{"id":"https://openalex.org/W1965138785","doi":"https://doi.org/10.1137/s0363012901387550","title":"General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations","display_name":"General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W1965138785","doi":"https://doi.org/10.1137/s0363012901387550","mag":"1965138785"},"language":"en","primary_location":{"id":"doi:10.1137/s0363012901387550","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012901387550","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/A5012568499","display_name":"Shanjian Tang","orcid":"https://orcid.org/0000-0003-3884-042X"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shanjian Tang","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5012568499"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.4393,"has_fulltext":false,"cited_by_count":191,"citation_normalized_percentile":{"value":0.89464705,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"42","issue":"1","first_page":"53","last_page":"75"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9998000264167786,"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":0.9998000264167786,"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/T12011","display_name":"Insurance, Mortality, Demography, Risk Management","score":0.9846000075340271,"subfield":{"id":"https://openalex.org/subfields/3317","display_name":"Demography"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11413","display_name":"Risk and Portfolio Optimization","score":0.9775999784469604,"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":0.8437855243682861},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.6330208778381348},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.6275734901428223},{"id":"https://openalex.org/keywords/stochastic-control","display_name":"Stochastic control","score":0.5238785147666931},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.48291847109794617},{"id":"https://openalex.org/keywords/brownian-motion","display_name":"Brownian motion","score":0.45455750823020935},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.44856899976730347},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.4268922209739685},{"id":"https://openalex.org/keywords/algebraic-riccati-equation","display_name":"Algebraic Riccati equation","score":0.42375028133392334},{"id":"https://openalex.org/keywords/optimal-control","display_name":"Optimal control","score":0.40589067339897156},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35591721534729004},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3383761942386627},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3373212218284607},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3307451605796814},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.30247962474823},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.09091934561729431},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08679059147834778},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07607874274253845}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8437855243682861},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.6330208778381348},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.6275734901428223},{"id":"https://openalex.org/C170131372","wikidata":"https://www.wikidata.org/wiki/Q7617811","display_name":"Stochastic control","level":3,"score":0.5238785147666931},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.48291847109794617},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.45455750823020935},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44856899976730347},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.4268922209739685},{"id":"https://openalex.org/C13847129","wikidata":"https://www.wikidata.org/wiki/Q4723989","display_name":"Algebraic Riccati equation","level":4,"score":0.42375028133392334},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.40589067339897156},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35591721534729004},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3383761942386627},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3373212218284607},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3307451605796814},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.30247962474823},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.09091934561729431},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08679059147834778},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07607874274253845},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0363012901387550","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0363012901387550","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":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W196003930","https://openalex.org/W1515432731","https://openalex.org/W1523024486","https://openalex.org/W1525883337","https://openalex.org/W1985050062","https://openalex.org/W1987397545","https://openalex.org/W1988310975","https://openalex.org/W1994791309","https://openalex.org/W2022977680","https://openalex.org/W2027827748","https://openalex.org/W2031392273","https://openalex.org/W2033603136","https://openalex.org/W2060664524","https://openalex.org/W2060861336","https://openalex.org/W2060878939","https://openalex.org/W2070504031","https://openalex.org/W2074988991","https://openalex.org/W2076846099","https://openalex.org/W2078895915","https://openalex.org/W2082094090","https://openalex.org/W2086139156","https://openalex.org/W2102316513","https://openalex.org/W2127061658","https://openalex.org/W2157883119","https://openalex.org/W4212780424","https://openalex.org/W4298228534","https://openalex.org/W4299411938"],"related_works":["https://openalex.org/W2005019312","https://openalex.org/W2105459831","https://openalex.org/W1546579375","https://openalex.org/W1563296412","https://openalex.org/W2127238600","https://openalex.org/W2114791549","https://openalex.org/W16457797","https://openalex.org/W2792258377","https://openalex.org/W1530596433","https://openalex.org/W2096105566"],"abstract_inverted_index":{"Consider":[0],"the":[1,4,20,23,52,65,69,87,93,109,118,131,142,179,183,189,192,196,207,225,228,231,235,238,243,270,276,303,316],"minimization":[2],"of":[3,22,57,133,178,227,230,237,262,269,289,295,305],"following":[5,24,66],"quadratic":[6],"cost":[7],"functional:":[8],"$$J(u):=E\\langle":[9],"Mx_T,x_T\\rangle":[10],"+E\\int_0^T(\\langle":[11],"Q_sx_s,x_s\\rangle":[12],"+\\langle":[13],"N_su_s,u_s\\rangle":[14],")\\,":[15],"ds,$$":[16],"where":[17],"x":[18],"is":[19,41,49,102,222,283,312],"solution":[21,229,314],"linear":[25],"stochastic":[26,53,105,184,197,208,239,272],"control":[27],"system:":[28],"$$":[29,39],"\\eq{dx_t=&(A_tx_t+B_tu_t)\\,":[30],"dt":[31],"+\\sum_{i=1}^d(C_t^ix_t+D_t^iu_t)\\,":[32],"dW_t^i,\\cr":[33],"x_0=&h\\in":[34],"\\mathbb{R}^n,\\qquad":[35],"u_t\\in":[36],"\\mathbb{R}^m;":[37],"\\cr}":[38],"u":[40],"a":[42,103,134,175,253,260,286,290,309,313],"square":[43],"integrable":[44],"adapted":[45,80],"process.":[46],"The":[47,219,279],"problem":[48,129],"conventionally":[50],"called":[51],"LQ":[54,209],"(the":[55,111],"abbreviation":[56],"\"linear":[58],"quadratic\")":[59],"problem.":[60,210],"We":[61,85,172],"are":[62,76,212],"concerned":[63],"with":[64,108,298],"general":[67,100],"case:":[68],"coefficients":[70],"A,B,Ci,Di,":[71],"Q,":[72],"N,":[73],"and":[74,89,125,163,195],"M":[75],"allowed":[77],"to":[78,223,234,259,284,315],"be":[79],"processes":[81],"or":[82],"random":[83],"matrices.":[84],"prove":[86],"existence":[88,226],"uniqueness":[90],"result":[91],"for":[92,206,293],"associated":[94,271],"Riccati":[95,180,193,232,265,296,318],"equation,":[96],"which":[97,137,299,311],"in":[98,117,155,216,267],"our":[99,217],"case":[101,132],"backward":[104],"differential":[106],"equation":[107,181,194,233],"generator":[110],"drift":[112],"term)":[113],"being":[114],"highly":[115],"nonlinear":[116],"two":[119,201,213],"unknown":[120],"variables.":[121],"This":[122,187],"solves":[123],"Bismut":[124,147],"Peng's":[126],"long-standing":[127],"open":[128],"(for":[130],"Brownian":[135],"filtration),":[136],"was":[138],"initially":[139],"proposed":[140],"by":[141],"French":[143],"mathematician":[144],"J.":[145],"M.":[146,164],"[in":[148],"S\u00e9minaire":[149],"de":[150],"Probabilit\u00e9s":[151],"XII,":[152],"Lecture":[153],"Notes":[154],"Math.":[156],"649,":[157],"C.":[158],"Dellacherie,":[159],"P.":[160],"A.":[161],"Meyer,":[162],"Weil,":[165],"eds.,":[166],"Springer-Verlag,":[167],"Berlin,":[168],"1978,":[169],"pp.":[170],"180--264].":[171],"also":[173],"provide":[174],"rigorous":[176],"derivation":[177],"from":[182,242],"Hamilton":[185,198,273],"system.":[186,246],"completes":[188],"interrelationship":[190],"between":[191],"system":[199],"as":[200],"different":[202],"but":[203],"equivalent":[204],"tools":[205],"There":[211],"key":[214,281],"points":[215],"arguments.":[218],"first":[220],"one":[221],"connect":[224],"homomorphism":[236],"flows":[240],"derived":[241],"optimally":[244],"controlled":[245],"Actually,":[247],"we":[248,255,300],"establish":[249,285],"their":[250],"equivalence.":[251],"As":[252],"consequence,":[254],"can":[256],"construct":[257],"solutions":[258,294,307],"sequence":[261,304],"suitably":[263],"modified":[264],"equations":[266],"terms":[268],"systems":[274],"(and":[275],"optimal":[277],"controls).":[278],"second":[280],"point":[282],"new":[287],"type":[288],"priori":[291],"estimate":[292],"equations,":[297],"show":[301],"that":[302],"constructed":[306],"has":[308],"limit":[310],"original":[317],"equation.":[319]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":11},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":15},{"year":2022,"cited_by_count":22},{"year":2021,"cited_by_count":14},{"year":2020,"cited_by_count":11},{"year":2019,"cited_by_count":18},{"year":2018,"cited_by_count":15},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":16},{"year":2014,"cited_by_count":15},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":4}],"updated_date":"2026-04-07T14:57:38.498316","created_date":"2025-10-10T00:00:00"}
