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We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(\\tau_x, K_x, L_x).$"},"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":"1603.02506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-08T12:57:47Z","cross_cats_sorted":[],"title_canon_sha256":"e2a1d328615df1bf98571efbea394fd77bb99596313ae82545a5d6150fa2911c","abstract_canon_sha256":"cca789e317d1d0cd36a04102c2d183fa23d960ce214d977156a39717bdad1aab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:23.052204Z","signature_b64":"47DJh8DncUuFJq5DOaWDt3HZ1/qLV1JjDYQlX+qvhaFWDF6OI/e24PevU4EDGmPN9FBruyFPEDqmiYXj5xOjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c159f5ec326062587fd985db595fc4cf71e04b966145d00f6fe185d75031d46","last_reissued_at":"2026-05-18T01:19:23.051547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:23.051547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Joint law of the hitting time, overshoot and undershoot for a L\\'evy process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Laure Coutin, Waly Ngom","submitted_at":"2016-03-08T12:57:47Z","abstract_excerpt":"Let be $(X_t, t\\geq 0)$ be a L\\'evy process which is the sum of a Brownian motion with drift and a compound Poisson process. 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Secondly, we calculate the joint law of $(\\tau_x, K_x, L_x).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02506","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.02506","created_at":"2026-05-18T01:19:23.051651+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.02506v1","created_at":"2026-05-18T01:19:23.051651+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02506","created_at":"2026-05-18T01:19:23.051651+00:00"},{"alias_kind":"pith_short_12","alias_value":"JQKZ6XWDEYDC","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"JQKZ6XWDEYDCLB75","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"JQKZ6XWD","created_at":"2026-05-18T12:30:25.849896+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/JQKZ6XWDEYDCLB75TBO3LFP4JT","json":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT.json","graph_json":"https://pith.science/api/pith-number/JQKZ6XWDEYDCLB75TBO3LFP4JT/graph.json","events_json":"https://pith.science/api/pith-number/JQKZ6XWDEYDCLB75TBO3LFP4JT/events.json","paper":"https://pith.science/paper/JQKZ6XWD"},"agent_actions":{"view_html":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT","download_json":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT.json","view_paper":"https://pith.science/paper/JQKZ6XWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.02506&json=true","fetch_graph":"https://pith.science/api/pith-number/JQKZ6XWDEYDCLB75TBO3LFP4JT/graph.json","fetch_events":"https://pith.science/api/pith-number/JQKZ6XWDEYDCLB75TBO3LFP4JT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT/action/storage_attestation","attest_author":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT/action/author_attestation","sign_citation":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT/action/citation_signature","submit_replication":"https://pith.science/pith/JQKZ6XWDEYDCLB75TBO3LFP4JT/action/replication_record"}},"created_at":"2026-05-18T01:19:23.051651+00:00","updated_at":"2026-05-18T01:19:23.051651+00:00"}