{"paper":{"title":"Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generalized multiple Fourier series enable strong approximation of iterated Ito stochastic integrals for arbitrary multiplicity.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dmitriy F. Kuznetsov","submitted_at":"2020-03-28T03:22:52Z","abstract_excerpt":"The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. The monograph opens up a new direction in researching of ISIs. For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in $L_2([t, T]^k)$ for the expansion and strong approximation of Ito ISIs of multiplicity $k,$ $k\\in{\\bf N}$ (Chapter 1). This result has been adapted for Stratonovich ISIs of multiplicities 1 to 8 (t"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in L_2([t, T]^k) for the expansion and strong approximation of Ito ISIs of multiplicity k, k in N.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The generalized multiple Fourier series of the integrand converges in L2 norm to the iterated integral itself for arbitrary multiplicity k, without additional restrictions on the weight functions beyond those stated for the Stratonovich case (Chapter 1).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops strong mean-square approximations for iterated stochastic integrals of multiplicity k using generalized multiple Fourier series expansions, with explicit error formulas and applications to numerical solution of Ito SDEs and semilinear SPDEs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized multiple Fourier series enable strong approximation of iterated Ito stochastic integrals for arbitrary multiplicity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a3e681664850ca58df86b68082aec1042b34eb3387c491ea40b1770eafbd2d7e"},"source":{"id":"2003.14184","kind":"arxiv","version":76},"verdict":{"id":"e0e8cbac-6f95-424e-ae80-c70db4a9e4ac","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T14:18:40.408647Z","strongest_claim":"For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in L_2([t, T]^k) for the expansion and strong approximation of Ito ISIs of multiplicity k, k in N.","one_line_summary":"Develops strong mean-square approximations for iterated stochastic integrals of multiplicity k using generalized multiple Fourier series expansions, with explicit error formulas and applications to numerical solution of Ito SDEs and semilinear SPDEs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The generalized multiple Fourier series of the integrand converges in L2 norm to the iterated integral itself for arbitrary multiplicity k, without additional restrictions on the weight functions beyond those stated for the Stratonovich case (Chapter 1).","pith_extraction_headline":"Generalized multiple Fourier series enable strong approximation of iterated Ito stochastic integrals for arbitrary multiplicity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.14184/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"}