{"id":"https://openalex.org/W2046955945","doi":"https://doi.org/10.1090/s0025-5718-04-01714-4","title":"A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization","display_name":"A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization","publication_year":2004,"publication_date":"2004-10-05","ids":{"openalex":"https://openalex.org/W2046955945","doi":"https://doi.org/10.1090/s0025-5718-04-01714-4","mag":"2046955945"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-04-01714-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01714-4","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-04-01714-4/S0025-5718-04-01714-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-04-01714-4/S0025-5718-04-01714-4.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010896193","display_name":"Yves Achdou","orcid":"https://orcid.org/0000-0002-0514-3727"},"institutions":[{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]},{"id":"https://openalex.org/I3017942884","display_name":"Institut de Math\u00e9matiques de Jussieu-Paris Rive Gauche","ror":"https://ror.org/03fk87k11","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I204730241","https://openalex.org/I3017942884","https://openalex.org/I39804081","https://openalex.org/I4210141950"]},{"id":"https://openalex.org/I39804081","display_name":"Sorbonne Universit\u00e9","ror":"https://ror.org/02en5vm52","country_code":"FR","type":"education","lineage":["https://openalex.org/I39804081"]},{"id":"https://openalex.org/I4210158291","display_name":"Laboratoire Jacques-Louis Lions","ror":"https://ror.org/04xmteb38","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I204730241","https://openalex.org/I39804081","https://openalex.org/I4210141950","https://openalex.org/I4210158291"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Yves Achdou","raw_affiliation_strings":["UFR Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France; and Laboratoire J.L. Lions, Universit\u00c3\u00a9 Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France","UFR Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France","Laboratoire J.L. Lions, Universit\u00c3\u00a9 Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France"],"affiliations":[{"raw_affiliation_string":"UFR Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France; and Laboratoire J.L. Lions, Universit\u00c3\u00a9 Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France","institution_ids":["https://openalex.org/I39804081","https://openalex.org/I204730241","https://openalex.org/I4210158291"]},{"raw_affiliation_string":"UFR Math\u00c3\u00a9matiques, Universit\u00c3\u00a9 Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I3017942884"]},{"raw_affiliation_string":"Laboratoire J.L. Lions, Universit\u00c3\u00a9 Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France","institution_ids":["https://openalex.org/I39804081","https://openalex.org/I4210158291"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078651576","display_name":"Bruno Franchi","orcid":"https://orcid.org/0000-0002-6337-956X"},"institutions":[{"id":"https://openalex.org/I9360294","display_name":"University of Bologna","ror":"https://ror.org/01111rn36","country_code":"IT","type":"education","lineage":["https://openalex.org/I9360294"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Bruno Franchi","raw_affiliation_strings":["Dipartimento di Matematica, Universit\u00c3\u00a0 di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00c3\u00a0 di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy","institution_ids":["https://openalex.org/I9360294"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5052413636","display_name":"Nicoletta Tchou","orcid":null},"institutions":[{"id":"https://openalex.org/I4210161663","display_name":"Institut de recherche math\u00e9matique de Rennes","ror":"https://ror.org/05y76vp22","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I2800285241","https://openalex.org/I28221208","https://openalex.org/I3124938446","https://openalex.org/I4210117091","https://openalex.org/I4210141950","https://openalex.org/I4210146044","https://openalex.org/I4210161663","https://openalex.org/I4399657933","https://openalex.org/I56067802","https://openalex.org/I56067802"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Nicoletta Tchou","raw_affiliation_strings":["IRMAR, Universit\u00c3\u00a9 de Rennes 1, Rennes, France"],"affiliations":[{"raw_affiliation_string":"IRMAR, Universit\u00c3\u00a9 de Rennes 1, Rennes, France","institution_ids":["https://openalex.org/I4210161663"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5010896193"],"corresponding_institution_ids":["https://openalex.org/I204730241","https://openalex.org/I3017942884","https://openalex.org/I39804081","https://openalex.org/I4210158291"],"apc_list":null,"apc_paid":null,"fwci":0.8432,"has_fulltext":true,"cited_by_count":12,"citation_normalized_percentile":{"value":0.79996041,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"74","issue":"251","first_page":"1291","last_page":"1322"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9994999766349792,"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":0.9994999766349792,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.983299970626831,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/mathematics","display_name":"Mathematics","score":0.8338069915771484},{"id":"https://openalex.org/keywords/stochastic-volatility","display_name":"Stochastic volatility","score":0.6838222742080688},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6213712692260742},{"id":"https://openalex.org/keywords/valuation-of-options","display_name":"Valuation of options","score":0.5773343443870544},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.48522260785102844},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.4725643992424011},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.4709304869174957},{"id":"https://openalex.org/keywords/black\u2013scholes-model","display_name":"Black\u2013Scholes model","score":0.4629247188568115},{"id":"https://openalex.org/keywords/semi-elliptic-operator","display_name":"Semi-elliptic operator","score":0.4515473246574402},{"id":"https://openalex.org/keywords/sabr-volatility-model","display_name":"SABR volatility model","score":0.4355127811431885},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.41365689039230347},{"id":"https://openalex.org/keywords/volatility","display_name":"Volatility (finance)","score":0.3730151653289795},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3693506121635437},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.258256196975708},{"id":"https://openalex.org/keywords/differential-operator","display_name":"Differential operator","score":0.23574355244636536}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8338069915771484},{"id":"https://openalex.org/C85393063","wikidata":"https://www.wikidata.org/wiki/Q596307","display_name":"Stochastic volatility","level":3,"score":0.6838222742080688},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6213712692260742},{"id":"https://openalex.org/C194483076","wikidata":"https://www.wikidata.org/wiki/Q380382","display_name":"Valuation of options","level":2,"score":0.5773343443870544},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48522260785102844},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.4725643992424011},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.4709304869174957},{"id":"https://openalex.org/C163128081","wikidata":"https://www.wikidata.org/wiki/Q1338307","display_name":"Black\u2013Scholes model","level":3,"score":0.4629247188568115},{"id":"https://openalex.org/C24902271","wikidata":"https://www.wikidata.org/wiki/Q7449297","display_name":"Semi-elliptic operator","level":3,"score":0.4515473246574402},{"id":"https://openalex.org/C187625094","wikidata":"https://www.wikidata.org/wiki/Q7388452","display_name":"SABR volatility model","level":4,"score":0.4355127811431885},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.41365689039230347},{"id":"https://openalex.org/C91602232","wikidata":"https://www.wikidata.org/wiki/Q756115","display_name":"Volatility (finance)","level":2,"score":0.3730151653289795},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3693506121635437},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.258256196975708},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.23574355244636536}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/s0025-5718-04-01714-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01714-4","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-04-01714-4/S0025-5718-04-01714-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.153.7379","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.7379","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2005-74-251/S0025-5718-04-01714-4/S0025-5718-04-01714-4.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-00019756v1","is_oa":false,"landing_page_url":"https://hal.science/hal-00019756","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Math. Comp., 2005, 74, pp.1291-1322","raw_type":"Journal articles"},{"id":"pmh:oai:cris.unibo.it:11585/1419","is_oa":false,"landing_page_url":"http://hdl.handle.net/11585/1419","pdf_url":null,"source":{"id":"https://openalex.org/S4306402579","display_name":"Archivio istituzionale della ricerca (Alma Mater Studiorum Universit\u00e0 di Bologna)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210117483","host_organization_name":"Istituto di Ematologia di Bologna","host_organization_lineage":["https://openalex.org/I4210117483"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-04-01714-4","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-04-01714-4","pdf_url":"https://www.ams.org/mcom/2005-74-251/S0025-5718-04-01714-4/S0025-5718-04-01714-4.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320311030","display_name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","ror":"https://ror.org/01vx64p53"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2046955945.pdf","grobid_xml":"https://content.openalex.org/works/W2046955945.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W1492326914","https://openalex.org/W1514815886","https://openalex.org/W1527296550","https://openalex.org/W1600624394","https://openalex.org/W1990723904","https://openalex.org/W1999418605","https://openalex.org/W2005436751","https://openalex.org/W2024626650","https://openalex.org/W2041523977","https://openalex.org/W2061956999","https://openalex.org/W2087812785","https://openalex.org/W2094206697","https://openalex.org/W2199169705","https://openalex.org/W2333476344","https://openalex.org/W4235458784","https://openalex.org/W4292166829","https://openalex.org/W6609710778","https://openalex.org/W6832625590"],"related_works":["https://openalex.org/W3093796783","https://openalex.org/W4301670871","https://openalex.org/W3123155432","https://openalex.org/W3123418236","https://openalex.org/W2886095353","https://openalex.org/W3124791345","https://openalex.org/W4210940217","https://openalex.org/W2013221838","https://openalex.org/W2410077963","https://openalex.org/W2755264433"],"abstract_inverted_index":{"This":[0,15],"paper":[1],"completes":[2],"a":[3,7,17,26,30,50,53,72,94,101,106],"previous":[4],"work":[5],"on":[6],"Black":[8],"and":[9,42,99],"Scholes":[10],"equation":[11],"with":[12,60,105,109,116],"stochastic":[13],"volatility.":[14],"is":[16,49],"degenerate":[18,118],"parabolic":[19],"equation,":[20],"which":[21,80],"gives":[22],"the":[23,33,36,39,44,47,61,75,78,87,112,117,121],"price":[24,37],"of":[25,32,35,38,43,52,74,77,89],"European":[27],"option":[28],"as":[29],"function":[31,51],"time,":[34],"underlying":[40,62],"asset,":[41],"volatility,":[45],"when":[46],"volatility":[48],"mean":[54],"reverting":[55],"Orstein\u2013Uhlenbeck":[56],"process,":[57],"possibly":[58],"correlated":[59],"asset.":[63],"The":[64],"analysis":[65],"involves":[66],"weighted":[67],"Sobolev":[68],"spaces.":[69],"We":[70,91],"give":[71],"characterization":[73],"domain":[76],"operator,":[79],"permits":[81],"us":[82],"to":[83,111,126],"use":[84],"results":[85],"from":[86],"theory":[88],"semigroups.":[90],"then":[92],"study":[93],"related":[95],"model":[96],"elliptic":[97],"problem":[98],"propose":[100],"finite":[102],"element":[103],"method":[104],"regular":[107],"mesh":[108],"respect":[110],"intrinsic":[113],"metric":[114],"associated":[115],"operator.":[119],"For":[120],"error":[122],"estimate,":[123],"we":[124],"need":[125],"prove":[127],"an":[128],"approximation":[129],"result.":[130]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2025-10-10T00:00:00"}