{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:PV2ZNR7WNA5WVQMOG5MLIXS6LY","short_pith_number":"pith:PV2ZNR7W","canonical_record":{"source":{"id":"2412.11126","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-15T09:16:26Z","cross_cats_sorted":[],"title_canon_sha256":"ce885f611c24273a1459032553efb73b328dfcc387eb705bcb2189ef41975f87","abstract_canon_sha256":"a83412f600bc7b1593291d6cc7aa9ac1bb11249afc7aa51c28e4de5729127f47"},"schema_version":"1.0"},"canonical_sha256":"7d7596c7f6683b6ac18e3758b45e5e5e2e928528a0fb933083be771d90cbbfe7","source":{"kind":"arxiv","id":"2412.11126","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.11126","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"arxiv_version","alias_value":"2412.11126v1","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11126","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_12","alias_value":"PV2ZNR7WNA5W","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_16","alias_value":"PV2ZNR7WNA5WVQMO","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_8","alias_value":"PV2ZNR7W","created_at":"2026-07-05T09:49:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:PV2ZNR7WNA5WVQMOG5MLIXS6LY","target":"record","payload":{"canonical_record":{"source":{"id":"2412.11126","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-15T09:16:26Z","cross_cats_sorted":[],"title_canon_sha256":"ce885f611c24273a1459032553efb73b328dfcc387eb705bcb2189ef41975f87","abstract_canon_sha256":"a83412f600bc7b1593291d6cc7aa9ac1bb11249afc7aa51c28e4de5729127f47"},"schema_version":"1.0"},"canonical_sha256":"7d7596c7f6683b6ac18e3758b45e5e5e2e928528a0fb933083be771d90cbbfe7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:49:27.870501Z","signature_b64":"I6kRdFwFmRw9dXUTQ3p3U3FJ7PXK3+ot5ccjhS/qMxe31noEtfoA4bu3Zr3QvJF2DJi+lTuORBcfPj8TDWpKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d7596c7f6683b6ac18e3758b45e5e5e2e928528a0fb933083be771d90cbbfe7","last_reissued_at":"2026-07-05T09:49:27.870051Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:49:27.870051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.11126","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:49:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S6qVUFS03cXkRcZPAh/w9FGYWKwBvl7VwdHIkkshqLZ7Wlj0p/JU0RI1ejD61I3xnCDi2xmEG8/w3c8ABbFIBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:03:24.232199Z"},"content_sha256":"b26fa7e2abc7e02312d3af932096765b5ab60914fb86efe0debc7daa86b9ae4d","schema_version":"1.0","event_id":"sha256:b26fa7e2abc7e02312d3af932096765b5ab60914fb86efe0debc7daa86b9ae4d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:PV2ZNR7WNA5WVQMOG5MLIXS6LY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hongjun Gao, Shanqi Liu, Yaozhong Hu","submitted_at":"2024-12-15T09:16:26Z","abstract_excerpt":"For stochastic Volterra equations driven by standard Brownian and with singular kernels $K(u)=u^{H-\\frac{1}{2}}/\\Gamma(H+1/2), H\\in (0,1/2)$, it is known that the Milstein scheme has a convergence rate of $n^{-2H}$. In this paper, we show that this rate is optimal. Moreover, we show that the error normalized by $n^{-2H}$ converge stably in law to the (nonzero) solution of a certain linear Volterra equation of random coefficients with the same fractional kernel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11126","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.11126/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:49:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q+BAhz6Q0NwQHfrwsH3C/x5KDgiP7JcK5Tzr1fFTPzJm64PcpT4EYdMPO7Ekqs1nLR4EANRTnfplqu+o2tjADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:03:24.232702Z"},"content_sha256":"407dd420e5518186f354160180f30635fae9621d08564a56301b8ac4747c04b8","schema_version":"1.0","event_id":"sha256:407dd420e5518186f354160180f30635fae9621d08564a56301b8ac4747c04b8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/bundle.json","state_url":"https://pith.science/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T15:03:24Z","links":{"resolver":"https://pith.science/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY","bundle":"https://pith.science/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/bundle.json","state":"https://pith.science/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PV2ZNR7WNA5WVQMOG5MLIXS6LY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:PV2ZNR7WNA5WVQMOG5MLIXS6LY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a83412f600bc7b1593291d6cc7aa9ac1bb11249afc7aa51c28e4de5729127f47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-15T09:16:26Z","title_canon_sha256":"ce885f611c24273a1459032553efb73b328dfcc387eb705bcb2189ef41975f87"},"schema_version":"1.0","source":{"id":"2412.11126","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.11126","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"arxiv_version","alias_value":"2412.11126v1","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.11126","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_12","alias_value":"PV2ZNR7WNA5W","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_16","alias_value":"PV2ZNR7WNA5WVQMO","created_at":"2026-07-05T09:49:27Z"},{"alias_kind":"pith_short_8","alias_value":"PV2ZNR7W","created_at":"2026-07-05T09:49:27Z"}],"graph_snapshots":[{"event_id":"sha256:407dd420e5518186f354160180f30635fae9621d08564a56301b8ac4747c04b8","target":"graph","created_at":"2026-07-05T09:49:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.11126/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For stochastic Volterra equations driven by standard Brownian and with singular kernels $K(u)=u^{H-\\frac{1}{2}}/\\Gamma(H+1/2), H\\in (0,1/2)$, it is known that the Milstein scheme has a convergence rate of $n^{-2H}$. In this paper, we show that this rate is optimal. Moreover, we show that the error normalized by $n^{-2H}$ converge stably in law to the (nonzero) solution of a certain linear Volterra equation of random coefficients with the same fractional kernel.","authors_text":"Hongjun Gao, Shanqi Liu, Yaozhong Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-15T09:16:26Z","title":"Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11126","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b26fa7e2abc7e02312d3af932096765b5ab60914fb86efe0debc7daa86b9ae4d","target":"record","created_at":"2026-07-05T09:49:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a83412f600bc7b1593291d6cc7aa9ac1bb11249afc7aa51c28e4de5729127f47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-12-15T09:16:26Z","title_canon_sha256":"ce885f611c24273a1459032553efb73b328dfcc387eb705bcb2189ef41975f87"},"schema_version":"1.0","source":{"id":"2412.11126","kind":"arxiv","version":1}},"canonical_sha256":"7d7596c7f6683b6ac18e3758b45e5e5e2e928528a0fb933083be771d90cbbfe7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d7596c7f6683b6ac18e3758b45e5e5e2e928528a0fb933083be771d90cbbfe7","first_computed_at":"2026-07-05T09:49:27.870051Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:49:27.870051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I6kRdFwFmRw9dXUTQ3p3U3FJ7PXK3+ot5ccjhS/qMxe31noEtfoA4bu3Zr3QvJF2DJi+lTuORBcfPj8TDWpKBw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:49:27.870501Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.11126","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b26fa7e2abc7e02312d3af932096765b5ab60914fb86efe0debc7daa86b9ae4d","sha256:407dd420e5518186f354160180f30635fae9621d08564a56301b8ac4747c04b8"],"state_sha256":"dc55c58515f4d4d80eee1992d1fe9b09e38414ed96110f718a208b345ba334bd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p0WuKq8/ZmfWeCtlgWiY26h/qbaxQRaFiZ0oRx5dCTvzzVOMuT0RboyeRnaIPxHbKhxyYw2YDKXpCV1kVzX3Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T15:03:24.236489Z","bundle_sha256":"00fa02053b2cfca1be11d0db7a3dfcac9160b08c1ebc69fc9b95a3dbb617e2c5"}}