{"id":"https://openalex.org/W1996171685","doi":"https://doi.org/10.1090/s0025-5718-00-01214-x","title":"On the complexity of stochastic integration","display_name":"On the complexity of stochastic integration","publication_year":2000,"publication_date":"2000-03-02","ids":{"openalex":"https://openalex.org/W1996171685","doi":"https://doi.org/10.1090/s0025-5718-00-01214-x","mag":"1996171685"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-00-01214-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01214-x","pdf_url":"https://www.ams.org/mcom/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.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/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5108821523","display_name":"G.W. Wasilkowski","orcid":null},"institutions":[{"id":"https://openalex.org/I143302722","display_name":"University of Kentucky","ror":"https://ror.org/02k3smh20","country_code":"US","type":"education","lineage":["https://openalex.org/I143302722"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"G. Wasilkowski","raw_affiliation_strings":["Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA","Department of Computer Science University of Kentucky Lexington KY 40506 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA","institution_ids":["https://openalex.org/I143302722"]},{"raw_affiliation_string":"Department of Computer Science University of Kentucky Lexington KY 40506 USA","institution_ids":["https://openalex.org/I143302722"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113538000","display_name":"H. Wo\u017aniakowski","orcid":null},"institutions":[{"id":"https://openalex.org/I4654613","display_name":"University of Warsaw","ror":"https://ror.org/039bjqg32","country_code":"PL","type":"education","lineage":["https://openalex.org/I4654613"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"H. Wo\u017aniakowski","raw_affiliation_strings":["Department of Computer Science, Columbia University, New York, NY 10027, USA and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Columbia University, New York, NY 10027, USA and Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland#TAB#","institution_ids":["https://openalex.org/I4654613"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.7859,"has_fulltext":true,"cited_by_count":18,"citation_normalized_percentile":{"value":0.79686013,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"70","issue":"234","first_page":"685","last_page":"698"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9986000061035156,"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":0.9986000061035156,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9980000257492065,"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"}},{"id":"https://openalex.org/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9950000047683716,"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/mathematics","display_name":"Mathematics","score":0.8549630641937256},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.6769365668296814},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5659751892089844},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.5511736273765564},{"id":"https://openalex.org/keywords/numerical-integration","display_name":"Numerical integration","score":0.5312799215316772},{"id":"https://openalex.org/keywords/integration-by-parts","display_name":"Integration by parts","score":0.5271134972572327},{"id":"https://openalex.org/keywords/lebesgue-integration","display_name":"Lebesgue integration","score":0.5043836832046509},{"id":"https://openalex.org/keywords/lebesgue-measure","display_name":"Lebesgue measure","score":0.5034219622612},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47071048617362976},{"id":"https://openalex.org/keywords/brownian-motion","display_name":"Brownian motion","score":0.4689962863922119},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.420663982629776},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3038939833641052},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2827906012535095},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07293885946273804}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8549630641937256},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.6769365668296814},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5659751892089844},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.5511736273765564},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.5312799215316772},{"id":"https://openalex.org/C118365987","wikidata":"https://www.wikidata.org/wiki/Q273328","display_name":"Integration by parts","level":2,"score":0.5271134972572327},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.5043836832046509},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.5034219622612},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47071048617362976},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.4689962863922119},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.420663982629776},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3038939833641052},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2827906012535095},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07293885946273804},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-00-01214-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01214-x","pdf_url":"https://www.ams.org/mcom/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.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:10.1.1.192.4223","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.4223","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/journals/mcom/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.97.7379","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.97.7379","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/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-00-01214-x","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01214-x","pdf_url":"https://www.ams.org/mcom/2001-70-234/S0025-5718-00-01214-X/S0025-5718-00-01214-X.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":[{"id":"https://openalex.org/G183198511","display_name":"Information-Based Complexity of Multivariate Problems","funder_award_id":"9729971","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4015677791","display_name":"Theory and Applications of Information-Based Complexity","funder_award_id":"9731858","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"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1996171685.pdf","grobid_xml":"https://content.openalex.org/works/W1996171685.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W324240392","https://openalex.org/W1525253579","https://openalex.org/W1575104486","https://openalex.org/W2016416695","https://openalex.org/W2017309389","https://openalex.org/W2031965247","https://openalex.org/W2073937734","https://openalex.org/W2081275435","https://openalex.org/W2137224975","https://openalex.org/W2141436719","https://openalex.org/W2527294977","https://openalex.org/W2569481888","https://openalex.org/W4285719527","https://openalex.org/W4302198360"],"related_works":["https://openalex.org/W2074721578","https://openalex.org/W928772053","https://openalex.org/W3008743878","https://openalex.org/W2032212755","https://openalex.org/W2917668491","https://openalex.org/W4287864225","https://openalex.org/W4233593527","https://openalex.org/W4236312217","https://openalex.org/W3184958942","https://openalex.org/W4244471727"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,35,92,97,106,114,123,126,130,150,154,157,165],"complexity":[3,36,151,166],"of":[4,27,38,71,117,125,156,167],"approximating":[5],"stochastic":[6],"integrals":[7],"with":[8,96],"error":[9],"<inline-formula":[10,40],"content-type=\"math/mathml\">":[11,41],"<mml:math":[12,42],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[13,43],"alttext=\"epsilon\">":[14],"<mml:semantics>":[15,48],"<mml:mi>":[16,50],"\u03b5":[17,51],"</mml:mi>":[18,52],"<mml:annotation":[19,61],"encoding=\"application/x-tex\">\\varepsilon</mml:annotation>":[20],"</mml:semantics>":[21,64],"</mml:math>":[22,65],"</inline-formula>":[23,66],"for":[24,69],"various":[25],"classes":[26,70,116],"functions.":[28,74,118],"For":[29,144],"Ito":[30,83,168],"integration,":[31,146],"we":[32,147],"show":[33,148],"that":[34,82,149],"is":[37,78,85,103,110,133,139],"order":[39],"alttext=\"epsilon":[44],"Superscript":[45],"negative":[46],"1\">":[47],"<mml:msup>":[49],"<mml:mrow":[53],"class=\"MJX-TeXAtom-ORD\">":[54],"<mml:mo>":[55],"\u2212":[56],"</mml:mo>":[57],"<mml:mn>1</mml:mn>":[58],"</mml:mrow>":[59],"</mml:msup>":[60],"encoding=\"application/x-tex\">\\varepsilon":[62],"^{-1}</mml:annotation>":[63],",":[67],"even":[68],"very":[72,140],"smooth":[73],"The":[75,100,119],"lower":[76],"bound":[77,102],"obtained":[79,104],"by":[80,105],"showing":[81],"integration":[84,90],"not":[86],"easier":[87],"than":[88,164],"Lebesgue":[89],"in":[91,113,135],"average":[93],"case":[94],"setting":[95],"Wiener":[98],"measure.":[99],"upper":[101],"Milstein":[107,120],"algorithm,":[108],"which":[109],"almost":[111],"optimal":[112],"considered":[115],"algorithm":[121],"uses":[122],"values":[124,137],"Brownian":[127],"motion":[128],"and":[129,138,159],"integrand.":[131],"It":[132],"bilinear":[134],"these":[136],"easy":[141],"to":[142],"implement.":[143],"Stratonovich":[145],"depends":[152],"on":[153],"smoothness":[155],"integrand":[158],"may":[160],"be":[161],"much":[162],"smaller":[163],"integration.":[169]},"counts_by_year":[{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
