{"id":"https://openalex.org/W2142188295","doi":"https://doi.org/10.1090/s0002-9939-10-10306-2","title":"On the uniqueness of classical solutions of Cauchy problems","display_name":"On the uniqueness of classical solutions of Cauchy problems","publication_year":2010,"publication_date":"2010-01-27","ids":{"openalex":"https://openalex.org/W2142188295","doi":"https://doi.org/10.1090/s0002-9939-10-10306-2","mag":"2142188295"},"language":"en","primary_location":{"id":"doi:10.1090/s0002-9939-10-10306-2","doi":"https://doi.org/10.1090/s0002-9939-10-10306-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0002-9939-10-10306-2","pdf_url":"https://www.ams.org/proc/2010-138-06/S0002-9939-10-10306-2/S0002-9939-10-10306-2.pdf","source":{"id":"https://openalex.org/S4210226894","display_name":"Proceedings of the American Mathematical Society","issn_l":"0002-9939","issn":["0002-9939","1088-6826"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-b","jufo-2","norway-1","poland-100","russia-white-list-2"],"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":"Proceedings of the American Mathematical Society","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/proc/2010-138-06/S0002-9939-10-10306-2/S0002-9939-10-10306-2.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005237318","display_name":"Erhan Bayraktar","orcid":"https://orcid.org/0000-0002-1926-4570","observed_orcids":["https://orcid.org/0000-0002-1926-4570"]},"institutions":[{"id":"https://openalex.org/I27837315","display_name":"University of Michigan","ror":"https://ror.org/00jmfr291","country_code":"US","type":"education","lineage":["https://openalex.org/I27837315"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Erhan Bayraktar","raw_affiliation_strings":["Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48104"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48104","institution_ids":["https://openalex.org/I27837315"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068974015","display_name":"Hao Xing","orcid":"https://orcid.org/0000-0002-6995-2486","observed_orcids":["https://orcid.org/0000-0002-6995-2486"]},"institutions":[{"id":"https://openalex.org/I111088046","display_name":"Boston University","ror":"https://ror.org/05qwgg493","country_code":"US","type":"education","lineage":["https://openalex.org/I111088046"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hao Xing","raw_affiliation_strings":["Department of Mathematics and Statistics, Boston University, Boston, Massachu- setts 02215"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Boston University, Boston, Massachu- setts 02215","institution_ids":["https://openalex.org/I111088046"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.2028,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":{"value":0.95274018,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"138","issue":"6","first_page":"2061","last_page":"2064"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.5414999723434448,"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":0.5414999723434448,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.32100000977516174,"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"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.057999998331069946,"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/cauchy-problem","display_name":"Cauchy problem","score":0.7950000166893005},{"id":"https://openalex.org/keywords/uniqueness-of-classical-solutions","display_name":"uniqueness of classical solutions","score":0.9190000295639038},{"id":"https://openalex.org/keywords/second-order-partial-differential-equations","display_name":"second-order partial differential equations","score":0.628000020980835},{"id":"https://openalex.org/keywords/necessary-and-sufficient-conditions","display_name":"necessary and sufficient conditions","score":0.6980000138282776},{"id":"https://openalex.org/keywords/stochastic-volatility","display_name":"stochastic volatility","score":0.4519999921321869}],"concepts":[{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.8281275033950806},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6894307732582092},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.666512668132782},{"id":"https://openalex.org/C153635880","wikidata":"https://www.wikidata.org/wiki/Q1412436","display_name":"Cauchy problem","level":3,"score":0.659276008605957},{"id":"https://openalex.org/C49344536","wikidata":"https://www.wikidata.org/wiki/Q726441","display_name":"Cauchy distribution","level":2,"score":0.5445918440818787},{"id":"https://openalex.org/C91602232","wikidata":"https://www.wikidata.org/wiki/Q756115","display_name":"Volatility (finance)","level":2,"score":0.5103997588157654},{"id":"https://openalex.org/C88246228","wikidata":"https://www.wikidata.org/wiki/Q830513","display_name":"Residue theorem","level":4,"score":0.5049780011177063},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5002784729003906},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4883084297180176},{"id":"https://openalex.org/C106947605","wikidata":"https://www.wikidata.org/wiki/Q3278575","display_name":"Cauchy boundary condition","level":4,"score":0.4488254189491272},{"id":"https://openalex.org/C189658257","wikidata":"https://www.wikidata.org/wiki/Q834025","display_name":"Cauchy's integral theorem","level":5,"score":0.44354158639907837},{"id":"https://openalex.org/C76835800","wikidata":"https://www.wikidata.org/wiki/Q859102","display_name":"Cauchy's convergence test","level":5,"score":0.4197115898132324},{"id":"https://openalex.org/C82006148","wikidata":"https://www.wikidata.org/wiki/Q913764","display_name":"Cauchy's integral formula","level":4,"score":0.3951519727706909},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0899597704410553},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.07440164685249329},{"id":"https://openalex.org/C154416045","wikidata":"https://www.wikidata.org/wiki/Q5500196","display_name":"Free boundary problem","level":3,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1090/s0002-9939-10-10306-2","doi":"https://doi.org/10.1090/s0002-9939-10-10306-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0002-9939-10-10306-2","pdf_url":"https://www.ams.org/proc/2010-138-06/S0002-9939-10-10306-2/S0002-9939-10-10306-2.pdf","source":{"id":"https://openalex.org/S4210226894","display_name":"Proceedings of the American Mathematical Society","issn_l":"0002-9939","issn":["0002-9939","1088-6826"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-b","jufo-2","norway-1","poland-100","russia-white-list-2"],"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":"Proceedings of the American Mathematical Society","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:arx:papers:0908.1086","doi":null,"is_oa":false,"landing_page_url":"http://arxiv.org/pdf/0908.1086","pdf_url":"http://arxiv.org/pdf/0908.1086","source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"preprint"},{"id":"pmh:oai:arXiv.org:0908.1086","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/0908.1086","pdf_url":"https://arxiv.org/pdf/0908.1086","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2142188295","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/0908.1086.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.249.49","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.249.49","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0908.1086v7.pdf","raw_type":"text"},{"id":"pmh:oai:eprints.lse.ac.uk:31871","doi":null,"is_oa":false,"landing_page_url":"http://eprints.lse.ac.uk/31871/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401593","display_name":"London School of Economics and Political Science Research Online (London School of Economics and Political Science)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I909854389","host_organization_name":"London School of Economics and Political Science","host_organization_lineage":["https://openalex.org/I909854389"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.0908.1086","doi":"https://doi.org/10.48550/arxiv.0908.1086","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.0908.1086","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.1090/s0002-9939-10-10306-2","doi":"https://doi.org/10.1090/s0002-9939-10-10306-2","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0002-9939-10-10306-2","pdf_url":"https://www.ams.org/proc/2010-138-06/S0002-9939-10-10306-2/S0002-9939-10-10306-2.pdf","source":{"id":"https://openalex.org/S4210226894","display_name":"Proceedings of the American Mathematical Society","issn_l":"0002-9939","issn":["0002-9939","1088-6826"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-b","jufo-2","norway-1","poland-100","russia-white-list-2"],"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":"Proceedings of the American Mathematical Society","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G6076913652","display_name":"AMC-SS:  Problems in Mathematical Finance","funder_award_id":"0906257","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2142188295.pdf","grobid_xml":"https://content.openalex.org/works/W2142188295.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W30805069","https://openalex.org/W588786186","https://openalex.org/W1483382787","https://openalex.org/W1494628843","https://openalex.org/W1577527822","https://openalex.org/W1580561995","https://openalex.org/W1613583603","https://openalex.org/W1977403568","https://openalex.org/W1994438934","https://openalex.org/W2003850034","https://openalex.org/W2025088845","https://openalex.org/W2042775870","https://openalex.org/W2120900212","https://openalex.org/W2507750056","https://openalex.org/W2952491190","https://openalex.org/W3100237169","https://openalex.org/W3104891884","https://openalex.org/W3125572697","https://openalex.org/W6944071609"],"related_works":["https://openalex.org/W3046028118","https://openalex.org/W2077642468","https://openalex.org/W2068276130","https://openalex.org/W1995777107","https://openalex.org/W2092141864","https://openalex.org/W2028355776","https://openalex.org/W2017035189","https://openalex.org/W2957185091","https://openalex.org/W2465352227","https://openalex.org/W2042348435"],"abstract_inverted_index":{"Given":[0],"that":[1,15,50],"the":[2,25,28,48],"terminal":[3],"condition":[4,46],"is":[5,12,31,51],"of":[6,35],"at":[7,36],"most":[8,37],"linear":[9,38],"growth,":[10],"it":[11],"well":[13],"known":[14],"a":[16,20,33,45,56,61],"Cauchy":[17,57],"problem":[18,58],"admits":[19],"unique":[21,62],"classical":[22],"solution":[23],"when":[24],"coefficient":[26],"multiplying":[27],"second":[29],"derivative":[30],"also":[32],"function":[34],"growth.":[39],"In":[40],"this":[41],"paper,":[42],"we":[43],"give":[44],"on":[47],"volatility":[49],"necessary":[52],"and":[53],"sufficient":[54],"for":[55],"to":[59],"admit":[60],"solution.":[63]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":3}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}