{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:EKR54BVI577UGYV5Y5H2O2PBWK","short_pith_number":"pith:EKR54BVI","canonical_record":{"source":{"id":"1011.1475","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-05T18:26:03Z","cross_cats_sorted":["math.CA","q-fin.CP"],"title_canon_sha256":"0620aed38856209306dbcd06a0c22a2aff18a4fc6e31fc53172600709f9ab909","abstract_canon_sha256":"1af67738e7c5a265688e1d897ac280e406e30d40e8139bdf454c15b619e4a327"},"schema_version":"1.0"},"canonical_sha256":"22a3de06a8efff4362bdc74fa769e1b2b0c7641716d5d6852d06ddb49c1341b0","source":{"kind":"arxiv","id":"1011.1475","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1475","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1475v2","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1475","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"pith_short_12","alias_value":"EKR54BVI577U","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EKR54BVI577UGYV5","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EKR54BVI","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:EKR54BVI577UGYV5Y5H2O2PBWK","target":"record","payload":{"canonical_record":{"source":{"id":"1011.1475","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-05T18:26:03Z","cross_cats_sorted":["math.CA","q-fin.CP"],"title_canon_sha256":"0620aed38856209306dbcd06a0c22a2aff18a4fc6e31fc53172600709f9ab909","abstract_canon_sha256":"1af67738e7c5a265688e1d897ac280e406e30d40e8139bdf454c15b619e4a327"},"schema_version":"1.0"},"canonical_sha256":"22a3de06a8efff4362bdc74fa769e1b2b0c7641716d5d6852d06ddb49c1341b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:13.428991Z","signature_b64":"5wnR7k7oYX3uTOpyMGIzbZXVa+UqzwxvsmQWC8V3Qn8A+JRWqvvVb5Do9ehbLNYgCQmKji1o3fbYgh4/4dMGDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22a3de06a8efff4362bdc74fa769e1b2b0c7641716d5d6852d06ddb49c1341b0","last_reissued_at":"2026-05-18T02:47:13.428393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:13.428393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.1475","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:47:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"We2OHFxRMZFWdA/o3VGBoQCNYyOQyfBx6ot7jZimfYdemAUWMqt9SKGL5Ekiuac829NMn6mEnLRz1LI6pfB4DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T12:01:34.498284Z"},"content_sha256":"fe160641c3acb1155a19fa8f1075dd959e3af092c354edf26dcc5a6ced402f9a","schema_version":"1.0","event_id":"sha256:fe160641c3acb1155a19fa8f1075dd959e3af092c354edf26dcc5a6ced402f9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:EKR54BVI577UGYV5Y5H2O2PBWK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock's formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","q-fin.CP"],"primary_cat":"math.PR","authors_text":"Hassan Allouba, Ramiro Fontes","submitted_at":"2010-11-05T18:26:03Z","abstract_excerpt":"In a 2006 article (\\cite{A1}), Allouba gave his quadratic covariation differentiation theory for It\\^o's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to It\\^o's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1475","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:47:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AmCf1Eq+Vnif7+CRROmunfIVltxB/XjVSECzaGLB7LmkcpzPOWLaotP4yZzIDgRJezjABLS+W6hsSbdamk8bAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T12:01:34.498636Z"},"content_sha256":"d73901d35fdec163c564fb74e6bebfc2a9dc0dfc2fa53cd3f9ced5721b774517","schema_version":"1.0","event_id":"sha256:d73901d35fdec163c564fb74e6bebfc2a9dc0dfc2fa53cd3f9ced5721b774517"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EKR54BVI577UGYV5Y5H2O2PBWK/bundle.json","state_url":"https://pith.science/pith/EKR54BVI577UGYV5Y5H2O2PBWK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EKR54BVI577UGYV5Y5H2O2PBWK/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":"2026-06-30T12:01:34Z","links":{"resolver":"https://pith.science/pith/EKR54BVI577UGYV5Y5H2O2PBWK","bundle":"https://pith.science/pith/EKR54BVI577UGYV5Y5H2O2PBWK/bundle.json","state":"https://pith.science/pith/EKR54BVI577UGYV5Y5H2O2PBWK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EKR54BVI577UGYV5Y5H2O2PBWK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EKR54BVI577UGYV5Y5H2O2PBWK","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":"1af67738e7c5a265688e1d897ac280e406e30d40e8139bdf454c15b619e4a327","cross_cats_sorted":["math.CA","q-fin.CP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-05T18:26:03Z","title_canon_sha256":"0620aed38856209306dbcd06a0c22a2aff18a4fc6e31fc53172600709f9ab909"},"schema_version":"1.0","source":{"id":"1011.1475","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1475","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1475v2","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1475","created_at":"2026-05-18T02:47:13Z"},{"alias_kind":"pith_short_12","alias_value":"EKR54BVI577U","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EKR54BVI577UGYV5","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EKR54BVI","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:d73901d35fdec163c564fb74e6bebfc2a9dc0dfc2fa53cd3f9ced5721b774517","target":"graph","created_at":"2026-05-18T02:47:13Z","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":"In a 2006 article (\\cite{A1}), Allouba gave his quadratic covariation differentiation theory for It\\^o's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to It\\^o's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory","authors_text":"Hassan Allouba, Ramiro Fontes","cross_cats":["math.CA","q-fin.CP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-05T18:26:03Z","title":"Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock's formulas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1475","kind":"arxiv","version":2},"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:fe160641c3acb1155a19fa8f1075dd959e3af092c354edf26dcc5a6ced402f9a","target":"record","created_at":"2026-05-18T02:47:13Z","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":"1af67738e7c5a265688e1d897ac280e406e30d40e8139bdf454c15b619e4a327","cross_cats_sorted":["math.CA","q-fin.CP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-05T18:26:03Z","title_canon_sha256":"0620aed38856209306dbcd06a0c22a2aff18a4fc6e31fc53172600709f9ab909"},"schema_version":"1.0","source":{"id":"1011.1475","kind":"arxiv","version":2}},"canonical_sha256":"22a3de06a8efff4362bdc74fa769e1b2b0c7641716d5d6852d06ddb49c1341b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"22a3de06a8efff4362bdc74fa769e1b2b0c7641716d5d6852d06ddb49c1341b0","first_computed_at":"2026-05-18T02:47:13.428393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:47:13.428393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5wnR7k7oYX3uTOpyMGIzbZXVa+UqzwxvsmQWC8V3Qn8A+JRWqvvVb5Do9ehbLNYgCQmKji1o3fbYgh4/4dMGDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:47:13.428991Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1475","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe160641c3acb1155a19fa8f1075dd959e3af092c354edf26dcc5a6ced402f9a","sha256:d73901d35fdec163c564fb74e6bebfc2a9dc0dfc2fa53cd3f9ced5721b774517"],"state_sha256":"48bec05cc5832d14554f9334349b737e7b3b6ba19ea73daa74a5f09f482d038a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rNyg9ZyoaYjFTBy3UlPczDbpJkNZJABHaX27dmb5a4k9Wq/3z+F2dw4ygV0XjwRzpRnBH+5bRPOa0bMd/sQkAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T12:01:34.500424Z","bundle_sha256":"d2955b17b9efb5fa1635e1f9c768bd1843e8b7609316fdf21cc667df9e82255a"}}