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In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\\int_0^t\\eta_sds$ or $\\int_0^t\\gamma_sd\\langle B\\rangle_s$, $\\eta, \\gamma \\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-It\\^o processes is unique: For $\\zeta\\in H^1_G(0,T)$, $\\eta\\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \\[\\int_0^t\\zeta_s dB_s+\\int_0^t\\eta_sds+K_t=L_t,\\ t\\in[0,T],\\] then we have $\\eta\\equiv0$, $\\zet"},"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":"1607.00616","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-07-03T08:54:47Z","cross_cats_sorted":[],"title_canon_sha256":"35fc8e12bfdd4bb00f1974ab0e2b8cd4bbf1413ec5def9619ced6b2c4d98d3a1","abstract_canon_sha256":"0819c6be4832dbe60a8ae0772dc8238fcc509ff4b641d5bf2584c238fcf781cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:34.605298Z","signature_b64":"mtDMR4dVm506uoYBPp8Q06R/qi+txbEDDtErSK/+lEIswSNmRzPTzVOC8GY1cU/K+hnxaxaaZ5MMj+3DLsgDAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"907dc8ae7038a53385287479e91c02770413547ca7ddea8aed571484ccfa7e82","last_reissued_at":"2026-05-18T01:11:34.604902Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:34.604902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Properties of $G$-martingales with finite variation and the application to $G$-Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yongsheng Song","submitted_at":"2016-07-03T08:54:47Z","abstract_excerpt":"As is known, a process of form $\\int_0^t\\eta_sd\\langle B\\rangle_s-\\int_0^t2G(\\eta_s)ds$, $\\eta\\in M^1_G(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\\int_0^t\\eta_sds$ or $\\int_0^t\\gamma_sd\\langle B\\rangle_s$, $\\eta, \\gamma \\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-It\\^o processes is unique: For $\\zeta\\in H^1_G(0,T)$, $\\eta\\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \\[\\int_0^t\\zeta_s dB_s+\\int_0^t\\eta_sds+K_t=L_t,\\ t\\in[0,T],\\] then we have $\\eta\\equiv0$, $\\zet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00616","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":"1607.00616","created_at":"2026-05-18T01:11:34.604975+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.00616v1","created_at":"2026-05-18T01:11:34.604975+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00616","created_at":"2026-05-18T01:11:34.604975+00:00"},{"alias_kind":"pith_short_12","alias_value":"SB64RLTQHCST","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SB64RLTQHCSTHBJI","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SB64RLTQ","created_at":"2026-05-18T12:30:44.179134+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/SB64RLTQHCSTHBJIOR46SHACO4","json":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4.json","graph_json":"https://pith.science/api/pith-number/SB64RLTQHCSTHBJIOR46SHACO4/graph.json","events_json":"https://pith.science/api/pith-number/SB64RLTQHCSTHBJIOR46SHACO4/events.json","paper":"https://pith.science/paper/SB64RLTQ"},"agent_actions":{"view_html":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4","download_json":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4.json","view_paper":"https://pith.science/paper/SB64RLTQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.00616&json=true","fetch_graph":"https://pith.science/api/pith-number/SB64RLTQHCSTHBJIOR46SHACO4/graph.json","fetch_events":"https://pith.science/api/pith-number/SB64RLTQHCSTHBJIOR46SHACO4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4/action/storage_attestation","attest_author":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4/action/author_attestation","sign_citation":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4/action/citation_signature","submit_replication":"https://pith.science/pith/SB64RLTQHCSTHBJIOR46SHACO4/action/replication_record"}},"created_at":"2026-05-18T01:11:34.604975+00:00","updated_at":"2026-05-18T01:11:34.604975+00:00"}