Weak approximation of stochastic differential dela

ISSN1360-1725UMISTWeakapproximationofstochasticdierentialdelayequationsEvelynBuckwarTonyShardlowNumericalAnalysisReportNo.439December2003ManchesterCentreforComputationalMathematicsNumericalAnalysisReportsDEPARTMENTSOFMATHEMATICSReportsavailablefrom:DepartmentofMathematicsUniversityofManchesterManchesterM139PLEnglandAndovertheWorld-WideWebfromURLs://ftp.ma.man.ac.uk/pub/narepWeakapproximationofstochasticdierentialdelayequationsEvelynBuckwarTonyShardlowyDecember2,2003AbstractAnumericalmethodforaclassofIt^ostochasticdierentialequationswithanitedelaytermisintroduced.ThemethodisbasedontheforwardEulerapproximationandisparameterisedbyitstimestep.WeakconvergencewithrespecttoaclassofsmoothtestfunctionalsisestablishedbyusingtheinnitedimensionalversionoftheKolmogorovequation.Withregularityassumptionsoncoecientsandinitialdata,therateofconvergenceisshowntobeproportionaltothetimestep.Somecomputationsarepresentedtodemonstratetherateofconvergence.Thisisanupdatedandcorrectedversionof[6].Keywords.Theoreticalapproximationofsolutions,Stochasticpartialdierentialequations,Stochasticdelayequations,Stabilityandconvergenceofnumericalapproxi-mations.AMSsubjectclassications.60H15,34K50,65L20,34A45.1IntroductionConsiderstochasticdierentialdelayequationsonRdoftheformdY(t)=hZ0a(ds)Y(t+s)+f(Y(t))idt+b(Y(t))dW(t);Y(0)=YS;Y(s)=YD(s)fors0,(1.1)forinitialconditionsYS2RdandYD2C([;0];Rd),wherea(
)isaddmatrixvaluedmeasureon[;0],f(
):Rd!Rd,b(
):Rd!Rdd,andW(
)isaBrownianmotiononRdwithcovarianceI.Thedelayis,whichshouldbeniteandpositive.TheequationshouldbeinterpretedinthesenseofIt^o.WenowdenetheforwardEulermethodfor(1.1).Denotetheoorfunctionbybtc,whichequalsthegreatestintegerlessthanorequaltot.Letai:=Z0a(ds)1[i
t;(i+1)
t)(s);i=b=
tc;:::;1;where1[t1;t2)(s)istheddidentitymatrixon[t1;t2)andiszerootherwise.Let
nbeindependentandnormallydistributedwithmeanzeroandvariance
tI.GenerateapproximationsYntoY(n
t)forn=1;2;:::byYn+1Yn=h1Xi=b=
tcaiYn+i+f(Yn)i
t+b(Yn)
n;(1.2)withinitialconditionsYi=YD(i
t)fori=b=
tc;:::;1andY0=YS.Humboldt-UniversitatzuBerlin,InstitutfurMathematik,BereichStochastik,UnterdenLinden6,10099Berlin,[e-mail],ManchesterUniveristy,OxfordRoad,ManchesterM139PL,[e-mail]rtbytheNueldFoundationNUF-NAL-00.1Inaseriesofpapers,strongapproximationmethodsforstochasticdierentialdelayequationswereconsideredbyC.andM.Tudor[25,26,27,28,29].Recentlythistopichasgainedmoreattention,see[1],[2],[14],and[16].Thetheorygivesconvergenceratesoforder
t1=2fortheforwardEulermethod,whichisoptimal,andappliestodelayequationsmoregeneralthan(1.1).TheaimofthisworkistounderstandtheweakconvergencepropertiesoftheforwardEulermethodfor(1.1).ThetheoreticalgroundingdevelopedfortheEulermethodinthispapershouldmakeitpossibletounderstandhigherorderweakapproximationmethodsforstochasticdierentialdelayequations.Therearetwobasicapproachestoachievingthisgoal.Asdevelopedin[15,17]forSDEs,wecanlookforhigherordermethods.Thedrawbackisthedicultyinimplementinghigherordermethodsforpracticalproblems.ThesecondapproachistouseRichardsonextrapolationbasedonalowerordernumericalmethodsuchasforwardEuler.Anunderstandingofthebehaviouroftheerrorasdevelopedin[24](wheretheerrorinweakapproximationisexpandedin
tpowerseries)isneededtojustifythisrigorously.Weleavetheseissuesasopenproblems.Wenowdescribethehypothesisneededforourweakconvergenceanalysis.Thehypothesisaremorerestrictivethanthoseneededforstrongconvergence,butgivebetterconvergencerates.Hypothesis1.1(i)Supposethatf:Rd!Rdisfourtimescontinuouslydierentiablewithf0,f00,f000,f0000bounded.Supposethatb:Rd!Rddisboundedwithfourboundedderivatives.(ii)Supposethata(ds)hasaC1densitya:[;0]!R.Foranintegerp4,introducethespacesGpoftestfunctions:Rd!Rthatarefourtimescontinuouslydierentiableandsatisfyk(n)(h)kL(Rdn;R)K(1+khkpnRd),forh2RdandsomeconstantK,forn=0;1;2;3;4(k
kL(Rm;R)isthestandardnorminducedonlinearoperatorsfromRmtoRbytheEuclideannorm).Inonedimension,thisspaceissucientlygeneraltoincludepolynomials.Forx=(YS;YD)T,writekxk:=(kYSk2Rd+kYDk2L2([;0];Rd))1=2.ForacontinuousfunctionYD:[;0]!Rd,letkYDkLip:=supt;t00kYD(t)YD(t0)kRdjtt0j:Theorem1.2LetHypothesis1.1hold.ConsiderYS2RdandaLipschitzfunctionYD:[;0]!Rd.LetY(t)(respectively,Yn)denotethesolutionof(1.1)(resp.,(1.2))correspondingtoinitialdatax=(YS;YD)T.ForT0and2Gp,p4,thereexistsaconstantKx0suchthatE(Y(T))E(YN)Kx
t;N
t=TandaconstantKindependentoftheinitialdatasuchthatKxK(1+kxkp)+K(1+kxkp1)kYDkLip:(1.3)Thisisthemainresultofthepresentpaper.Thetheoremmakesanumberofassumptionsontheregularityoftheproblem.ComparedtotheresultsavailableforSDEs,thehypothesisonf,b,andcomeasnosurpriseasfourderivativesarerequiredtoderivetheanalogousresultforSDEs.TheassumptionscanberelaxedforSDEsunderanon-degeneracyassumptiononthenoisebyuseoftheMalliavincalculus[4],butsuchtechniquesarenotusedinthispaper.Theassumptiononthedelaytermismorerestrictivean