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We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and $P(T_b<t,T_a>T_b)$ and we determine a system of integral equations verified by thes"},"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":"1403.1756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-03-07T14:08:04Z","cross_cats_sorted":[],"title_canon_sha256":"ffef07453e9895fe9b8415e3d946c3535c4bc5bfc4fc2ff3adc8ac527c03dc66","abstract_canon_sha256":"9f61f3dbbb39f0d027ae9a322c8f50ec2f60afe3dd35a00ee97e593905296fe1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:52.610082Z","signature_b64":"DZO9iasAXjFHl/xQjB8eBsI1BhNW/nSvHqFpth1KYoFzHGOXNBnxRHc2Odg8QN+UfUBK4qVrFuw+X+9A9PecCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"530dfafb35136b89dcdab566a10fb26b45a82c6c52e934c4e6361ceeba3113a5","last_reissued_at":"2026-05-18T02:56:52.609441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:52.609441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Joint densities of first hitting times of a diffusion process through two time dependent boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Cristina Zucca, Laura Sacerdote, Ottavia Telve","submitted_at":"2014-03-07T14:08:04Z","abstract_excerpt":"Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\\in I$, $\\forall t>t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and $P(T_b<t,T_a>T_b)$ and we determine a system of integral equations verified by thes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.1756","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":"1403.1756","created_at":"2026-05-18T02:56:52.609536+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.1756v1","created_at":"2026-05-18T02:56:52.609536+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.1756","created_at":"2026-05-18T02:56:52.609536+00:00"},{"alias_kind":"pith_short_12","alias_value":"KMG7V6ZVCNVY","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KMG7V6ZVCNVYTXG2","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KMG7V6ZV","created_at":"2026-05-18T12:28:35.611951+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/KMG7V6ZVCNVYTXG2WVTKCD5SNN","json":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN.json","graph_json":"https://pith.science/api/pith-number/KMG7V6ZVCNVYTXG2WVTKCD5SNN/graph.json","events_json":"https://pith.science/api/pith-number/KMG7V6ZVCNVYTXG2WVTKCD5SNN/events.json","paper":"https://pith.science/paper/KMG7V6ZV"},"agent_actions":{"view_html":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN","download_json":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN.json","view_paper":"https://pith.science/paper/KMG7V6ZV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.1756&json=true","fetch_graph":"https://pith.science/api/pith-number/KMG7V6ZVCNVYTXG2WVTKCD5SNN/graph.json","fetch_events":"https://pith.science/api/pith-number/KMG7V6ZVCNVYTXG2WVTKCD5SNN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN/action/storage_attestation","attest_author":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN/action/author_attestation","sign_citation":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN/action/citation_signature","submit_replication":"https://pith.science/pith/KMG7V6ZVCNVYTXG2WVTKCD5SNN/action/replication_record"}},"created_at":"2026-05-18T02:56:52.609536+00:00","updated_at":"2026-05-18T02:56:52.609536+00:00"}