1 Riccati Differential and Algebraic Equations fro

1RiccatiDifferentialandAlgebraicEquationsfromStochasticDifferentialGamesMichaelMcAseyandLibinMouDepartmentofMathematicsBradleyUniversityPeoria,IL61625Abstract.InthispaperwestudyaclassofmatrixdifferentialandalgebraicRiccatiequationsarisingfromstochasticdifferentialgameswithaquadraticcost.Wewillusethenotionofupperandlowersolutionstogivenecessaryandsufficientconditionsfortheexistenceofsolutionstotheequations.Wealsoobtainamonotonicity,acomparisontheoremandinterpretationtheoremsforupperandlowersolutions.Inaddition,themean-squarestabilityofsolutionsisalsodiscussed.§1.IntroductionThefocusofthispaperisaclassofmatrixdifferentialandalgebraicequationsofRiccatitype,whicharisefromstochasticdifferentialgamesandothercontrolproblemswithaquadraticcost.Tomotivatethediscussion,weconsiderthefollowingsetting.Fixabcd=ßD­ß‚V[[!#8.LetandbetwostandardBrownianmotionsonaindependentprobabilityspaceovercdabcd==ß[=œ!ßVwithalmostsurely.Letbethesetof-valued,square335hintegrableprocessesadaptedwiththe-fieldgeneratedby,,respectively.Associated5[†3œß#3abwitheachisaabcdcdcd?߭ߴ߂ß?#?===Nß?##hhhquadraticcost:abNÐ?ß?ќIBB€ÐBKB€#BW€V€#BW?€?V?Ñ.#=#####šabab(›XXXXXXXXR???(1)whereisthesolutiontotheequationB.BœEB€F€F?.€GB€H.[€GB€H?.[ßababab######??B=œDab,(2)andIÖ×3œß#ßrepresentstheexpectationoftheenclosedrandomvariable.WeassumethatforEßFßGßHßßRßVW33333Kßandarematrixfunctionsonsatisfyingcondition(4)giveninSection2.cd!Motivatedbythissetting,weconsiderthefollowingzero-sumdifferentialgameproblem.Problem1.Fsuchthatindab??ßss#ß­hcd=MaxMinMinMax?#?###??#NߜNߜNßababab????ss??,orequivalently,forall.N?ß?Ÿ??N?ß??ß?­ßssssabababcd####Nߟabh=Thatis,therearetwoplayersforthedifferentialgame.Player1choosescontroltominimize?stheobjectivewhilePlayer2choosescontroltomaximize.Theorem1anditsproofshowsthatNN?s#asolutiontoProblem1canbeconstructedbyusingsolutionsofthefollowingRiccatidifferential2equation.T€ET€TE€GTG€Kw3œ#33XXXXXXXXabababab3œ#33333333333FT€HTG€WV€HTHFT€HTG€Wœ!ß­MTœ;.(3)RInequation(2),thecoefficientsof,andmaycontainadditionaltermsthatare..[.[#knownfunctionsof.Onotherhand,theproblemwillleadtothesameRiccatiequation(3)plussomeauxiliaryequations.Theseauxiliaryequationsarelinearmatrixdifferentialequations(dependingon)Tthatalwayshavesolutions.Therefore,equation(3)anditsassociatedalgebraicequation(32)areourfocusinthispaper.TheexistingworkonRiccatiequationsfromdifferentialgamesoriginatesfromstudyingdeterministicgamesorstochasticgameswithnoiseindependentofthestatecontrols.Assuchthesepapersusuallyconsiderthespecialcasesof(3)with.Forthesecases,GœGœHœHœ!##equation(3)reducestoT€ET€TE€KwXX3œ#333TFVFTœ!ÞSee[2,Chapters4and9],[3,Chapter6],and[7,Chapter14],and[5]forsomeclassicalresultsonthisequation.WewillintroducetheconceptsofupperandlowersolutionsforthedifferentialRiccatiequation(3)andthealgebraicRiccatiequation(32).Theorems1and9inSections2and4,respectively,providemeaningfulinterpretationsofthenotionofupperandlowersolutions.Forexample,Theorem1showsthatifisalower(upper)solutionto(3)thengivesalower(upper)boundfortheTDTÐ=ÑDXobjectiveoverappropriatecontrols.Theorem9givesinterpretationsforupperandlowersolutionsNto(32)similarinspirittoTheorem1.Equation(3)appearstobequitecomplicated,butarepresentationofthecostrevealsitssimplestructure(Proposition3).InSection3,weuseNProposition3toproveacomparisontheorem(Theorem6)whichgivesconditionssothatanuppersolutiondominatesalowersolution.Thistheoremleadstonecessaryandsufficientconditionsforexistenceofsolutions(Theorem7)toequation(3).InSection4,weturntothealgebraicRiccatiequation(32)associatedwith(3),whicharisesfromstochasticgameproblemswithaninfinite-horizon.Theproblemandsomeconceptsrelatedtoms-stabilityareintroducedhere.InSection5,weproveamonotonicityresult(Theorem10)forsolutionsto(3),whichleadstonecessaryandsufficientconditions(Theorem11)forexistenceofsolutionsto(32).Generalizingatheoremin[2,Theorem9.7],weprovethat(32)hasasolutionwithfurtherproperties(Theorem12)relatedtomean-squarestability.InTheorem13weprovideaconditionthatguaranteesthatasolutionto(32)isms-stable.Theconceptsofupperandlowersolutionshavebeenintroducedandusedin[10]and[11]fortheRiccatiequationsarisingfromtheoptimalcontrolproblemsMaxandMin?##NßNßabab!??!.We?willseeseveralapplicationsofresultsfrom[10]and[11]inthispaper.Infact,suchapplicationspartiallymotivatedthesettingof[10]and[11].§2.PreliminaryResults3Notations.Denoteby’8thesetofallrealsymmetricmatrices.Denote()if8‚8Q RQžRQRQR\M,andpositivesemidefinite(definite).ForaHilbertspaceandaninterval,­’8isaPMßM\_ab\bethespaceofallboundedandmeasurablefunctionsfromto.Inaddition,wedefinePMߜÖT­PMßÀT­PMß×ß__w_ababab\\\.Unlessotherwisestated,theintervaliseitherorandisconsideredfixed.AllMßЁ_ßÓcd!equationsandinequalitiesinvolvingmatrixfunctionsarepointwisein.Forbrevity,­Mwewillwrite,forexample,“”or“in”insteadof“forevery.”KKK ! !M !­MabAssumption.WeassumethatEßFßGßHßßVßW33333KßRin(3)satisfyEßG­PMßàFßHßW­PÐMßÑàV­PÐMßÑ­PMß­3_333_3_5_abab‘‘’’’8‚88‚588X;;KÞR(4)Itwillbeusefultosplitequation(3)intoasumofitslinearpart