{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:GRNX2GJMJ53CR3DN4YIXA5HEA7","short_pith_number":"pith:GRNX2GJM","schema_version":"1.0","canonical_sha256":"345b7d192c4f7628ec6de6117074e407ec899cef45538fa7d6c824ce52907372","source":{"kind":"arxiv","id":"math/0601035","version":2},"attestation_state":"computed","paper":{"title":"G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shige Peng","submitted_at":"2006-01-03T08:20:24Z","abstract_excerpt":"We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expe"},"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":"math/0601035","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-01-03T08:20:24Z","cross_cats_sorted":[],"title_canon_sha256":"e9139eb32952b0bdaff91b5d890693b61d92f2372530e576078e5d822d5159f9","abstract_canon_sha256":"c8e3bf22fb477c2ed864e3cc22f6c3835abf484fad9229ecfb64d09b65e5388e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:58:17.456619Z","signature_b64":"8iBbVa96PfgShOJkYJKTbm8yz+TkyDXa5lkhZBt5I15reiAT4rqnHpSRPAfPLsZSmTewCq7iUOF1SQW4mM4eDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"345b7d192c4f7628ec6de6117074e407ec899cef45538fa7d6c824ce52907372","last_reissued_at":"2026-07-04T14:58:17.456261Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:58:17.456261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shige Peng","submitted_at":"2006-01-03T08:20:24Z","abstract_excerpt":"We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601035","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0601035/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0601035","created_at":"2026-07-04T14:58:17.456329+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0601035v2","created_at":"2026-07-04T14:58:17.456329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0601035","created_at":"2026-07-04T14:58:17.456329+00:00"},{"alias_kind":"pith_short_12","alias_value":"GRNX2GJMJ53C","created_at":"2026-07-04T14:58:17.456329+00:00"},{"alias_kind":"pith_short_16","alias_value":"GRNX2GJMJ53CR3DN","created_at":"2026-07-04T14:58:17.456329+00:00"},{"alias_kind":"pith_short_8","alias_value":"GRNX2GJM","created_at":"2026-07-04T14:58:17.456329+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.07867","citing_title":"Regularity of Solutions of Mean-Field $G$-SDEs","ref_index":23,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7","json":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7.json","graph_json":"https://pith.science/api/pith-number/GRNX2GJMJ53CR3DN4YIXA5HEA7/graph.json","events_json":"https://pith.science/api/pith-number/GRNX2GJMJ53CR3DN4YIXA5HEA7/events.json","paper":"https://pith.science/paper/GRNX2GJM"},"agent_actions":{"view_html":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7","download_json":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7.json","view_paper":"https://pith.science/paper/GRNX2GJM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0601035&json=true","fetch_graph":"https://pith.science/api/pith-number/GRNX2GJMJ53CR3DN4YIXA5HEA7/graph.json","fetch_events":"https://pith.science/api/pith-number/GRNX2GJMJ53CR3DN4YIXA5HEA7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7/action/storage_attestation","attest_author":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7/action/author_attestation","sign_citation":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7/action/citation_signature","submit_replication":"https://pith.science/pith/GRNX2GJMJ53CR3DN4YIXA5HEA7/action/replication_record"}},"created_at":"2026-07-04T14:58:17.456329+00:00","updated_at":"2026-07-04T14:58:17.456329+00:00"}