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We study the first-hitting time of the process $D$, namely, the process $E(t) = \\inf \\{s: D(s) > t \\}$, $t \\geq 0$.\n  The process $E$ is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the $n$-time tail distribution function $P[E(t_1) > s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution given natural boundary conditions."},"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":"0906.5083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-06-27T14:51:34Z","cross_cats_sorted":[],"title_canon_sha256":"8c2ed8b009fb897c07d3214e69368d7d21f301768d298bf78eb33f7c601c0fe5","abstract_canon_sha256":"6447c1fdf43aca1c204c2d6fa1614cdf77bdce4a05a2523a051d8902aee5ee3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:45:16.223982Z","signature_b64":"+JFKOnpR/ee/kdZkkWuKtdCTpkHo6KGDZ/CBfME9wxQQgIedf+b8WISg3PyQCRYuM7Fa/gK9GcNxC3LCHNFuAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6866ff0f484196d319ed6e0ff589dbffb5312c808e36bbbeed7cdd35841c3fa4","last_reissued_at":"2026-07-04T15:45:16.223642Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:45:16.223642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mark S. 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We derive a partial differential equation for the Laplace transform of the $n$-time tail distribution function $P[E(t_1) > s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution given natural boundary conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.5083","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0906.5083/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0906.5083","created_at":"2026-07-04T15:45:16.223703+00:00"},{"alias_kind":"arxiv_version","alias_value":"0906.5083v1","created_at":"2026-07-04T15:45:16.223703+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.5083","created_at":"2026-07-04T15:45:16.223703+00:00"},{"alias_kind":"pith_short_12","alias_value":"NBTP6D2IIGLN","created_at":"2026-07-04T15:45:16.223703+00:00"},{"alias_kind":"pith_short_16","alias_value":"NBTP6D2IIGLNGGPN","created_at":"2026-07-04T15:45:16.223703+00:00"},{"alias_kind":"pith_short_8","alias_value":"NBTP6D2I","created_at":"2026-07-04T15:45:16.223703+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376","json":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376.json","graph_json":"https://pith.science/api/pith-number/NBTP6D2IIGLNGGPNNYH7LCO376/graph.json","events_json":"https://pith.science/api/pith-number/NBTP6D2IIGLNGGPNNYH7LCO376/events.json","paper":"https://pith.science/paper/NBTP6D2I"},"agent_actions":{"view_html":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376","download_json":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376.json","view_paper":"https://pith.science/paper/NBTP6D2I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0906.5083&json=true","fetch_graph":"https://pith.science/api/pith-number/NBTP6D2IIGLNGGPNNYH7LCO376/graph.json","fetch_events":"https://pith.science/api/pith-number/NBTP6D2IIGLNGGPNNYH7LCO376/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376/action/storage_attestation","attest_author":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376/action/author_attestation","sign_citation":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376/action/citation_signature","submit_replication":"https://pith.science/pith/NBTP6D2IIGLNGGPNNYH7LCO376/action/replication_record"}},"created_at":"2026-07-04T15:45:16.223703+00:00","updated_at":"2026-07-04T15:45:16.223703+00:00"}