dRQf9CgKNz69+kaApBDNTZwgp16XA7FkgcXnqvkW9N+0FAGW+7a++TmvAzzdXHBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a03d77a51a75eb4322aefcaa254226f1a7621dba25267e0508fbee8419fb2d84","last_reissued_at":"2026-05-18T04:17:16.041442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:16.041442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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}. 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