Table Of ContentATLANTISSTUDIESINPROBABILITYANDSTATISTICS
VOLUME2
SERIESEDITOR: CHRISP.TSOKOS
Atlantis Studies in Probability and Statistics
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UniversityofSouthFloridaTampa,
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(ISSN:1879-6893)
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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
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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
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