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Let $m$ be an odd integer. Under the assumption that the quadratic variation $\\left[ M\\right] $ of $M$ is differentiable with $\\mathbf{E}\\left[ \\left\\vert d\\left[ M\\right] (t)/dt\\right\\vert ^{m}\\right] $ finite, it is shown that the $m$th power variation $$ \\lim_{\\varepsilon\\rightarrow0}\\varepsilon^{-1}\\int_{0}^{T}ds\\left( X\\left( s+\\varepsilon\\right) -X\\l"},"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":"1407.4568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-07-17T06:09:49Z","cross_cats_sorted":[],"title_canon_sha256":"2c76d272eb419a0d277432406f789a30ac2ac56cdb201823b62a747a4dd022bb","abstract_canon_sha256":"16a04d87a43f450c1499f84cf5509ca4b53669d0a2f048879fe4be3204219058"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:23.722539Z","signature_b64":"ljwY9N7oZ0Y9PYOfzP3ogHnSPi3vmVxGIXwoM6RxTEJb41z5Yvz2tWm6XhKayZ4RS92RsbbWp/QGU+NfYo1FBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a915402315f256692ca6a812e5d0c998cda68e10e1c5573c649b069d486e843","last_reissued_at":"2026-05-18T02:47:23.722177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:23.722177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Francesco Russo (UMA), Frederi Viens","submitted_at":"2014-07-17T06:09:49Z","abstract_excerpt":"We consider a class of stochastic processes $X$ defined by $X\\left( t\\right) =\\int_{0}^{T}G\\left( t,s\\right) dM\\left( s\\right) $ for $t\\in\\lbrack0,T]$, where $M$ is a square-integrable continuous martingale and $G$ is a deterministic kernel. Let $m$ be an odd integer. Under the assumption that the quadratic variation $\\left[ M\\right] $ of $M$ is differentiable with $\\mathbf{E}\\left[ \\left\\vert d\\left[ M\\right] (t)/dt\\right\\vert ^{m}\\right] $ finite, it is shown that the $m$th power variation $$ \\lim_{\\varepsilon\\rightarrow0}\\varepsilon^{-1}\\int_{0}^{T}ds\\left( X\\left( s+\\varepsilon\\right) -X\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.4568","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":"1407.4568","created_at":"2026-05-18T02:47:23.722233+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.4568v1","created_at":"2026-05-18T02:47:23.722233+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.4568","created_at":"2026-05-18T02:47:23.722233+00:00"},{"alias_kind":"pith_short_12","alias_value":"HKIVIARRL4SW","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"HKIVIARRL4SWNEWK","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"HKIVIARR","created_at":"2026-05-18T12:28:30.664211+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/HKIVIARRL4SWNEWKNKAS4XIMTG","json":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG.json","graph_json":"https://pith.science/api/pith-number/HKIVIARRL4SWNEWKNKAS4XIMTG/graph.json","events_json":"https://pith.science/api/pith-number/HKIVIARRL4SWNEWKNKAS4XIMTG/events.json","paper":"https://pith.science/paper/HKIVIARR"},"agent_actions":{"view_html":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG","download_json":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG.json","view_paper":"https://pith.science/paper/HKIVIARR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.4568&json=true","fetch_graph":"https://pith.science/api/pith-number/HKIVIARRL4SWNEWKNKAS4XIMTG/graph.json","fetch_events":"https://pith.science/api/pith-number/HKIVIARRL4SWNEWKNKAS4XIMTG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG/action/storage_attestation","attest_author":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG/action/author_attestation","sign_citation":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG/action/citation_signature","submit_replication":"https://pith.science/pith/HKIVIARRL4SWNEWKNKAS4XIMTG/action/replication_record"}},"created_at":"2026-05-18T02:47:23.722233+00:00","updated_at":"2026-05-18T02:47:23.722233+00:00"}