Stochastic Volterra integro-differential equations

StochasticVolterraintegro-differentialequations:stabilityandnumericalmethodsLeonidE.Shaikhet∗&JasonA.Roberts†March9,2004AbstractWeconsiderthereliabilityofsomenumericalmethodsinpreservingthestabilitypropertiesofthelinearstochasticfunctionaldifferentialequation˙x(t)=αx(t)+βZt0x(s)ds+σx(t−τ)˙W(t),whereα,β,σ,τ≥0arerealconstants,andW(t)isastandardWienerprocess.Weadopttheshorthandnotationof˙x(t)torepresentthedifferentialdx(t)etc.OurchoiceoftestequationisastochasticperturbationoftheclassicaldeterministicBrunner&Lamberttestequationforσ=0andsoourinvesti-gationmaybethoughtofasanextensionoftheirwork.SufficientconditionsfortheasymptoticmeansquarestabilityofsolutionstoboththedifferentialequationanddiscreteanaloguesarederivedusingthegeneralmethodofLyapunovfunctionalsconstructionproposedbyKol-manovskii&Shaikhetwhichhaspreviouslybeensuccessfullyemployedfordeterministicandstochasticdifferentialanddifferenceequationswithdelay.Theareasoftheregionsofasymptoticstabilityforeachθmethod,in-dicatedbythesufficientconditionsforthediscretesystem,areshowntobeequalandweshowthatanupperboundcanbeputonthetime-stepparameterforthenumericalmethodfowhichthesystemisasymptoticallymean-squarestable.Weillustrateourresultsbymeansofnumericalexperimentsandvariousstabilitydiagrams.Weexaminetheextenttowhichthecontinuoussystemcantoleratestochasticperturbationsbeforelosingitsstabilitypropertiesandweillustratehowonemayaccuratelychooseanumericalmethodtopreservethestabilitypropertiesoftheoriginalprobleminthenumericalsolution.Ournumericalexperimentsalsoindicatethatthequalityofthesufficientconditionsisveryhigh.∗DonetskStateAcademyofManagement,Ukraine,[e-mail],collaborationsup-portedbyNATO,grantref.PST.EV.979727†MathematicsDepartment,UniversityCollegeChester,England,[e-mail]11IntroductionVolterraintegro-differentialequationsariseinthemodellingofhereditarysystems(i.e.systemswherethepastinfluencesthepresent)suchaspopulationgrowth,pollution,financialmarketsandmechanicalsystems(see[4],[1]forexample).Thelong-termbehaviourandstabilityofsuchsystemsisanimportantareaforinvesti-gation.Forexample-willapopulationdeclinetodangerouslylowlevels?Couldasmallchangeintheenvironmentalconditionshavedrasticconsequencesonthelong-termsurvivalofthepopulation?Thereisagrowingbodyofworkdevotedtosuchinvestigations(see[19],[6]forexample).Analyticalsolutionstosuchproblemsaregenerallyunavailableandnumericalmethodsareadoptedforobtainingapprox-imatesolutions.Anaturalquestiontoaskis“dothenumericalsolutionspreservethestabilitypropertiesoftheexactsolution?”.Wereferthereadertoanumberofworkswheretheanswerstosuchquestionsareinvestigated:[2],[3],[7],[8],[5],[21].Manyreal-worldphenomenaaresubjecttorandomnoiseorperturbations(forex-ample,freakweatherconditionsmayadverselyaffectthesupportsofabridge,possiblychangingthelong-termintegrityofthestructure).Itisanaturalexten-sionofthedeterministicworkcarriedoutbyourselvesandotherstoconsiderthestablilityofstochasticsystemsandofnumericalsolutionstosuchsystems.Were-ferthereaderstoanumberoftextswhichdiscusstheroleofstochasticsystemsinmathematicalmodelling:[9],[1],[20].Inthispaperweconsiderthescalarlineartestequation˙x(t)=αx(t)+βZt0x(s)ds+σx(t−τ)˙W(t),(1.1)x(s)=ϕ(s),s∈[−τ,0],whereα,β,σ,τ≥0arerealconstants,andW(t)isastandardWienerprocess.Inparticularifσ=0thenthisequationreducestothedeterministiclineartestequationofBrunnerandLambert[2].Whenconsideringstabilityofasystemwemustdecideonasuitabledefinitionforstability.Thereareanumberofdefinitionsforthestabilityofstochasticsystems.Acommonchoiceofdefinitionamongstnumericalanalystsinvestigatingstochasticdifferentialequationsisthatofmeansquarestabilityandasymptoticmeansquarestability.Wederiveasymptoticmeansquarestabilityconditionsforthelineartestequation(1.1).Ananalagousapproachisusedtoderiveconditionsforasymptoticmeansquarestabilityofalinearstochasticdifferenceequation.Itisshownthatourchoiceofnumericalmethodsarespecialcasesofthisparticulardifferenceequation,therebyallowingustoproducestabilityconditionsforthenumericalsolutionstotheoriginalproblem.Finally,wepresentsomestabilitydiagramsandnumericalexperimentstoillustrateourresults.2Themainconclusionofourinvestigationherecanbeformulatedinthefollowingway:ifthetrivialsolutionoftheinitialfunctionaldifferentialequationisasymp-toticallymeansquarestablethenthereexistawayandastepofdiscretizationofthisequationthatthetrivialsolutionofthecorrespondingdifferenceequationisasymptoticallymeansquarestabletoo.Moreover,itispossibletofindanupperboundforthestepofdiscretizationforwhichthecorrespondingdiscreteanaloguepreservesthepropertiesofstability.TheconditionsforasymptoticmeansquarestabilityareobtainedherebyvirtueofKolmanovskiiandShaikhet’sgeneralmethodofLyapunovfunctionalsconstruction([11]to[17])whichisapplicableforbothdifferentialanddifferenceequations,bothfordeterministicandstochasticsystemswithdelay.Letusremindourselveshereofsomedefinitionsandstatementswhichwillbeused.Let{Ω,F,P}beabasicprobabilityspacewithafamilyofσ-algebrasft⊂F,t≥0,Hbeaspaceoff0-adaptedfunctionsϕ(s),s≤0,Eisthesignofexpecta