{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:JUETYCJ3BD4BTL3M6BIZNVH5YW","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":"5ac99c8dd84fd32fb313a2d828d89113ec82439058036300125d94001d79653f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-02-11T22:58:34Z","title_canon_sha256":"674c4288f07d7086ec74b7758627d329b4475a1ba87401d23aab1d6de440cc94"},"schema_version":"1.0","source":{"id":"1002.2446","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.2446","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"arxiv_version","alias_value":"1002.2446v5","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.2446","created_at":"2026-05-18T03:34:46Z"},{"alias_kind":"pith_short_12","alias_value":"JUETYCJ3BD4B","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JUETYCJ3BD4BTL3M","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JUETYCJ3","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:11f6e7fb4241f32a5439116643b6973d9c817ef7452cd8b4a6702773b0a84162","target":"graph","created_at":"2026-05-18T03:34:46Z","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"},"paper":{"abstract_excerpt":"We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticip","authors_text":"David-Antoine Fourni\\'e, Rama Cont","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-02-11T22:58:34Z","title":"Functional It\\^{o} calculus and stochastic integral representation of martingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.2446","kind":"arxiv","version":5},"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:fae57a4d34495414d44180ddccbfd2d9c4b0f660a4ff2de377081852f5ba0dd8","target":"record","created_at":"2026-05-18T03:34:46Z","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":"5ac99c8dd84fd32fb313a2d828d89113ec82439058036300125d94001d79653f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-02-11T22:58:34Z","title_canon_sha256":"674c4288f07d7086ec74b7758627d329b4475a1ba87401d23aab1d6de440cc94"},"schema_version":"1.0","source":{"id":"1002.2446","kind":"arxiv","version":5}},"canonical_sha256":"4d093c093b08f819af6cf05196d4fdc5a5b58d17767000bee81013c819b31564","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d093c093b08f819af6cf05196d4fdc5a5b58d17767000bee81013c819b31564","first_computed_at":"2026-05-18T03:34:46.641636Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:46.641636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xl8p6hxaTXQNiFKRW90oL5Ilh+tjxwYqP3Zx2f20lqjmaodQbqrBxc0/mNW9ZyEk90IX0X+FeCOH1RzAjS87Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:46.642160Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.2446","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fae57a4d34495414d44180ddccbfd2d9c4b0f660a4ff2de377081852f5ba0dd8","sha256:11f6e7fb4241f32a5439116643b6973d9c817ef7452cd8b4a6702773b0a84162"],"state_sha256":"c2494ef0f3f5e7bdac5fadc1ea972dcb47616aa73ac9c8ef6c9c8664f3df30f4"}