{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:E2H7ZRPMKHNVWN75U6ZIE44BOO","short_pith_number":"pith:E2H7ZRPM","canonical_record":{"source":{"id":"1205.0295","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-02T00:21:52Z","cross_cats_sorted":[],"title_canon_sha256":"7372602c3807ac56db14954a277c17c30f3c216ecf70088f073556f411549c5b","abstract_canon_sha256":"7d616751b3450202416915706dc54b6d6f4cfaaed637d93fb87577f5d7b3fc0b"},"schema_version":"1.0"},"canonical_sha256":"268ffcc5ec51db5b37fda7b282738173bfba7f373b8c94bed9df145f3211a02d","source":{"kind":"arxiv","id":"1205.0295","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0295","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0295v2","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0295","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"pith_short_12","alias_value":"E2H7ZRPMKHNV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E2H7ZRPMKHNVWN75","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E2H7ZRPM","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:E2H7ZRPMKHNVWN75U6ZIE44BOO","target":"record","payload":{"canonical_record":{"source":{"id":"1205.0295","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-02T00:21:52Z","cross_cats_sorted":[],"title_canon_sha256":"7372602c3807ac56db14954a277c17c30f3c216ecf70088f073556f411549c5b","abstract_canon_sha256":"7d616751b3450202416915706dc54b6d6f4cfaaed637d93fb87577f5d7b3fc0b"},"schema_version":"1.0"},"canonical_sha256":"268ffcc5ec51db5b37fda7b282738173bfba7f373b8c94bed9df145f3211a02d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:46.898851Z","signature_b64":"fE4Xe4VKFYym3NFUt6bKnxhSqMZVwtA9iepMNMYkX+JDi/zBQu8WoYbHwdNPREbpapQUDhOrp/Xr3uNVwy2/BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"268ffcc5ec51db5b37fda7b282738173bfba7f373b8c94bed9df145f3211a02d","last_reissued_at":"2026-05-18T01:27:46.898159Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:46.898159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.0295","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":"2026-05-18T01:27:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jgi/PC+4bau8hoKevLWWXxHgSohek/mBjZLkYxDL8tE6fT8VqQXsRqBN38nJ/ybzQzLlBxtDTiUQvLW1AVMMDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T02:36:11.162575Z"},"content_sha256":"3ce114f2d547a660c63a4b3c536e1bd78efed01c6149a97dc59ce87f38663fc0","schema_version":"1.0","event_id":"sha256:3ce114f2d547a660c63a4b3c536e1bd78efed01c6149a97dc59ce87f38663fc0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:E2H7ZRPMKHNVWN75U6ZIE44BOO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Representation Theorem for Smooth Brownian Martingales","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Henry Schellhorn","submitted_at":"2012-05-02T00:21:52Z","abstract_excerpt":"We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin derivative. This result can also be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Founi\\'e (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0295","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":"2026-05-18T01:27:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IKSMKOH8cabz49Akft54al2Uei59it+3vo33q4LgIUzjRiaFCZZ4ZFAfHpxVl4SpyIhLGkoFT9tQoPJMXIgOAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T02:36:11.162999Z"},"content_sha256":"2bbebd7705531284d824eb4af7e1c387eb80dcf3be902ca2782172e4d5e078b3","schema_version":"1.0","event_id":"sha256:2bbebd7705531284d824eb4af7e1c387eb80dcf3be902ca2782172e4d5e078b3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/bundle.json","state_url":"https://pith.science/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/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":"2026-06-07T02:36:11Z","links":{"resolver":"https://pith.science/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO","bundle":"https://pith.science/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/bundle.json","state":"https://pith.science/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E2H7ZRPMKHNVWN75U6ZIE44BOO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:E2H7ZRPMKHNVWN75U6ZIE44BOO","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":"7d616751b3450202416915706dc54b6d6f4cfaaed637d93fb87577f5d7b3fc0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-02T00:21:52Z","title_canon_sha256":"7372602c3807ac56db14954a277c17c30f3c216ecf70088f073556f411549c5b"},"schema_version":"1.0","source":{"id":"1205.0295","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.0295","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"arxiv_version","alias_value":"1205.0295v2","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.0295","created_at":"2026-05-18T01:27:46Z"},{"alias_kind":"pith_short_12","alias_value":"E2H7ZRPMKHNV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E2H7ZRPMKHNVWN75","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E2H7ZRPM","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:2bbebd7705531284d824eb4af7e1c387eb80dcf3be902ca2782172e4d5e078b3","target":"graph","created_at":"2026-05-18T01:27:46Z","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 show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin derivative. This result can also be seen as a generalization of the semi-group theory of parabolic partial differential equations to the parabolic path-dependent partial differential equations introduced by Dupire (2009) and Cont and Founi\\'e (2011). The exponential operator can be calculated explicitly in a series expansion, which resembles t","authors_text":"Henry Schellhorn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-02T00:21:52Z","title":"A Representation Theorem for Smooth Brownian Martingales"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0295","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":"sha256:3ce114f2d547a660c63a4b3c536e1bd78efed01c6149a97dc59ce87f38663fc0","target":"record","created_at":"2026-05-18T01:27:46Z","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":"7d616751b3450202416915706dc54b6d6f4cfaaed637d93fb87577f5d7b3fc0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-02T00:21:52Z","title_canon_sha256":"7372602c3807ac56db14954a277c17c30f3c216ecf70088f073556f411549c5b"},"schema_version":"1.0","source":{"id":"1205.0295","kind":"arxiv","version":2}},"canonical_sha256":"268ffcc5ec51db5b37fda7b282738173bfba7f373b8c94bed9df145f3211a02d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"268ffcc5ec51db5b37fda7b282738173bfba7f373b8c94bed9df145f3211a02d","first_computed_at":"2026-05-18T01:27:46.898159Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:46.898159Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fE4Xe4VKFYym3NFUt6bKnxhSqMZVwtA9iepMNMYkX+JDi/zBQu8WoYbHwdNPREbpapQUDhOrp/Xr3uNVwy2/BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:46.898851Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.0295","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ce114f2d547a660c63a4b3c536e1bd78efed01c6149a97dc59ce87f38663fc0","sha256:2bbebd7705531284d824eb4af7e1c387eb80dcf3be902ca2782172e4d5e078b3"],"state_sha256":"f61dacafb1129a8a72830bbc5d614cc00ab9ba7a6fc47286181968357ab73456"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3ZaCczdBX/7sR0TNvfbVGAIO6sSO1kSYcBp2VO4VXKW4FQ/9URhPZpYTvyCM1N4D2dmsGemdyz4/ofddyOf8DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T02:36:11.167400Z","bundle_sha256":"a77723ab9eabeab5f519cbfccddf5bee33db60a91e6b2d0765ee155546dd6e6d"}}