{"id":"https://openalex.org/W4386835794","doi":"https://doi.org/10.3390/axioms12090888","title":"Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations","display_name":"Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations","publication_year":2023,"publication_date":"2023-09-18","ids":{"openalex":"https://openalex.org/W4386835794","doi":"https://doi.org/10.3390/axioms12090888"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12090888","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090888","pdf_url":"https://www.mdpi.com/2075-1680/12/9/888/pdf?version=1695010408","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/9/888/pdf?version=1695010408","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026591937","display_name":"Mahvish Samar","orcid":null},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Mahvish Samar","raw_affiliation_strings":["School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China","School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China","institution_ids":["https://openalex.org/I135237710"]},{"raw_affiliation_string":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China","institution_ids":["https://openalex.org/I135237710"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020757596","display_name":"Kutorzi Edwin Yao","orcid":"https://orcid.org/0000-0002-9728-2504"},"institutions":[{"id":"https://openalex.org/I154099455","display_name":"Shandong University","ror":"https://ror.org/0207yh398","country_code":"CN","type":"education","lineage":["https://openalex.org/I154099455"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Kutorzi Yao","raw_affiliation_strings":["School of Mathematics and Institute for Financial Studies, Shandong University, Jinan 250100, China"],"raw_orcid":"https://orcid.org/0000-0002-9728-2504","affiliations":[{"raw_affiliation_string":"School of Mathematics and Institute for Financial Studies, Shandong University, Jinan 250100, China","institution_ids":["https://openalex.org/I154099455"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100671239","display_name":"Xinzhong Zhu","orcid":"https://orcid.org/0000-0002-0033-5260"},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinzhong Zhu","raw_affiliation_strings":["School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China","institution_ids":["https://openalex.org/I135237710"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026591937"],"corresponding_institution_ids":["https://openalex.org/I135237710"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.9686,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.76363141,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":97},"biblio":{"volume":"12","issue":"9","first_page":"888","last_page":"888"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10282","display_name":"Financial Risk and Volatility Modeling","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9797999858856201,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/volterra-integral-equation","display_name":"Volterra integral equation","score":0.7107378840446472},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6682021021842957},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6350250840187073},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.620644748210907},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.5983243584632874},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.583397626876831},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5566205978393555},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5505695939064026},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5175844430923462},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.5032245516777039},{"id":"https://openalex.org/keywords/collocation-method","display_name":"Collocation method","score":0.5021369457244873},{"id":"https://openalex.org/keywords/numerical-analysis","display_name":"Numerical analysis","score":0.44375649094581604},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.41263580322265625},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.356057345867157},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25843456387519836},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.24054524302482605},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.15533357858657837}],"concepts":[{"id":"https://openalex.org/C21965488","wikidata":"https://www.wikidata.org/wiki/Q2632613","display_name":"Volterra integral equation","level":3,"score":0.7107378840446472},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6682021021842957},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6350250840187073},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.620644748210907},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.5983243584632874},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.583397626876831},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5566205978393555},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5505695939064026},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5175844430923462},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.5032245516777039},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.5021369457244873},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.44375649094581604},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.41263580322265625},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.356057345867157},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25843456387519836},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.24054524302482605},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.15533357858657837},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms12090888","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090888","pdf_url":"https://www.mdpi.com/2075-1680/12/9/888/pdf?version=1695010408","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:0e537acea8bf4193ac6aac88f6822e18","is_oa":true,"landing_page_url":"https://doaj.org/article/0e537acea8bf4193ac6aac88f6822e18","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 9, p 888 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms12090888","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090888","pdf_url":"https://www.mdpi.com/2075-1680/12/9/888/pdf?version=1695010408","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3380160785","display_name":"\u590d\u6742\u6761\u4ef6\u4e0b\u591a\u6a21\u6001\u6570\u636e\u5b66\u4e60\u7684\u57fa\u7840\u7406\u8bba\u4e0e\u65b9\u6cd5\u7814\u7a76","funder_award_id":"LZ22F030003","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"},{"id":"https://openalex.org/G4333678435","display_name":null,"funder_award_id":"61976196","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6049826414","display_name":null,"funder_award_id":"ZC304022938","funder_id":"https://openalex.org/F4320322819","funder_display_name":"Zhejiang Normal University"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322819","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4386835794.pdf"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W626109261","https://openalex.org/W1994005646","https://openalex.org/W1994791309","https://openalex.org/W1995400364","https://openalex.org/W1995644520","https://openalex.org/W2010836875","https://openalex.org/W2019013909","https://openalex.org/W2020442214","https://openalex.org/W2042330694","https://openalex.org/W2046331573","https://openalex.org/W2048693776","https://openalex.org/W2063999201","https://openalex.org/W2078895915","https://openalex.org/W2090187181","https://openalex.org/W2095445700","https://openalex.org/W2157883119","https://openalex.org/W2563778816","https://openalex.org/W2768331096","https://openalex.org/W2795296448","https://openalex.org/W2901089401","https://openalex.org/W2947856316","https://openalex.org/W2963045823","https://openalex.org/W2996399199","https://openalex.org/W3012154301","https://openalex.org/W3044716300","https://openalex.org/W3121912870","https://openalex.org/W3123798732","https://openalex.org/W3196235134","https://openalex.org/W4236109675","https://openalex.org/W4246136706","https://openalex.org/W4309332109","https://openalex.org/W4323847894","https://openalex.org/W4378907085"],"related_works":["https://openalex.org/W2517687681","https://openalex.org/W2118560105","https://openalex.org/W2358959122","https://openalex.org/W2078318892","https://openalex.org/W4225984655","https://openalex.org/W2031624200","https://openalex.org/W2791210196","https://openalex.org/W1987365920","https://openalex.org/W4220906687","https://openalex.org/W2183342624"],"abstract_inverted_index":{"This":[0],"work":[1],"uses":[2],"the":[3,31,43,49,52,63,102],"collocation":[4],"approximation":[5],"method":[6,87],"to":[7,41,47],"solve":[8],"a":[9,73,78,90,96],"specific":[10],"type":[11],"of":[12,36,51,67,104],"backward":[13],"stochastic":[14,33],"Volterra":[15],"integral":[16,98],"equations":[17],"(BSVIEs).":[18],"Using":[19],"Newton\u2019s":[20],"method,":[21,106],"BSVIEs":[22],"can":[23],"be":[24],"solved":[25],"using":[26,77,95],"block":[27],"pulse":[28],"functions":[29],"and":[30,46,69],"corresponding":[32],"operational":[34],"matrix":[35],"integration.":[37],"We":[38,82],"present":[39],"examples":[40],"illustrate":[42],"estimate":[44],"analysis":[45],"demonstrate":[48],"convergence":[50],"two":[53],"approximating":[54],"sequences":[55],"separately.":[56],"To":[57,100],"measure":[58],"their":[59],"accuracy,":[60],"we":[61,107],"compare":[62],"solutions":[64,71],"with":[65,111],"values":[66],"exact":[68],"approximative":[70],"at":[72],"few":[74],"selected":[75],"locations":[76],"specified":[79],"absolute":[80],"error.":[81],"also":[83],"propose":[84],"an":[85],"efficient":[86],"for":[88],"solving":[89],"triangular":[91],"linear":[92],"algebraic":[93],"problem":[94],"single":[97],"equation.":[99],"confirm":[101],"effectiveness":[103],"our":[105],"conduct":[108],"numerical":[109],"experiments":[110],"issues":[112],"from":[113],"real-world":[114],"applications.":[115]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}