{"id":"https://openalex.org/W2939797022","doi":"https://doi.org/10.1137/19m1261912","title":"An Optimal Polynomial Approximation of Brownian Motion","display_name":"An Optimal Polynomial Approximation of Brownian Motion","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2939797022","doi":"https://doi.org/10.1137/19m1261912","mag":"2939797022"},"language":"en","primary_location":{"id":"doi:10.1137/19m1261912","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1261912","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1904.06998","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"James Foster","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"James Foster","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Terry Lyons","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Terry Lyons","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Harald Oberhauser","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Harald Oberhauser","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.3868,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.8933012,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"58","issue":"3","first_page":"1393","last_page":"1421"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.6305000185966492,"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":0.6305000185966492,"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":0.05990000069141388,"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/T11413","display_name":"Risk and Portfolio Optimization","score":0.052400000393390656,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/fractional-brownian-motion","display_name":"Fractional Brownian motion","score":0.677299976348877},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6003000140190125},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.4997999966144562},{"id":"https://openalex.org/keywords/brownian-motion","display_name":"Brownian motion","score":0.4977000057697296},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.4661000072956085},{"id":"https://openalex.org/keywords/geometric-brownian-motion","display_name":"Geometric Brownian motion","score":0.453900009393692},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4510999917984009},{"id":"https://openalex.org/keywords/gaussian-process","display_name":"Gaussian process","score":0.4203999936580658},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.36410000920295715}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.878600001335144},{"id":"https://openalex.org/C108819105","wikidata":"https://www.wikidata.org/wiki/Q1143293","display_name":"Fractional Brownian motion","level":3,"score":0.677299976348877},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6003000140190125},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.588699996471405},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.4997999966144562},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.4977000057697296},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.4661000072956085},{"id":"https://openalex.org/C101615488","wikidata":"https://www.wikidata.org/wiki/Q1503307","display_name":"Geometric Brownian motion","level":4,"score":0.453900009393692},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4510999917984009},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.4203999936580658},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.392300009727478},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.36410000920295715},{"id":"https://openalex.org/C78651780","wikidata":"https://www.wikidata.org/wiki/Q628309","display_name":"Brownian bridge","level":3,"score":0.3625999987125397},{"id":"https://openalex.org/C116779161","wikidata":"https://www.wikidata.org/wiki/Q7307228","display_name":"Reflected Brownian motion","level":5,"score":0.3434000015258789},{"id":"https://openalex.org/C68710425","wikidata":"https://www.wikidata.org/wiki/Q5275442","display_name":"Diffusion process","level":3,"score":0.3240000009536743},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.3231000006198883},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.319599986076355},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.3151000142097473},{"id":"https://openalex.org/C53920899","wikidata":"https://www.wikidata.org/wiki/Q4976524","display_name":"Brownian excursion","level":5,"score":0.3122999966144562},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.30390000343322754},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.3027999997138977},{"id":"https://openalex.org/C181789720","wikidata":"https://www.wikidata.org/wiki/Q4812191","display_name":"Asymptotically optimal algorithm","level":2,"score":0.29670000076293945},{"id":"https://openalex.org/C127349201","wikidata":"https://www.wikidata.org/wiki/Q753445","display_name":"Numerical integration","level":2,"score":0.2939999997615814},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.28870001435279846},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.27970001101493835},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.27250000834465027},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C60391097","wikidata":"https://www.wikidata.org/wiki/Q1056809","display_name":"Wiener process","level":2,"score":0.2605000138282776}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/19m1261912","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1261912","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1904.06998","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.06998","pdf_url":"https://arxiv.org/pdf/1904.06998","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1904.06998","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.06998","pdf_url":"https://arxiv.org/pdf/1904.06998","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3207594375","display_name":"DTP 2016-2017 University of Oxford","funder_award_id":"EP/N509711/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G5005012656","display_name":null,"funder_award_id":"EP/S026347/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320313467","display_name":"Alan Turing Institute","ror":"https://ror.org/035dkdb55"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":15,"referenced_works":["https://openalex.org/W1727830821","https://openalex.org/W1966128159","https://openalex.org/W1981593978","https://openalex.org/W1984865765","https://openalex.org/W1986101505","https://openalex.org/W2034555762","https://openalex.org/W2037760662","https://openalex.org/W2068251904","https://openalex.org/W2076201245","https://openalex.org/W2131838761","https://openalex.org/W2145137020","https://openalex.org/W2165867509","https://openalex.org/W2896207634","https://openalex.org/W2898589565","https://openalex.org/W2909632592"],"related_works":[],"abstract_inverted_index":{"In":[0,96],"this":[1,33],"paper,":[2],"we":[3],"will":[4,138],"present":[5],"a":[6,16,91,105,151],"strong":[7],"(or":[8,159],"pathwise)":[9],"approximation":[10,87,147],"of":[11,18,29,57,63,79,143,154],"standard":[12],"Brownian":[13,30,58,81,99,132],"motion":[14,31,59,133],"by":[15,128],"class":[17],"orthogonal":[19,73],"polynomials.":[20],"The":[21],"coefficients":[22],"that":[23,60],"are":[24,36],"obtained":[25],"from":[26],"the":[27,80,85,119,141,144,155],"expansion":[28],"in":[32,90],"polynomial":[34],"basis":[35],"independent":[37,48],"Gaussian":[38,49],"random":[39],"variables.":[40],"Therefore,":[41],"it":[42],"is":[43,88],"practical":[44],"(i.e.,":[45],"requires":[46],"$N$":[47],"coefficients)":[50],"to":[51,118,150],"generate":[52],"an":[53],"approximate":[54],"sample":[55],"path":[56],"respects":[61],"integration":[62],"polynomials":[64,74],"with":[65],"degree":[66],"less":[67],"than":[68],"$N$.":[69],"Moreover,":[70],"since":[71],"these":[72,126],"appear":[75],"naturally":[76],"as":[77,101],"eigenfunctions":[78],"bridge":[82],"covariance":[83],"function,":[84],"proposed":[86],"optimal":[89],"certain":[92],"weighted":[93],"$L^{2}(\\mathbb{P})$":[94],"sense.":[95],"addition,":[97],"discretizing":[98],"paths":[100],"piecewise":[102,120,145],"parabolas":[103],"gives":[104],"locally":[106],"higher":[107],"order":[108],"numerical":[109,136],"method":[110],"for":[111],"stochastic":[112],"differential":[113],"equations":[114],"(SDEs)":[115],"when":[116,148],"compared":[117,149],"linear":[121],"approach.":[122],"We":[123],"shall":[124],"demonstrate":[125],"ideas":[127],"simulating":[129],"inhomogeneous":[130],"geometric":[131],"(IGBM).":[134],"This":[135],"example":[137],"also":[139],"illustrate":[140],"deficiencies":[142],"parabola":[146],"new":[152],"version":[153],"asymptotically":[156],"efficient":[157],"log-ODE":[158],"Castell--Gaines)":[160],"method.":[161]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2019-04-25T00:00:00"}