{"id":"https://openalex.org/W2158284238","doi":"https://doi.org/10.1214/11-aap762","title":"Malliavin calculus for backward stochastic differential equations and application to numerical solutions","display_name":"Malliavin calculus for backward stochastic differential equations and application to numerical solutions","publication_year":2011,"publication_date":"2011-11-23","ids":{"openalex":"https://openalex.org/W2158284238","doi":"https://doi.org/10.1214/11-aap762","mag":"2158284238"},"language":"en","primary_location":{"id":"doi:10.1214/11-aap762","doi":"https://doi.org/10.1214/11-aap762","is_oa":true,"landing_page_url":"https://doi.org/10.1214/11-aap762","pdf_url":"https://projecteuclid.org/journals/annals-of-applied-probability/volume-21/issue-6/Malliavin-calculus-for-backward-stochastic-differential-equations-and-application-to/10.1214/11-AAP762.pdf","source":{"id":"https://openalex.org/S9093621","display_name":"The Annals of Applied Probability","issn_l":"1050-5164","issn":["1050-5164","2168-8737"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-b","doyens","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Applied Probability","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://projecteuclid.org/journals/annals-of-applied-probability/volume-21/issue-6/Malliavin-calculus-for-backward-stochastic-differential-equations-and-application-to/10.1214/11-AAP762.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074793555","display_name":"Yaozhong Hu","orcid":"https://orcid.org/0000-0003-4977-6099","observed_orcids":["https://orcid.org/0000-0003-4977-6099","https://orcid.org/0000-0001-6070-4630"]},"institutions":[{"id":"https://openalex.org/I146416000","display_name":"University of Kansas","ror":"https://ror.org/001tmjg57","country_code":"US","type":"education","lineage":["https://openalex.org/I146416000"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yaozhong Hu","raw_affiliation_strings":["University of Kansas"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Kansas","institution_ids":["https://openalex.org/I146416000"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033842662","display_name":"David Nualart","orcid":"https://orcid.org/0000-0002-4978-637X","observed_orcids":["https://orcid.org/0000-0002-4978-637X"]},"institutions":[{"id":"https://openalex.org/I146416000","display_name":"University of Kansas","ror":"https://ror.org/001tmjg57","country_code":"US","type":"education","lineage":["https://openalex.org/I146416000"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David Nualart","raw_affiliation_strings":["University of Kansas"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Kansas","institution_ids":["https://openalex.org/I146416000"]}]},{"author_position":"last","author":{"id":null,"display_name":"Xiaoming Song","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I146416000","display_name":"University of Kansas","ror":"https://ror.org/001tmjg57","country_code":"US","type":"education","lineage":["https://openalex.org/I146416000"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xiaoming Song","raw_affiliation_strings":["DEPARTMENT OF MATHEMATICS UNIVERSITY OF KANSAS LAWRENCE , KANSAS 66045 USA","University of Kansas"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"DEPARTMENT OF MATHEMATICS UNIVERSITY OF KANSAS LAWRENCE , KANSAS 66045 USA","institution_ids":["https://openalex.org/I146416000"]},{"raw_affiliation_string":"University of Kansas","institution_ids":["https://openalex.org/I146416000"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I146416000"],"apc_list":null,"apc_paid":null,"fwci":2.3713,"has_fulltext":true,"cited_by_count":45,"citation_normalized_percentile":{"value":0.91573808,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"21","issue":"6","first_page":"2379","last_page":"2423"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":1.0,"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":1.0,"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/T12135","display_name":"Fuzzy Systems and Optimization","score":0.0,"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"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/backward-stochastic-differential-equations","display_name":"backward stochastic differential equations","score":0.9789999723434448},{"id":"https://openalex.org/keywords/malliavin-calculus","display_name":"Malliavin calculus","score":0.9990000128746033},{"id":"https://openalex.org/keywords/numerical-approximation-schemes","display_name":"numerical approximation schemes","score":0.9330000281333923},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.9879999756813049},{"id":"https://openalex.org/keywords/random-number-generators","display_name":"random number generators","score":0.7390000224113464},{"id":"https://openalex.org/keywords/terminal-values","display_name":"terminal values","score":0.6589999794960022}],"concepts":[{"id":"https://openalex.org/C11505638","wikidata":"https://www.wikidata.org/wiki/Q6744166","display_name":"Malliavin calculus","level":4,"score":0.8259999752044678},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7437000274658203},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.6571000218391418},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.5867000222206116},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.48410001397132874},{"id":"https://openalex.org/C84629840","wikidata":"https://www.wikidata.org/wiki/Q16979017","display_name":"Stochastic partial differential equation","level":3,"score":0.4830000102519989},{"id":"https://openalex.org/C63331456","wikidata":"https://www.wikidata.org/wiki/Q1308570","display_name":"Stochastic calculus","level":4,"score":0.4406999945640564},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.4291999936103821},{"id":"https://openalex.org/C47096860","wikidata":"https://www.wikidata.org/wiki/Q575542","display_name":"Terminal value","level":4,"score":0.4074000120162964},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.397599995136261},{"id":"https://openalex.org/C97826883","wikidata":"https://www.wikidata.org/wiki/Q7069659","display_name":"Numerical partial differential equations","level":3,"score":0.390500009059906},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3792000114917755},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.3682999908924103},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3472000062465668},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.34610000252723694},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3443000018596649},{"id":"https://openalex.org/C2779664074","wikidata":"https://www.wikidata.org/wiki/Q3518405","display_name":"Terminal (telecommunication)","level":2,"score":0.3382999897003174},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3133000135421753},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.2775999903678894},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.2752000093460083},{"id":"https://openalex.org/C152522140","wikidata":"https://www.wikidata.org/wiki/Q795746","display_name":"Backward differentiation formula","level":5,"score":0.2705000042915344}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1214/11-aap762","doi":"https://doi.org/10.1214/11-aap762","is_oa":true,"landing_page_url":"https://doi.org/10.1214/11-aap762","pdf_url":"https://projecteuclid.org/journals/annals-of-applied-probability/volume-21/issue-6/Malliavin-calculus-for-backward-stochastic-differential-equations-and-application-to/10.1214/11-AAP762.pdf","source":{"id":"https://openalex.org/S9093621","display_name":"The Annals of Applied Probability","issn_l":"1050-5164","issn":["1050-5164","2168-8737"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-b","doyens","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Applied Probability","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1202.4625","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1202.4625","pdf_url":"https://arxiv.org/pdf/1202.4625","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":"pmh:oai:CULeuclid:euclid.aoap/1322057325","doi":null,"is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.aoap/1322057325","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"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":"","raw_type":"Text"},{"id":"pmh:oai:kuscholarworks.ku.edu:1808/17013","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/1808/17013","pdf_url":null,"source":{"id":"https://openalex.org/S4306400421","display_name":"KU ScholarWorks (The University of Kansas)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I146416000","host_organization_name":"University of Kansas","host_organization_lineage":["https://openalex.org/I146416000"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"doi:10.48550/arxiv.1202.4625","doi":"https://doi.org/10.48550/arxiv.1202.4625","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1202.4625","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":"article-journal"}],"best_oa_location":{"id":"doi:10.1214/11-aap762","doi":"https://doi.org/10.1214/11-aap762","is_oa":true,"landing_page_url":"https://doi.org/10.1214/11-aap762","pdf_url":"https://projecteuclid.org/journals/annals-of-applied-probability/volume-21/issue-6/Malliavin-calculus-for-backward-stochastic-differential-equations-and-application-to/10.1214/11-AAP762.pdf","source":{"id":"https://openalex.org/S9093621","display_name":"The Annals of Applied Probability","issn_l":"1050-5164","issn":["1050-5164","2168-8737"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["abdc-a","anvur-class-a","anvur-scientific","cwts-core","dongbi-b","doyens","jufo-3","norway-2","poland-140","russia-white-list-1"],"host_organization":"https://openalex.org/P4310319881","host_organization_name":"Institute of Mathematical Statistics","host_organization_lineage":["https://openalex.org/P4310319881"],"host_organization_lineage_names":["Institute of Mathematical Statistics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Annals of Applied Probability","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G5700677368","display_name":"Stochastic Analysis of Gaussian Fractional Noises","funder_award_id":"0904538","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G787286172","display_name":"Nonlinear Functionals of Fractional Brownian Motion","funder_award_id":"0504783","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/W2158284238.pdf","grobid_xml":"https://content.openalex.org/works/W2158284238.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W10636524","https://openalex.org/W659339155","https://openalex.org/W1977946148","https://openalex.org/W2009905602","https://openalex.org/W2039366144","https://openalex.org/W2043281212","https://openalex.org/W2046331573","https://openalex.org/W2046553535","https://openalex.org/W2078895915","https://openalex.org/W2095445700","https://openalex.org/W2104540101","https://openalex.org/W2129289929","https://openalex.org/W2157883119","https://openalex.org/W2167018969"],"related_works":["https://openalex.org/W7096298199","https://openalex.org/W2111750212","https://openalex.org/W2978098184","https://openalex.org/W1777099553","https://openalex.org/W2129397196","https://openalex.org/W3191051037","https://openalex.org/W4364378734","https://openalex.org/W2167018969","https://openalex.org/W2883018908","https://openalex.org/W2322386831"],"abstract_inverted_index":{"In":[0,17],"this":[1],"paper":[2],"we":[3,19,55,64],"study":[4],"backward":[5,71],"stochastic":[6,72],"differential":[7,73],"equations":[8,74],"with":[9],"general":[10,14],"terminal":[11,24],"value":[12,25],"and":[13,75],"random":[15],"generator.":[16],"particular,":[18],"do":[20],"not":[21,39],"require":[22],"the":[23,36,57,61,77,82,86,95],"be":[26,42],"given":[27],"by":[28],"a":[29,44],"forward":[30,45],"diffusion":[31],"equation.":[32],"The":[33,91],"randomness":[34],"of":[35,60,79,81],"generator":[37],"does":[38],"need":[40],"to":[41,51],"from":[43,49],"equation,":[46],"either.":[47],"Motivated":[48],"applications":[50],"numerical":[52,67],"simulations,":[53],"first":[54],"obtain":[56,76],"Lp-H\u00f6lder":[58,88],"continuity":[59,89],"solution.":[62],"Then":[63],"construct":[65],"several":[66],"approximation":[68],"schemes":[69,83],"for":[70],"rate":[78],"convergence":[80],"based":[84],"on":[85],"obtained":[87],"results.":[90],"main":[92],"tool":[93],"is":[94],"Malliavin":[96],"calculus.":[97]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2016-06-24T00:00:00"}