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1RiccatiDifferentialandAlgebraicEquationsfromStochasticDifferentialGamesMichaelMcAseyandLibinMouDepartmentofMathematicsBradleyUniversityPeoria,IL61625Abstract.InthispaperwestudyaclassofmatrixdifferentialandalgebraicRiccatiequationsarisingfromstochasticdifferentialgameswithaquadraticcost.Wewillusethenotionofupperandlowersolutionstogivenecessaryandsufficientconditionsfortheexistenceofsolutionstotheequations.Wealsoobtainamonotonicity,acomparisontheoremandinterpretationtheoremsforupperandlowersolutions.Inaddition,themean-squarestabilityofsolutionsisalsodiscussed.§1.IntroductionThefocusofthispaperisaclassofmatrixdifferentialandalgebraicequationsofRiccatitype,whicharisefromstochasticdifferentialgamesandothercontrolproblemswithaquadraticcost.Tomotivatethediscussion,weconsiderthefollowingsetting.Fixabcd=ßDßV[[!#8.LetandbetwostandardBrownianmotionsonaindependentprobabilityspaceovercdabcd==ß[=!ßVwithalmostsurely.Letbethesetof-valued,square335hintegrableprocessesadaptedwiththe-fieldgeneratedby,,respectively.Associated5[3ß#3abwitheachisaabcdcdcd?ßß´ßß?#?===Nß?##hhhquadraticcost:abNÐ?ß?ÑIBBÐBKB#BWV#BW??V?Ñ.#=#####abab(XXXXXXXXR???(1)whereisthesolutiontotheequationB.BEBFF?.GBH.[GBH?.[ßababab######??B=Dab,(2)andIÖ×3ß#ßrepresentstheexpectationoftheenclosedrandomvariable.WeassumethatforEßFßGßHßßRßVW33333Kßandarematrixfunctionsonsatisfyingcondition(4)giveninSection2.cd!Motivatedbythissetting,weconsiderthefollowingzero-sumdifferentialgameproblem.Problem1.Fsuchthatindab??ßss#ßhcd=MaxMinMinMax?#?###??#NßNßNßababab????ss??,orequivalently,forall.N?ß???N?ß??ß?ßssssabababcd####Nßabh=Thatis,therearetwoplayersforthedifferentialgame.Player1choosescontroltominimize?stheobjectivewhilePlayer2choosescontroltomaximize.Theorem1anditsproofshowsthatNN?s#asolutiontoProblem1canbeconstructedbyusingsolutionsofthefollowingRiccatidifferential2equation.TETTEGTGKw3#33XXXXXXXXabababab3#33333333333FTHTGWVHTHFTHTGW!ßMT;.(3)RInequation(2),thecoefficientsof,andmaycontainadditionaltermsthatare..[.[#knownfunctionsof.Onotherhand,theproblemwillleadtothesameRiccatiequation(3)plussomeauxiliaryequations.Theseauxiliaryequationsarelinearmatrixdifferentialequations(dependingon)Tthatalwayshavesolutions.Therefore,equation(3)anditsassociatedalgebraicequation(32)areourfocusinthispaper.TheexistingworkonRiccatiequationsfromdifferentialgamesoriginatesfromstudyingdeterministicgamesorstochasticgameswithnoiseindependentofthestatecontrols.Assuchthesepapersusuallyconsiderthespecialcasesof(3)with.Forthesecases,GGHH!##equation(3)reducestoTETTEKwXX3#333TFVFT!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.Denoteby8thesetofallrealsymmetricmatrices.Denote()if88Q RQRQRQR\M,andpositivesemidefinite(definite).ForaHilbertspaceandaninterval,8isaPMßM\_ab\bethespaceofallboundedandmeasurablefunctionsfromto.Inaddition,wedefinePMßÖTPMßÀTPMß×ß__w_ababab\\\.Unlessotherwisestated,theintervaliseitherorandisconsideredfixed.AllMßÐ_ßÓcd!equationsandinequalitiesinvolvingmatrixfunctionsarepointwisein.Forbrevity,Mwewillwrite,forexample,“”or“in”insteadof“forevery.”KKK ! !M !MabAssumption.WeassumethatEßFßGßHßßVßW33333KßRin(3)satisfyEßGPMßàFßHßWPÐMßÑàVPÐMßÑPMß3_333_3_5_abab888588X;;KÞR(4)Itwillbeusefultosplitequation(3)intoasumofitslinearpart