{"id":"https://openalex.org/W7202344265","doi":"https://doi.org/10.3934/math.20261002","title":"A regularized semi-discrete optimal control framework for recovering local volatility surfaces","display_name":"A regularized semi-discrete optimal control framework for recovering local volatility surfaces","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7202344265","doi":"https://doi.org/10.3934/math.20261002"},"language":"en","primary_location":{"id":"doi:10.3934/math.20261002","doi":"https://doi.org/10.3934/math.20261002","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.20261002","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/8/PDF/math-11-08-1002.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"http://www.aimspress.com/aimspress-data/math/2026/8/PDF/math-11-08-1002.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100321927","display_name":"Yun Liu","orcid":"https://orcid.org/0000-0001-8000-1641","observed_orcids":["https://orcid.org/0000-0001-8000-1641"]},"institutions":[{"id":"https://openalex.org/I3129293551","display_name":"Tarim University","ror":"https://ror.org/05202v862","country_code":"CN","type":"education","lineage":["https://openalex.org/I3129293551"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yun Liu","raw_affiliation_strings":["College of Information Engineering , Tarim University , Alar , Xinjiang 843300 , China","Key Laboratory of Tarim Oasis Agriculture ( Tarim University) Ministry of Education , Alar , Xinjiang 843300 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Information Engineering , Tarim University , Alar , Xinjiang 843300 , China","institution_ids":["https://openalex.org/I3129293551"]},{"raw_affiliation_string":"Key Laboratory of Tarim Oasis Agriculture ( Tarim University) Ministry of Education , Alar , Xinjiang 843300 , China","institution_ids":["https://openalex.org/I3129293551"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101964912","display_name":"Lifeng Guo","orcid":"https://orcid.org/0000-0003-0519-4791","observed_orcids":["https://orcid.org/0000-0003-0519-4791"]},"institutions":[{"id":"https://openalex.org/I3129293551","display_name":"Tarim University","ror":"https://ror.org/05202v862","country_code":"CN","type":"education","lineage":["https://openalex.org/I3129293551"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lifeng Guo","raw_affiliation_strings":["College of Information Engineering , Tarim University , Alar , Xinjiang 843300 , China","Key Laboratory of Tarim Oasis Agriculture ( Tarim University) Ministry of Education , Alar , Xinjiang 843300 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Information Engineering , Tarim University , Alar , Xinjiang 843300 , China","institution_ids":["https://openalex.org/I3129293551"]},{"raw_affiliation_string":"Key Laboratory of Tarim Oasis Agriculture ( Tarim University) Ministry of Education , Alar , Xinjiang 843300 , China","institution_ids":["https://openalex.org/I3129293551"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5110568586","display_name":"Xijuan Liu","orcid":"https://orcid.org/0000-0003-0988-1505","observed_orcids":["https://orcid.org/0000-0003-0988-1505"]},"institutions":[{"id":"https://openalex.org/I3129293551","display_name":"Tarim University","ror":"https://ror.org/05202v862","country_code":"CN","type":"education","lineage":["https://openalex.org/I3129293551"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xijuan Liu","raw_affiliation_strings":["College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China","Key Laboratory of Tarim Oasis Agriculture (Tarim University) Ministry of Education, Alar, Xinjiang 843300, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China","institution_ids":["https://openalex.org/I3129293551"]},{"raw_affiliation_string":"Key Laboratory of Tarim Oasis Agriculture (Tarim University) Ministry of Education, Alar, Xinjiang 843300, China","institution_ids":["https://openalex.org/I3129293551"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I3129293551"],"apc_list":{"value":1200,"currency":"USD","value_usd":1200},"apc_paid":{"value":1200,"currency":"USD","value_usd":1200},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"11","issue":"8","first_page":"24928","last_page":"24957"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.5522000193595886,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.5522000193595886,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.4336000084877014,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/local-volatility-surface","display_name":"local volatility surface","score":0.847000002861023},{"id":"https://openalex.org/keywords/volatility-calibration","display_name":"volatility calibration","score":0.8019999861717224},{"id":"https://openalex.org/keywords/black-scholes-model","display_name":"Black-Scholes model","score":0.9700000286102295},{"id":"https://openalex.org/keywords/dupire-model","display_name":"Dupire model","score":0.8040000200271606},{"id":"https://openalex.org/keywords/tikhonov-regularization","display_name":"Tikhonov regularization","score":0.8230000138282776},{"id":"https://openalex.org/keywords/coefficient-identification","display_name":"coefficient identification","score":0.8119999766349792},{"id":"https://openalex.org/keywords/ill-posed-inverse-problems","display_name":"ill-posed inverse problems","score":0.8009999990463257},{"id":"https://openalex.org/keywords/a-priori-estimates","display_name":"a priori estimates","score":0.7390000224113464},{"id":"https://openalex.org/keywords/lipschitz-stability","display_name":"Lipschitz stability","score":0.9150000214576721}],"concepts":[{"id":"https://openalex.org/C2779834249","wikidata":"https://www.wikidata.org/wiki/Q6664586","display_name":"Local volatility","level":4,"score":0.7401999831199646},{"id":"https://openalex.org/C152442038","wikidata":"https://www.wikidata.org/wiki/Q2778212","display_name":"Tikhonov regularization","level":3,"score":0.7379999756813049},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6987000107765198},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.6152999997138977},{"id":"https://openalex.org/C91602232","wikidata":"https://www.wikidata.org/wiki/Q756115","display_name":"Volatility (finance)","level":2,"score":0.6100000143051147},{"id":"https://openalex.org/C43711488","wikidata":"https://www.wikidata.org/wiki/Q7534783","display_name":"Skew","level":2,"score":0.6021999716758728},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.5514000058174133},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5444999933242798},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.49149999022483826},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4693000018596649},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.40149998664855957},{"id":"https://openalex.org/C27872270","wikidata":"https://www.wikidata.org/wiki/Q4839810","display_name":"Backus\u2013Gilbert method","level":5,"score":0.39590001106262207},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.3808000087738037},{"id":"https://openalex.org/C1893757","wikidata":"https://www.wikidata.org/wiki/Q3653001","display_name":"Inversion (geology)","level":3,"score":0.3750999867916107},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.3296999931335449},{"id":"https://openalex.org/C24189920","wikidata":"https://www.wikidata.org/wiki/Q1660345","display_name":"Implied volatility","level":3,"score":0.32670000195503235},{"id":"https://openalex.org/C2778153524","wikidata":"https://www.wikidata.org/wiki/Q4241335","display_name":"Optimality criterion","level":2,"score":0.26840001344680786},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2603999972343445},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.25769999623298645},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.2565999925136566}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/math.20261002","doi":"https://doi.org/10.3934/math.20261002","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.20261002","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/8/PDF/math-11-08-1002.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/math.20261002","doi":"https://doi.org/10.3934/math.20261002","is_oa":true,"landing_page_url":"https://doi.org/10.3934/math.20261002","pdf_url":"http://www.aimspress.com/aimspress-data/math/2026/8/PDF/math-11-08-1002.pdf","source":{"id":"https://openalex.org/S4210179087","display_name":"AIMS Mathematics","issn_l":"2473-6988","issn":["2473-6988"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","jufo-1","norway-1","oarl","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"AIMS Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2771635034","display_name":null,"funder_award_id":"415105003","funder_id":"https://openalex.org/F4320325208","funder_display_name":"Tarim University"},{"id":"https://openalex.org/G5086252408","display_name":null,"funder_award_id":"TDZKSS201904","funder_id":"https://openalex.org/F4320325208","funder_display_name":"Tarim University"},{"id":"https://openalex.org/G8889017783","display_name":null,"funder_award_id":"TDZKQN201823","funder_id":"https://openalex.org/F4320325208","funder_display_name":"Tarim University"}],"funders":[{"id":"https://openalex.org/F4320325208","display_name":"Tarim University","ror":"https://ror.org/05202v862"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7202344265.pdf","grobid_xml":"https://content.openalex.org/works/W7202344265.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1965101614","https://openalex.org/W2005749610","https://openalex.org/W2017655526","https://openalex.org/W2021980249","https://openalex.org/W2032595633","https://openalex.org/W2059862231","https://openalex.org/W2075917008","https://openalex.org/W2077791698","https://openalex.org/W2103975365","https://openalex.org/W2126853883","https://openalex.org/W2157707465","https://openalex.org/W2328121935","https://openalex.org/W2525737271","https://openalex.org/W2890075946","https://openalex.org/W2978191791","https://openalex.org/W2999920797","https://openalex.org/W3122526168","https://openalex.org/W4230960434","https://openalex.org/W4281699170","https://openalex.org/W4298228534","https://openalex.org/W4307125783","https://openalex.org/W4395456663","https://openalex.org/W7163990374"],"related_works":["https://openalex.org/W1965101614","https://openalex.org/W3122761224","https://openalex.org/W2084684086","https://openalex.org/W1929465443","https://openalex.org/W1872879586","https://openalex.org/W2328121935","https://openalex.org/W2157707465","https://openalex.org/W2406809191","https://openalex.org/W1971260280","https://openalex.org/W2592678834"],"abstract_inverted_index":{"This":[0,150],"paper":[1,151],"uses":[2],"standard":[3],"time-stepping":[4],"optimal":[5],"control":[6],"to":[7,45,105,159],"construct":[8],"a":[9,29,41,52,78],"supplementary":[10],"regularized":[11,112,139],"method":[12,140],"for":[13],"recovering":[14],"$\\sigma(S,":[15],"t)$":[16],"in":[17],"the":[18,24,47,63,67,106,110,120,163],"Black-Scholes":[19],"model.":[20,168],"With":[21],"Dupire":[22,69],"transformation,":[23],"volatility":[25,115,133,144,166],"calibration":[26],"problem":[27,32,50],"becomes":[28],"coefficient":[30],"identification":[31],"of":[33,54,66,109],"non-divergence":[34],"parabolic":[35],"equations.":[36],"Combining":[37],"Tikhonov":[38],"regularization":[39],"with":[40,89],"semi-discrete":[42,102],"time":[43,59,121],"stepping":[44],"break":[46],"two-dimensional":[48],"inversion":[49,167],"into":[51],"series":[53],"one-dimensional":[55],"subproblems":[56],"at":[57],"each":[58,84],"layer,":[60],"it":[61],"reduces":[62],"inherent":[64],"ill-posedness":[65],"direct":[68],"inversion.":[70],"We":[71],"derive":[72],"forward-adjoint":[73],"optimality":[74],"conditions":[75],"and":[76,81,93,131,146,156,161],"uniform":[77],"priori":[79],"bounds":[80],"prove":[82],"that":[83,101,136],"discrete":[85],"subproblem":[86],"admits":[87],"minimizers":[88,103],"existence,":[90],"local":[91,114],"uniqueness,":[92],"Lipschitz":[94],"stability.":[95],"Our":[96],"main":[97],"theoretical":[98,154],"result":[99],"shows":[100],"converge":[104],"weak":[107],"solution":[108],"continuous":[111],"implied":[113],"equation":[116],"(RILVE)":[117],"system":[118],"when":[119],"step":[122],"approaches":[123],"zero.":[124],"Numerical":[125],"simulations":[126],"on":[127],"synthetic":[128],"flat,":[129],"smile,":[130],"skew":[132],"data":[134],"show":[135],"this":[137],"layered":[138],"can":[141],"produce":[142],"accurate":[143],"results":[145],"resist":[147],"observational":[148],"noise.":[149],"provides":[152],"full":[153],"deductions":[155],"numerical":[157],"tests":[158],"support":[160],"improve":[162],"existing":[164],"RILVE":[165]},"counts_by_year":[],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2026-08-14T00:00:00"}