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ATLANTISSTUDIESINPROBABILITYANDSTATISTICS VOLUME2 SERIESEDITOR: CHRISP.TSOKOS Atlantis Studies in Probability and Statistics SeriesEditor: ChrisP.Tsokos, UniversityofSouthFloridaTampa, Tampa,USA (ISSN:1879-6893) Aimsandscopeoftheseries TheSeries‘AtlantisStudiesinProbabilityandStatistics’publishesstudiesofhigh-quality throughouttheareasofprobabilityandstatisticsthathavethepotentialtomakeasignifi- cantimpactontheadvancementinthesefields.Emphasisisgiventobroadinterdisciplinary areasatthefollowingthreelevels: (I)Advanced undergraduate textbooks, i.e., aimedatthe3rdand4thyearsofundergrad- uatestudy,inprobability,statistics,biostatistics,businessstatistics,engineeringstatistics, operationsresearch,etc.; (II)Graduatelevelbooks,andresearchmonographsintheaboveareas,plusBayesian,non- parametric,survivalanalysis,reliabilityanalysis,etc.; (III)FullConferenceProceedings,aswellasSelectedtopicsfromConferenceProceedings, coveringfrontierareasofthefield,togetherwithinvitedmonographsinspecialareas. AllproposalssubmittedinthisserieswillbereviewedbytheEditor-in-Chief,inconsulta- tionwithEditorialBoardmembersandotherexpertreviewers Formoreinformationonthisseriesandourotherbookseries,pleasevisitourwebsiteat: www.atlantis-press.com/publications/books PARIS–AMSTERDAM–BEIJING (cid:2)c ATLANTISPRESS Stochastic Differential Games TheoryandApplications KandethodyM.Ramachandran, ChrisP.Tsokos UniversityofSouthFlorida, DepartmentofMathematicsandStatistics 4202E.FowlerAvenue, Tampa,FL33620-5700,USA PARIS–AMSTERDAM–BEIJING AtlantisPress 8,squaredesBouleaux 75019Paris,France ForinformationonallAtlantisPresspublications,visitourwebsiteat:www.atlantis-press.com Copyright ThisbookispublishedundertheCreativeCommonsAttribution-Non-commerciallicense,meaning thatcopying,distribution,transmittingandadaptingthebookispermitted,providedthatthisisdone fornon-commercialpurposesandthatthebookisattributed. Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorby anymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorage andretrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. AtlantisStudiesinProbabilityandStatistics Volume1:BayesianTheoryandMethodswithApplications-VladimirP.Savchuk,C.P.Tsokos ISBNs Print: 978-94-91216-46-6 E-Book: 978-94-91216-47-3 ISSN: 1879-6893 (cid:2)c 2012ATLANTISPRESS Dedicationstoourfamilies: Usha,Vikas,VilasandVarshaRamachandran and Debbie,Mathew,Jonathan,andMariaTsokos Preface Conflicts in the form of wars, or competition among countries and industrial institutions areplentyinhumanhistory.Theintroductionofgametheoryinthemiddleofthetwentieth century shed insights and enabled researchers to analyze this subject with mathematical rigor. From the ground-breaking work of VonNeumann and Morgenston, modern game theory evolved enormously. In the last few decades, Dynamic game theory framework hasbeendeepenedandgeneralizedfromthepioneeringworkondifferentialgamesbyR. Isaacs, L.S. Pontryagin and his school, and on stochastic games by Shapley. This book willexposethereadertosomeofthefundamentalmethodologyinnon-cooperativegame theory,andhighlightsomenumericalmethods,alongwithsomerelevantapplications. Since the early development days, differential game theory has had a significant impact insuchdiversedisciplinesasappliedmathematics, economics, systemstheory, engineer- ing,operations,research,biology,ecology,environmentalsciences,amongothers.Modern gametheorynowreliesonwiderangingmathematicalandcomputationalmethods,andrel- evantapplicationsthatarerichandchallenging. Gametheoryhasbeenwidelyrecognized asanimportanttoolinmanyfields. Importanceofgametheorytoeconomicsisillustrated bythefactthatnumerousgametheorists,suchasJohnForbesNash,Jr.,RobertJ.Aumann and Thomas C. Schelling, have won the Nobel Memorial Prize in Economics Sciences. Simplyput,game-theoryhasthepotentialtoreshapetheanalysisofhumaninteraction. InChapter1,wewillpresentageneralintroduction,survey,andbackgroundmaterialfor stochastic differential games. A brief introduction of Linear pursuit-Evation differential gameswillbegiveninChapter2forabetterunderstandingofthesubjectconcepts. Chap- ter3willdealwithtwopersonZero-sumstochasticdifferentialgamesandvarioussolution methods. Wewillalsointroducegameswithmultiplemodes. Formalsolutionsforsome classesofstochasticlinearpursuit-evasiongameswillbegiveninChapter4.InChapter5, we will discuss N-person stochastic differential games. Diffusion models are in general vii viii StochasticDifferentialGames notverygoodapproximationsforrealworldproblems. Inordertodealwiththoseissues, wewillintroduceweakconvergencemethodsfortwopersontothestochasticdifferential gamesinChapter6. InChapter7,willcoverweakconvergencemethodsformanyplayer games. InChapter8, wewillintroducesomeusefulnumericalmethodsfortwodifferent payoffstructure;discountedpayoffandergodicpayoffaswellasthecaseofnonzerosum games. WewillconcludethebookinChapter9bygivingsomerealworldapplicationsof stochasticdifferentialgamestofinanceandcompetitiveadvertising. Wewishtoexpressoursincereappreciationtothereviewersofthepreliminarymanuscript ofthebookfortheirexcellentcommentsandsuggestions. Dr. M. Sambandham, Professor of Mathematics, Chief Editor, International Journal of Systemsandapplications. Dr.G.R.Aryal,AssistantProfessorofStatistics,PurdueUniversity,Calumet,Indiana. Dr.RebeccaWooten,AssistantProfessorofMathematics&Statistics,UniversityofSouth Florida,Tampa,Florida. Dr.V.Laksmikatham,DistinguishedProfessorofMathematics,Emeritus,FloridaInstitute ofTechnology. Dr.YongXu,AssistantProfessorofMathematicsatRadfordUniversity, Dr.Kannan,ProfessorofMathematics–Emeritus,UniversityofGeorgia. Dr. GeoffreyO.Okogbaa, ProfessorofIndustrialEngineeringandManagementScience, UniversityofSouthFlorida,Tampa,Florida. We would also like to thank the editorial staff of Atlantis Press, in particular, the project managerMr.WillievanBerkum. Finally,averyspecialthankstoBeverlyDeVine-Hoffmeyerforherexcellentworkintyp- ingthisbook. K.M.Ramachandran C.P.Tsokos Contents Preface vii 1. Introduction,SurveyandBackgroundMaterial 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 DeterministicDifferentialGames: ABriefSurvey . . . . . . . . . . . . . 5 1.3 StochasticDifferentialGames: DefinitionandBriefDiscussion . . . . . . 14 1.4 FormulationoftheProblem. . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2. StochasticLinearPursuit-EvasionGame 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 PreliminariesandanExistenceTheorem . . . . . . . . . . . . . . . . . . 26 2.3 ExistenceofaSolutionforaStochasticLinearPursuit-EvasionGame . . 30 2.4 TheSolutionofaStochasticLinearPursuit-EvasionGameWith NonrandomControls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3. TwoPersonZero-SumDifferentialGames-GeneralCase 47 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 TwoPersonZero-sumGames:Martingalemethods . . . . . . . . . . . . 47 3.3 TwoPersonZero-sumGamesandViscositySolutions . . . . . . . . . . . 58 3.4 Stochasticdifferentialgameswithmultiplemodes . . . . . . . . . . . . . 61 4. Formal Solutions for Some Classes of Stochastic Linear Pursuit- EvasionGames 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 FormalsolutionforaStochasticLinearPursuit-Evasiongamewith perfectinformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ix x StochasticDifferentialGames 4.4 OnStochasticPursuit-Evasiongameswithimperfectinformation . . . . . 69 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5. N-PersonNoncooperativeDifferentialGames 73 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 AstochasticPursuit-EvasionGame . . . . . . . . . . . . . . . . . . . . 73 5.3 Generalsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6. WeakConvergenceinTwoPlayerStochasticDifferentialGames 95 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.2 WeakConvergencePreliminaries . . . . . . . . . . . . . . . . . . . . . . 96 6.3 SomePopularPayoffStructures . . . . . . . . . . . . . . . . . . . . . . 98 6.4 Two Person Zero-sum Stochastic Differential Game with Multiple Modes,WeakConvergence . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5 PartiallyObservedStochasticDifferentialGames . . . . . . . . . . . . . 125 6.6 DeterministicApproximationsinTwo-PersonDifferentialGames . . . . 135 7. WeakConvergenceinManyPlayerGames 147 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 SomePopularPayoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3 DeterministicApproximationsinN-PersonDifferentialGames . . . . . . 157 8. SomeNumericalMethods 165 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.2 DiscountedPayoffCase. . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.3 ErgodicPayoffcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 8.4 Non-zeroSumCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 9. ApplicationstoFinance 215 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 9.2 StochasticEquityInvestmentModelwithInstitutionalInvestror Speculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 9.3 CompetitiveAdvertisingunderUncertainty . . . . . . . . . . . . . . . . 221 References 233 Chapter1 Introduction, Survey and Background Material 1.1 Introduction Game theory has emerged out of the growing need for scientists and economists to have bettergraspoftherealworldintoday’stechnologicalrevolution. Gametheorydealswith tactical interactions among multiple decision makers. These interactions can range from completely non-cooperative to completely cooperative. These decision makers are usu- allyreferredasplayersoragents. Eachplayereithertriestomaximize(inwhichcasethe objectivefunctionisautilityfunctionorbenefitfunction)orminimize(inwhichcasethe objective function is called a cost function or a loss function) using multiple alternatives (actions,orequivalentlydecisionvariable). Iftheplayerswereabletoenterintoacooper- ativeagreementsothattheselectionofactionsordecisionsisdonecollectivelyandwith fulltrust,sothatallplayerswouldbenefittotheextentpossible,andnoinefficiencywould arise,thenwewouldbeintherealmofcooperativegametheory.Theissuesofbargaining, coalitionformation,excessutilitydistribution,etc. areofimportanceincooperativegame theory. Howevercooperativegametheorywillnotbecoveredinthisbook. Thisbookwill onlydealwithnon-cooperativegametheory, wherenocooperationisallowedamongthe players. Theoriginofgametheoryandtheirdevelopmentcouldbetracedtothepioneeringworkof JohnVonNeumannandOskarMorgenston[201]publishedin1944. Duetotheintroduc- tionofguidedinterceptormissilesin1950s,thequestionsofpursuitandevasiontookcenter stage. Themathematicalformulationandstudyofthedifferentialgameswasinitiatedby RufusIsaacs,whowasthenwiththeMathematicsdepartmentoftheRANDCorporation, inaseriesofRANDCorporationmemorandathatappearedin1954,[90]. Thisworkand hisfurtherresearcheswereincorporatedintoabook[91]whichinspiredmuchfurtherwork andinterestinthisarea. AftertheOscarfilmcalled“ABeautifulMind”wasreleasedby 1

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