{"id":"https://openalex.org/W3002302304","doi":"https://doi.org/10.1137/19m1239404","title":"Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs","display_name":"Comparison of Viscosity Solutions of Semilinear Path-Dependent PDEs","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3002302304","doi":"https://doi.org/10.1137/19m1239404","mag":"3002302304"},"language":"en","primary_location":{"id":"doi:10.1137/19m1239404","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1239404","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044750709","display_name":"Zhenjie Ren","orcid":"https://orcid.org/0000-0003-4656-4074"},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210107641","display_name":"Centre de Math\u00e9matiques Appliqu\u00e9es de l'\u00c9cole polytechnique","ror":"https://ror.org/012e1xn46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210107641","https://openalex.org/I4210141950","https://openalex.org/I4210145102"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Zhenjie Ren","raw_affiliation_strings":["Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique"],"raw_orcid":"https://orcid.org/0000-0003-4656-4074","affiliations":[{"raw_affiliation_string":"Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5105990961","display_name":"Nizar Touzi","orcid":null},"institutions":[{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210107641","display_name":"Centre de Math\u00e9matiques Appliqu\u00e9es de l'\u00c9cole polytechnique","ror":"https://ror.org/012e1xn46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210107641","https://openalex.org/I4210141950","https://openalex.org/I4210145102"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Nizar Touzi","raw_affiliation_strings":["Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100374706","display_name":"Jianfeng Zhang","orcid":"https://orcid.org/0000-0001-9784-7803"},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]},{"id":"https://openalex.org/I2800817003","display_name":"California Southern University","ror":"https://ror.org/058zz0t50","country_code":"US","type":"education","lineage":["https://openalex.org/I2800817003"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jianfeng Zhang","raw_affiliation_strings":["University of Southern California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I2800817003","https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5044750709"],"corresponding_institution_ids":["https://openalex.org/I142476485","https://openalex.org/I4210107641"],"apc_list":null,"apc_paid":null,"fwci":0.9548,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.77439468,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"58","issue":"1","first_page":"277","last_page":"302"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.996399998664856,"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":0.996399998664856,"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/T12137","display_name":"Economic theories and models","score":0.9807999730110168,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"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/T11413","display_name":"Risk and Portfolio Optimization","score":0.9524000287055969,"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/mathematics","display_name":"Mathematics","score":0.8647724986076355},{"id":"https://openalex.org/keywords/viscosity-solution","display_name":"Viscosity solution","score":0.6923970580101013},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.6841364502906799},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.6344252228736877},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5152273774147034},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.5058250427246094},{"id":"https://openalex.org/keywords/viscosity","display_name":"Viscosity","score":0.4684082865715027},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.43680691719055176},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.41422486305236816},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.3821362853050232},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3608853220939636},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3474341928958893},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10339364409446716}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8647724986076355},{"id":"https://openalex.org/C88731125","wikidata":"https://www.wikidata.org/wiki/Q7935676","display_name":"Viscosity solution","level":2,"score":0.6923970580101013},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.6841364502906799},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.6344252228736877},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5152273774147034},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.5058250427246094},{"id":"https://openalex.org/C127172972","wikidata":"https://www.wikidata.org/wiki/Q128709","display_name":"Viscosity","level":2,"score":0.4684082865715027},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.43680691719055176},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.41422486305236816},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3821362853050232},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3608853220939636},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3474341928958893},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10339364409446716},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/19m1239404","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1239404","pdf_url":null,"source":{"id":"https://openalex.org/S897311980","display_name":"SIAM Journal on Control and Optimization","issn_l":"0363-0129","issn":["0363-0129","1095-7138"],"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 Control and Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4787758955","display_name":null,"funder_award_id":"DMS-1413717","funder_id":"https://openalex.org/F4320337380","funder_display_name":"Division of Mathematical Sciences"}],"funders":[{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":53,"referenced_works":["https://openalex.org/W130965628","https://openalex.org/W1485598997","https://openalex.org/W1558902221","https://openalex.org/W1559997346","https://openalex.org/W1569706336","https://openalex.org/W1577527822","https://openalex.org/W1580561995","https://openalex.org/W1646613531","https://openalex.org/W1826259864","https://openalex.org/W1971594473","https://openalex.org/W1972547274","https://openalex.org/W1977895389","https://openalex.org/W1980753586","https://openalex.org/W1994791309","https://openalex.org/W2012244785","https://openalex.org/W2018706083","https://openalex.org/W2024309428","https://openalex.org/W2032316144","https://openalex.org/W2075819923","https://openalex.org/W2078895915","https://openalex.org/W2090929784","https://openalex.org/W2097100146","https://openalex.org/W2103141287","https://openalex.org/W2104152840","https://openalex.org/W2114457482","https://openalex.org/W2123140996","https://openalex.org/W2128802446","https://openalex.org/W2131071649","https://openalex.org/W2136011614","https://openalex.org/W2143365754","https://openalex.org/W2157765339","https://openalex.org/W2164282683","https://openalex.org/W2168474627","https://openalex.org/W2233427088","https://openalex.org/W2236516011","https://openalex.org/W2565858504","https://openalex.org/W2749408143","https://openalex.org/W2962718871","https://openalex.org/W2962917010","https://openalex.org/W2962944597","https://openalex.org/W2963078692","https://openalex.org/W2963748055","https://openalex.org/W2964221211","https://openalex.org/W3025390840","https://openalex.org/W3101566353","https://openalex.org/W3102177650","https://openalex.org/W3121223669","https://openalex.org/W3125380953","https://openalex.org/W4210791600","https://openalex.org/W4230022934","https://openalex.org/W4233510726","https://openalex.org/W4240136462","https://openalex.org/W4297854194"],"related_works":["https://openalex.org/W4285277090","https://openalex.org/W4327738859","https://openalex.org/W2112641952","https://openalex.org/W2092311672","https://openalex.org/W2914542058","https://openalex.org/W4385382439","https://openalex.org/W4286896588","https://openalex.org/W2029188814","https://openalex.org/W3136792315","https://openalex.org/W4387635768"],"abstract_inverted_index":{"This":[0,116,179],"paper":[1,218],"provides":[2,110],"a":[3,59,130,138],"probabilistic":[4],"proof":[5,113],"of":[6,13,21,62,70,102,105,114,123,140,188,216,236],"the":[7,19,53,68,106,121,124,145,149,186,189,225,237],"comparison":[8,198],"result":[9,99,181,192,199,215],"for":[10,133,193],"viscosity":[11,22,72,151],"solutions":[12,23,73,152],"path-dependent":[14,126],"semilinear":[15],"PDEs.":[16],"We":[17],"consider":[18],"notion":[20,69,139],"introduced":[24],"in":[25,48,75,120,148,203],"[I.":[26,204],"Ekren,":[27],"et":[28,206],"al.,":[29,207],"Ann.":[30,208],"Probab.,":[31,209],"42":[32,210],"(2014),":[33,211],"pp.":[34,87,95,212],"204--236],":[35],"which":[36,45],"considers":[37],"as":[38,185],"test":[39,63,107],"functions":[40,64,108],"all":[41],"those":[42],"smooth":[43],"processes":[44],"are":[46,174],"tangent":[47],"mean.":[49],"When":[50],"restricted":[51],"to":[52,67,144],"Markovian":[54],"case,":[55],"this":[56,103,217],"definition":[57],"induces":[58],"larger":[60],"set":[61],"and":[65,78,109,156,169],"reduces":[66],"stochastic":[71],"analyzed":[74],"[E.":[76],"Bayraktar":[77],"M.":[79],"Sirbu,":[80],"Proc.":[81],"Amer.":[82,159],"Math.":[83,160],"Soc.,":[84],"140":[85],"(2012),":[86],"3645--3654;":[88],"SIAM":[89],"J.":[90],"Control":[91],"Optim.,":[92],"51":[93],"(2013),":[94],"4274--4294].":[96],"Our":[97],"main":[98],"takes":[100],"advantage":[101],"enlargement":[104],"an":[111],"easier":[112],"comparison.":[115],"is":[117,219],"most":[118],"remarkable":[119],"context":[122],"linear":[125],"heat":[127],"equation.":[128],"As":[129],"key":[131],"ingredient":[132],"our":[134],"methodology,":[135],"we":[136,170,228],"introduce":[137],"punctual":[141],"differentiation,":[142],"similar":[143,197],"corresponding":[146],"concept":[147],"standard":[150],"[L.":[153],"A.":[154],"Caffarelli":[155],"X.":[157],"Cabre,":[158],"Soc.":[161],"Colloq.":[162],"Publ.,":[163],"43,":[164],"AMS,":[165],"Providence,":[166],"RI,":[167],"1995],":[168],"prove":[171],"that":[172,227],"semimartingales":[173],"almost":[175],"everywhere":[176],"punctually":[177],"differentiable.":[178],"smoothness":[180,191],"can":[182],"be":[183],"viewed":[184],"counterpart":[187],"Aleksandroff":[190],"convex":[194],"functions.":[195],"A":[196],"was":[200],"established":[201],"earlier":[202],"Ekren":[205],"204--236].":[213],"The":[214],"more":[220,223],"general":[221],"and,":[222],"importantly,":[224],"arguments":[226],"develop":[229],"do":[230],"not":[231],"rely":[232],"on":[233],"any":[234],"representation":[235],"solution.":[238]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
