{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:TNY573U6562DILG6C457BFJJ7E","short_pith_number":"pith:TNY573U6","schema_version":"1.0","canonical_sha256":"9b71dfee9eefb4342cde173bf09529f90dc3e43d40c11766b2ab9bfd43833d78","source":{"kind":"arxiv","id":"2605.12950","version":1},"attestation_state":"computed","paper":{"title":"Stochastic Mean-Field LQ Stackelberg Differential Games with Random Coefficients: Theory and a Deep FBSDE Picard Solver","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Stackelberg optimal controls in mean-field LQ games with random coefficients are characterized by a Riccati-free coupled FBSDE system solved by a deep Picard method.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jie Xiong, Ying Yang, Zhouyu Wang","submitted_at":"2026-05-13T03:34:11Z","abstract_excerpt":"This paper studies a stochastic mean-field linear-quadratic Stackelberg differential game with random coefficients. The interaction between mean-field terms and random coefficients precludes the direct use of conventional decoupling techniques. We apply an extended Lagrange multiplier method to derive an affine operator representation of the follower's optimal response. The induced leader problem is then formulated as a generalized stochastic LQ control problem with operator-valued coefficients, and the Stackelberg optimal control is characterized through a Riccati-free coupled FBSDE system. W"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.12950","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-05-13T03:34:11Z","cross_cats_sorted":[],"title_canon_sha256":"85c3babc31c40e7ba9014030a8da2746499f32503a3b8c1d582e074ce0986efb","abstract_canon_sha256":"f10d050a7a6ba9a3e4c209588986d493e086e66c9621090fecf9a671111ed1b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:09.421003Z","signature_b64":"fQnwHjR7bA7VJpPcmujYoc2eD8+AyjQqp1C5vFEN3k9wu/xGEyVvcubh9BU9BXGU+U0PTxy0DoCSL4Aa5dZIAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b71dfee9eefb4342cde173bf09529f90dc3e43d40c11766b2ab9bfd43833d78","last_reissued_at":"2026-05-18T03:09:09.420261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:09.420261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stochastic Mean-Field LQ Stackelberg Differential Games with Random Coefficients: Theory and a Deep FBSDE Picard Solver","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Stackelberg optimal controls in mean-field LQ games with random coefficients are characterized by a Riccati-free coupled FBSDE system solved by a deep Picard method.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jie Xiong, Ying Yang, Zhouyu Wang","submitted_at":"2026-05-13T03:34:11Z","abstract_excerpt":"This paper studies a stochastic mean-field linear-quadratic Stackelberg differential game with random coefficients. The interaction between mean-field terms and random coefficients precludes the direct use of conventional decoupling techniques. We apply an extended Lagrange multiplier method to derive an affine operator representation of the follower's optimal response. The induced leader problem is then formulated as a generalized stochastic LQ control problem with operator-valued coefficients, and the Stackelberg optimal control is characterized through a Riccati-free coupled FBSDE system. W"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The Stackelberg optimal control is characterized through a Riccati-free coupled FBSDE system, and the Deep FBSDE Picard Solver preserves the Stackelberg order through follower-response learning, response-sensitivity extraction, leader optimization, and neural augmented Lagrangian enforcement of mean-field consistency constraints.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the extended Lagrange multiplier method successfully produces an affine operator representation of the follower's optimal response despite the interaction between mean-field terms and random coefficients, and that the resulting FBSDE system admits solutions that the deep Picard solver can approximate reliably.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives a Riccati-free coupled FBSDE characterization for mean-field LQ Stackelberg games with random coefficients and proposes a Deep FBSDE Picard Solver that learns follower responses and enforces mean-field consistency.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Stackelberg optimal controls in mean-field LQ games with random coefficients are characterized by a Riccati-free coupled FBSDE system solved by a deep Picard method.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3c9c94e6a8e8f3be3695703ccb7dc4072929a486451ed12cb538beef3b5fa9d4"},"source":{"id":"2605.12950","kind":"arxiv","version":1},"verdict":{"id":"b04fa08a-47e2-4c8d-bd1f-231959f350ae","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:21:59.461890Z","strongest_claim":"The Stackelberg optimal control is characterized through a Riccati-free coupled FBSDE system, and the Deep FBSDE Picard Solver preserves the Stackelberg order through follower-response learning, response-sensitivity extraction, leader optimization, and neural augmented Lagrangian enforcement of mean-field consistency constraints.","one_line_summary":"Derives a Riccati-free coupled FBSDE characterization for mean-field LQ Stackelberg games with random coefficients and proposes a Deep FBSDE Picard Solver that learns follower responses and enforces mean-field consistency.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the extended Lagrange multiplier method successfully produces an affine operator representation of the follower's optimal response despite the interaction between mean-field terms and random coefficients, and that the resulting FBSDE system admits solutions that the deep Picard solver can approximate reliably.","pith_extraction_headline":"Stackelberg optimal controls in mean-field LQ games with random coefficients are characterized by a Riccati-free coupled FBSDE system solved by a deep Picard method."},"references":{"count":39,"sample":[{"doi":"","year":1985,"title":"H. Abou-Kandil and P. Bertrand,Analytical solution for an open-loop Stackelberg game, IEEE, 1985","work_id":"9540531e-d4ce-44f6-a782-02db239b172a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"B. Acciaio, J. Backhoff-Veraguas, and R. Carmona,Extended mean field control prob- lems: stochastic maximum principle and transport perspective, SIAM, 2019","work_id":"c1a77c7d-eb7d-4c4e-b12a-7f7d81ddba5b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"C. Beck, W. E, A. Jentzen, Machine learning approximation algorithms for high- dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, J","work_id":"bd9a22aa-a3d0-4d82-b698-8a70df15c6cb","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A. Bensoussan, M. H. M. Chau, and S. C. P. Yam,Mean field Stackelberg games: Ag- gregation of delayed instructions, SIAM, 2015","work_id":"79f0f280-22b3-404f-816b-54a4c1ee750c","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"A. Bensoussan, M. H. M. Chau, Y. Lai, and S. C. P. Yam,Linear-quadratic mean field Stackelberg games with state and control delays, SIAM, 2017","work_id":"33e4cce2-c074-4a6a-965c-a929d7e606f8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":39,"snapshot_sha256":"2e953ba9469447dad5ac9cca3f2b1659af84c28eb142811fc1aaacc8128ff1ab","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"3794223bfb64f81132851eceb989c3b5993b2ed63a5446dd532aa5a1ae7b2e9d"},"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":"2605.12950","created_at":"2026-05-18T03:09:09.420391+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.12950v1","created_at":"2026-05-18T03:09:09.420391+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12950","created_at":"2026-05-18T03:09:09.420391+00:00"},{"alias_kind":"pith_short_12","alias_value":"TNY573U6562D","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"TNY573U6562DILG6","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"TNY573U6","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E","json":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E.json","graph_json":"https://pith.science/api/pith-number/TNY573U6562DILG6C457BFJJ7E/graph.json","events_json":"https://pith.science/api/pith-number/TNY573U6562DILG6C457BFJJ7E/events.json","paper":"https://pith.science/paper/TNY573U6"},"agent_actions":{"view_html":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E","download_json":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E.json","view_paper":"https://pith.science/paper/TNY573U6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.12950&json=true","fetch_graph":"https://pith.science/api/pith-number/TNY573U6562DILG6C457BFJJ7E/graph.json","fetch_events":"https://pith.science/api/pith-number/TNY573U6562DILG6C457BFJJ7E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E/action/storage_attestation","attest_author":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E/action/author_attestation","sign_citation":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E/action/citation_signature","submit_replication":"https://pith.science/pith/TNY573U6562DILG6C457BFJJ7E/action/replication_record"}},"created_at":"2026-05-18T03:09:09.420391+00:00","updated_at":"2026-05-18T03:09:09.420391+00:00"}