Stochastic Impulse Control of Non-Markovian Proces

arXiv:0806.2761v1[math.PR]17Jun2008StochasticImpulseControlofNon-MarkovianProcessesBoualemDjehiche∗SaidHamad`ene†andIbtissamHdhiri‡June17,2008AbstractWeconsideraclassofstochasticimpulsecontrolproblemsofgeneralstochasticprocessesi.e.notnecessarilyMarkovian.Underfairlygeneralconditionsweestablishexistenceofanoptimalimpulsecontrol.Wealsoproveexistenceofcombinedoptimalstochasticandimpulsecontrolofafairlygeneralclassofdiffusionswithrandomcoefficients.Unlike,intheMarkovianframework,wecannotapplyquasi-variationalinequalitiestechniques.WeratherderivethemainresultsusingtechniquesinvolvingreflectedBSDEsandtheSnellenvelope.AMSsubjectClassifications:60G40;60H10;62L15;93E20;49N25.Keywords:stochasticimpulsecontrol;Snellenvelope;Stochasticcontrol;backwardstochasticdifferentialequations;optimalstoppingtime.1IntroductionFindingastochasticimpulsecontrolpolicyamountstodeterminingthesequenceofrandomdatesatwhichthepolicyisexercisedandthesequenceofimpulsesdescribingthemagnitudeoftheappliedpolicies,whichmaximizesagivenrewardfunction.Giventhegeneralapplicability∗DepartmentofMathematics,TheRoyalInstituteofTechnology,S-10044Stockholm,Sweden.e-mail:[e-mail]†Universit´eduMaine,D´epartementdeMath´ematiques,EquipeStatistiqueetProcessus,AvenueOlivierMessiaen,72085LeMans,Cedex9,France.e-mail:[e-mail]‡Universit´eduMaine,D´epartementdeMath´ematiques,EquipeStatistiqueetProcessus,AvenueOlivierMessiaen,72085LeMans,Cedex9,France.e-mail:[e-mail]1ofstochasticimpulsecontrolmodelsinvariousfieldssuchasfinance,e.g.cashmanagement(seeKorn(1999)foranexcellentsurveyandthetextbookbyJeanblancetal.(2005)andthereferencestherein),andmanagementofrenewableresources(seee.g.Alvarez(2004),AlvarezandKoskel(2007)andthereferencestherein),itisnotsurprisingthatthemathematicalframeworkofsuchproblemsiswellestablished(seeLepeltier-Marchal(1984),ØksendalandSulem(2006)andthereferencesthereinandtheseminaltextbookbyBensoussanandLions(1984)onquasi-variationalinequalitiesandimpulsecontrol).Indeed,inmostcases,theimpulsecontrolproblemisstudiedrelyingonquasi-variationalinequalities,whichispossibleonlythroughtacitlyassumingthattheunderlyingdynamicsofthecontrolledsystemisMarkovianandtheinstantaneouspartoftherewardfunctionadeterministicfunctionofthevalueoftheprocessatacertaininstant.Theseassumptionsareobviouslynotrealisticinmostapplications,suchasincertainmodelsincommoditiestrading.EveniftheunderlyingprocessisMarkov,theinstantaneouspartoftherewardfunctionmaydependonthewholepathoftheprocessorissimplyrandom.InthisstudyweconsideraclassofstochasticimpulsecontrolproblemswheretheunderlyingdynamicsofthecontrolledsystemistypicallynotMarkovandwheretheinstantaneousrewardfunctionalisrandom,inwhichcase,wecannotrelyonthewellestablishedquasi-variationalinequalitiestechniquetosolveit.Instead,wesolvetheproblemusingtechniquesinvolvingreflectedBSDEsandtheSnellenvelopethatseemsuitwellthisgeneralsituation.Themainideaistoexpressthevalue-processofthecontrolproblemasaSnellenvelopeandshowthatitsolvesareflectedBSDE,whoseexistenceanduniquenessareguaranteedprovidedsomemildintegrabilityconditionsoftheinvolvedcoefficients.ThisisdonethroughanappropriateapproximationschemeofthesystemofreflectedBSDEsthatisshowntoconvergetoourvalueprocess.Theunderlyingapproximatingsequenceisshowntobethevalueprocessofanimpulsecontroloverstrategieswhichhaveonlyaboundednumberofimpulses,forwhichanoptimalpolicyisalsoshowntoexist.Finally,passingtothelimit,lettingthenumberofimpulsesbecomelarge,weproveexistenceofanoptimalpolicyofourstochasticimpulsecontrolproblem.Thepaperisorganizedasfollows.InSection2werecallthemaintoolsonreflectedBSDEsandSnellenvelopewewillusetoestablishthemainresults.InSection3,weformulatetheconsideredstochasticimpulsecontrol.InSection4,weconsideranappropriateapproximationschemeofthesystemofreflectedBSDEsthatisshowntoconvergetoourvalueprocess.In2Section5,weestablishexistenceofanoptimalimpulsecontroloverstrategieswithaboundednumberofimpulses,inSection6,weproveexistenceofanoptimalimpulsecontroloveralladmissiblestrategies.Moreover,thecorrespondingvalueprocessisthelimitofthesequenceofvalueprocessesassociatedwiththeoptimalimpulsecontroloverfinitestrategies,astheirnumberbecomeslarge.Finally,inSection7,weconsideramixedstochasticcontrolandimpulsecontrolproblemofafairlylargeclassofdiffusionprocessesthatarenotnecessarilyMarkovian.UsingaBeneˇs-typeselectiontheorem,wederiveanoptimalpolicyusingsimilartools.2PreliminariesandnotationThroughoutthispaper(Ω,F,IP)isafixedprobabilityspaceonwhichisdefinedastandardd-dimensionalBrownianmotionB=(Bt)0≤t≤Twhosenaturalfiltrationis(F0t:=σ{Bs,s≤t})0≤t≤T;(Ft)0≤t≤Tisthecompletedfiltrationof(F0t)0≤t≤TwiththeIP-nullsetsofF,hence(Ft)0≤t≤Tsatisfiestheusualconditions,i.e.,itisrightcontinuousandcomplete.Let•Pbetheσ-algebraon[0,T]×ΩofFt-progressivelymeasurableprocesses.•foranyp≤2,Hp,kbethesetofP-measurableprocessesv=(vt)0≤t≤TwithvaluesinRksuchthatE[RT0|vs|pds]∞.•S2(resp.S2c)bethesetofP-measurableandc`adl`ag(abbreviationofrightcontinuousandleftlimited)(resp.continuous)processesY=(Yt