{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:22EX6ZQXNMY7VUANQAV2XG6WQB","short_pith_number":"pith:22EX6ZQX","schema_version":"1.0","canonical_sha256":"d6897f66176b31fad00d802bab9bd6806672550d8906d26287908442d3e3fb4b","source":{"kind":"arxiv","id":"1603.01720","version":1},"attestation_state":"computed","paper":{"title":"The quadratic covariation for a weighted fractional Brownian motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Litan Yan, Qinghua Zhang, Xichao Sun","submitted_at":"2016-03-05T12:38:12Z","abstract_excerpt":"Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property $$ E[(B^{a,b}_{s+\\varepsilon}-B^{a,b}_s)^2] =O(\\varepsilon^{1+b})\\not\\sim \\varepsilon^{1+a+b}=E[(B^{a,b}_{\\varepsilon})^2]\\qquad (\\varepsilon\\to 0) $$ for all $s>0$, we consider the generalized quadratic covariation $\\bigl[f(B^{a,b}),B^{a,b}\\bigr]^{(a,b)}$ defined by $$ \\bigl[f(B^{a,b}),B^{a,b}\\bigr]^{(a,b)}_t=\\lim_{\\varepsilon\\downarrow 0}\\frac{1+a+b}{\\varepsilon^{1+b}}\\int_\\varepsilon^{t+\\varepsilon} \\left\\{f(B^{a,b}_{s+\\varepsilon}) "},"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":"1603.01720","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2016-03-05T12:38:12Z","cross_cats_sorted":[],"title_canon_sha256":"8798bcd9ca9ca166ec53ae6422621839caf5d59a8ac67bbc2b20efd4644ac251","abstract_canon_sha256":"9d923b41e29ee73bb8b318e7db028dc80506e5b38698c85d15e626404be9097a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:34.936899Z","signature_b64":"c3z/baALB/zoUpyRbZ6eqUq5xmXpiEUcAcJQrGQbXmQsxMt/0xUrEIb9GfsTnoTN3lOSuEaNsbjfS+mHN33MBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6897f66176b31fad00d802bab9bd6806672550d8906d26287908442d3e3fb4b","last_reissued_at":"2026-05-18T01:19:34.936388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:34.936388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The quadratic covariation for a weighted fractional Brownian motion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Litan Yan, Qinghua Zhang, Xichao Sun","submitted_at":"2016-03-05T12:38:12Z","abstract_excerpt":"Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property $$ E[(B^{a,b}_{s+\\varepsilon}-B^{a,b}_s)^2] =O(\\varepsilon^{1+b})\\not\\sim \\varepsilon^{1+a+b}=E[(B^{a,b}_{\\varepsilon})^2]\\qquad (\\varepsilon\\to 0) $$ for all $s>0$, we consider the generalized quadratic covariation $\\bigl[f(B^{a,b}),B^{a,b}\\bigr]^{(a,b)}$ defined by $$ \\bigl[f(B^{a,b}),B^{a,b}\\bigr]^{(a,b)}_t=\\lim_{\\varepsilon\\downarrow 0}\\frac{1+a+b}{\\varepsilon^{1+b}}\\int_\\varepsilon^{t+\\varepsilon} \\left\\{f(B^{a,b}_{s+\\varepsilon}) "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01720","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":"1603.01720","created_at":"2026-05-18T01:19:34.936463+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.01720v1","created_at":"2026-05-18T01:19:34.936463+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01720","created_at":"2026-05-18T01:19:34.936463+00:00"},{"alias_kind":"pith_short_12","alias_value":"22EX6ZQXNMY7","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"22EX6ZQXNMY7VUAN","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"22EX6ZQX","created_at":"2026-05-18T12:29:52.810259+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/22EX6ZQXNMY7VUANQAV2XG6WQB","json":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB.json","graph_json":"https://pith.science/api/pith-number/22EX6ZQXNMY7VUANQAV2XG6WQB/graph.json","events_json":"https://pith.science/api/pith-number/22EX6ZQXNMY7VUANQAV2XG6WQB/events.json","paper":"https://pith.science/paper/22EX6ZQX"},"agent_actions":{"view_html":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB","download_json":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB.json","view_paper":"https://pith.science/paper/22EX6ZQX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.01720&json=true","fetch_graph":"https://pith.science/api/pith-number/22EX6ZQXNMY7VUANQAV2XG6WQB/graph.json","fetch_events":"https://pith.science/api/pith-number/22EX6ZQXNMY7VUANQAV2XG6WQB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB/action/storage_attestation","attest_author":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB/action/author_attestation","sign_citation":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB/action/citation_signature","submit_replication":"https://pith.science/pith/22EX6ZQXNMY7VUANQAV2XG6WQB/action/replication_record"}},"created_at":"2026-05-18T01:19:34.936463+00:00","updated_at":"2026-05-18T01:19:34.936463+00:00"}