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Probability Theory and Stochastic Modelling 84 René Carmona François Delarue Probabilistic Theory of Mean Field Games with Applications II Mean Field Games with Common Noise and Master Equations Probability Theory and Stochastic Modelling Volume 84 Editors-in-chief PeterW.Glynn,Stanford,CA,USA AndreasE. Kyprianou,Bath,UK YvesLeJan,Orsay,France AdvisoryBoard SørenAsmussen,Aarhus,Denmark MartinHairer,Coventry,UK PeterJagers,Gothenburg,Sweden IoannisKaratzas,NewYork,NY,USA FrankP.Kelly,Cambridge,UK BerntØksendal,Oslo,Norway GeorgePapanicolaou,Stanford,CA,USA EtiennePardoux,Marseille,France EdwinPerkins,Vancouver,Canada HalilMeteSoner,Zürich,Switzerland The Probability Theory and Stochastic Modelling series is a merger and continuation of Springer’s two well established series Stochastic Modelling and Applied Probability and Probability and Its Applications series. It publishes researchmonographsthatmakeasignificantcontributiontoprobabilitytheoryoran applications domain in which advanced probability methods are fundamental. Booksinthisseriesareexpectedtofollowrigorousmathematicalstandards,while alsodisplayingtheexpositoryqualitynecessarytomakethemusefulandaccessible toadvancedstudentsaswellasresearchers.Theseriescoversallaspectsofmodern probabilitytheoryincluding (cid:129) Gaussianprocesses (cid:129) Markovprocesses (cid:129) Randomfields,pointprocessesandrandomsets (cid:129) Randommatrices (cid:129) Statisticalmechanicsandrandommedia (cid:129) Stochasticanalysis aswellasapplicationsthatinclude(butarenotrestrictedto): (cid:129) Branchingprocessesandothermodelsofpopulationgrowth (cid:129) Communicationsandprocessingnetworks (cid:129) Computational methods in probability and stochastic processes, including simulation (cid:129) Geneticsandotherstochasticmodelsinbiologyandthelifesciences (cid:129) Informationtheory,signalprocessing,andimagesynthesis (cid:129) Mathematicaleconomicsandfinance (cid:129) Statisticalmethods(e.g.empiricalprocesses,MCMC) (cid:129) Statisticsforstochasticprocesses (cid:129) Stochasticcontrol (cid:129) Stochasticmodelsinoperationsresearchandstochasticoptimization (cid:129) Stochasticmodelsinthephysicalsciences Moreinformationaboutthisseriesathttp://www.springer.com/series/13205 René Carmona • François Delarue Probabilistic Theory of Mean Field Games with Applications II Mean Field Games with Common Noise and Master Equations 123 RenéCarmona FrançoisDelarue ORFEDepartment InstitutUniversitairedeFrance PrograminAppliedand &LaboratoireJ.A.Dieudonné ComputationalMathematics UniversitéNiceSophiaAntipolis PrincetonUniversity Nice,France Princeton,NJ,USA ISSN2199-3130 ISSN2199-3149 (electronic) ProbabilityTheoryandStochasticModelling ISBN978-3-319-56435-7 ISBN978-3-319-56436-4 (eBook) DOI10.1007/978-3-319-56436-4 LibraryofCongressControlNumber:2017940847 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword Sinceitsinceptionaboutadecadeago,thetheoryofMeanFieldGameshasrapidly developed into one of the most significant and exciting sources of progress in the studyofthedynamicalandequilibriumbehavioroflargesystems.Theintroduction of ideas from statistical physics to identify approximate equilibria for sizeable dynamic games created a new wave of interest in the study of large populations of competitive individuals with “mean field” interactions. This two-volume book grew out of series of lectures and short courses given by the authors over the last fewyearsonthemathematicaltheoryofMeanFieldGamesandtheirapplications in social sciences, economics, engineering, and finance. While this is indeed the object of the book, by taste, background, and expertise, we chose to focus on the probabilisticapproachtothesegamemodels. Inatrailblazingcontribution,LasryandLionsproposedin2006amethodology to produce approximate Nash equilibria for stochastic differential games with symmetric interactions and a large number of players. In their models, a given playerfeelsthepresenceandthebehavioroftheotherplayersthroughtheempirical distribution of their private states. This type of interaction was studied in the statistical physics literature under the name of mean field interaction, hence the terminology Mean Field Game coined by Lasry and Lions. The theory of these new game models was developed in lectures given by Pierre-Louis Lions at the Collège de France which were video-taped and made available on the internet. Simultaneously,Caines,Huang,andMalhaméproposedasimilarapproachtolarge games under the name of Nash Certainty Equivalence principle. This terminology fellfromgraceandthestandardreferencetothesegamemodelsisnowMeanField Games. While slow to pick up momentum, the subject has seen a renewed wave of interest over the last seven years. The mean field game paradigm has evolved fromitsseminalprinciplesintoafull-fledgedfieldattractingtheoreticallyinclined investigators as well as applied mathematicians, engineers, and social scientists. The number of lectures, workshops, conferences, and publications devoted to the subjecthasgrownexponentially,andwethoughtitwastimetoprovidetheapplied mathematics community interested in the subject with a textbook presenting the stateoftheart,asweseeit.Becauseofourpersonaltaste,wechosetofocusonwhat v vi Foreword we like to call the probabilistic approach to mean field games. While a significant portionofthetextisbasedonoriginalresearchbytheauthors,greatcarewastaken toincludemodelsandresultscontributedbyothers,whetherornottheywereaware ofthefacttheywereworkingwithmeanfieldgames.Sothebookshouldfeeland readlikeatextbook,notaresearchmonograph. Mostof thematerial and examples found in thetext appear for the firsttimein book form. In fact, a good part of the presentation is original, and the lion’s share oftheargumentsusedinthetexthavebeendesignedespeciallyforthepurposeof thebook.Ourconcernforpedagogyjustifies(oratleastexplains)whywechoseto dividethematerialintwovolumesandpresentmeanfieldgameswithoutacommon noisefirst.Weeasetheintroductionofthetechnicalitiesneededtotreatmodelswith a common noise in a crescendo of sophistication in the complexity of the models. Also,weincludedattheendofeachvolumefourextensiveindexes(authorindex, notationindex,subjectindex,andassumptionindex)tomakenavigationthroughout thebookseamless. Acknowledgments First and foremost, we want to thank our wives Debbie and Mélanie for their understandingandunwaveringsupport.Theintensityoftheresearchcollaboration whichledtothistwo-volumebookincreaseddramaticallyovertheyears,invading our academic lives as well as our social lives, pushing us to the brink of sanity at times.Weshallneverbeabletothankthemenoughfortheirpatienceandtolerance. This book project would not have been possible without them: our gratitude is limitless. Next we would like to thank Pierre-Louis Lions, Jean-Michel Lasry, Peter Caines, Minyi Huang, and Roland Malhamé for their incredible insight in intro- ducing the concept of mean field games. Working independently on both sides of the pond, their original contributions broke the grounds for an entirely new and fertile field of research. Next in line is Pierre Cardaliaguet, not only for numerous private conversations on game theory but also for the invaluable service provided by the notes he wrote from Pierre-Louis Lions’ lectures at the Collège de France. Althoughtheywereneverpublishedinprintedform,thesenoteshadatremendous impactonthemathematicalcommunitytryingtolearnaboutthesubject,especially atatimewhenwritingsonmeanfieldgameswerefewandfarbetween. Wealsoexpressourgratitudetotheorganizersofthe2013and2015conferences onmeanfieldgamesinPadovaandParis:YvesAchdou,PierreCardaliaguet,Italo Capuzzo-Dolcetta,PaoloDaiPra,andJean-MichelLasry. While we like to cast ourselves as proponents of the probabilistic approach to mean field games, it is fair to say that we were far from being the only ones followingthispath.Infact,someofourpaperswerepostedessentiallyatthesame timeaspapersofBensoussan,Frehse,andYam,addressingsimilarquestions,with thesametypeofmethods.Webenefittedgreatlyfromthisstimulatingandhealthy competition. Foreword vii Wealsothankourcoauthors,especiallyJean-FrançoisChasagneux,DanCrisan, Jean-Pierre Fouque, Daniel Lacker, Peiqi Wang, and Geoffrey Zhu. We used our jointworksasthebasisforpartsofthetextwhichtheywillrecognizeeasily. Also, we would like to express our gratitude to the many colleagues and studentswhogracefullytoleratedourrelentlesspromotionofthisemergingfieldof researchthroughcourses,seminar,andlectureseries.Inparticular,wewouldliketo thankJean-FrançoisChasagneux,RamaCont,DanCrisan,RomualdElie,Josselin Garnier,MarcelNutz,HuyenPham,andNizarTouziforgivingustheopportunity todojustthat. Finally, we would like to thank Paul-Eric Chaudru de Raynal, Mark Cerenzia, Christy Graves, Dan Lacker, Victor Marx, and Peiqi Wang for their help in proofreadinganearlierversionofthemanuscript. The work of R. Carmona was partially supported by NSF DMS-1211928 and DMS-1716673, and by ARO W911NF-17-1-0578. The work of F. Delarue was partially supported by the Institut Universitaire de France and by the Agence NationaledelaRecherche(projectMFG,ANR-16-CE40-0015-01). Princeton,NJ,USA RenéCarmona Nice,France FrançoisDelarue July29,2016 Preface of Volume II While the first volume of the book only addressed mean field games where the sourcesofrandomshockswereidiosyncratictotheindividualplayers,thissecond volumetacklestheanalysisofmeanfieldgamesinwhichtheplayersaresubjectto a common source of random shocks. We call these models games with a common noise.Generalsolvabilityresults,aswellasuniquenessconditions,areestablished in Part I. Part II is devoted to the study of the so-called master equation. In the finalchapterofthispart,weconnectthevariousasymptoticmodelsinvestigatedin thebook,startingfromgameswithfinitelymanyplayers.Evenformodelswithouta commonnoise,thistypeofanalysisheavilyreliesonthetoolsdevelopedthroughout thissecondvolume.LikeinthecaseofVolumeI,VolumeIIendswithanepilogue. There, we discuss several extensions which fit naturally the framework of models withacommonnoise.Theyincludegameswithmajorandminorplayers,andgames oftiming. Thereaderwillfindonpagexiiibelowadiagramsummarizingtheinterconnec- tionsbetweenthedifferentchaptersandpartsofthebook. Chapter 1 is the cornerstone of the second volume. It must be seen as a preparationfortheanalysisofmeanfieldgameswithacommonnoise.Asequilibria become random in the presence of a common noise, we need to revisit the tools introducedanddevelopedinthefirstvolumeforthesolutionofoptimalstochastic control problems and establish a similar technology for stochastic dynamics and cost depending on an additional source of randomness. To that effect, we provide a general introduction to forward-backward systems in random environment and possiblynon-Brownianfiltrations.Akeypointinouranalysisistoallowtherandom environment not to be adapted to the Brownian motions driving the controlled dynamics. This forces us to impose compatibility conditions constraining the correlation between the noise carrying the random environment and the general filtration to which the solution of the forward-backward system is required to be adapted.Althoughthesecompatibilityconditionsareratherdifficulttohandle,they turnouttobeabsolutelycrucialastheyplayafundamentalrolethroughoutthetext, inparticularforthenotionofweakequilibriadefinedinthesubsequentChapter2. Asfarasweknow,thismaterialdoesnotexistinbookform. ix x PrefaceofVolumeII Thenotionofsolutionformeanfieldgameswithacommonnoiseendsupbeing more subtle than one could expect. In contrast with the case addressed in Volume I, the fixed point problem underpinning the definition of an equilibrium cannot be tackled by means of Schauder’s fixed point theorem. In order to account for the dependence of the equilibria upon the realization of the common noise, it is indeed necessary to enlarge the space in which the fixed point has to be sought. Unfortunately, proceeding in this way increases dramatically the complexity of the problem, as the new space of solutions becomes so big that it becomes very difficult to identify tractable compactness criteria to use with Schauder’s theorem. ThepurposesofChapters2and3ispreciselytoovercomethisissue.Thefirststep of our strategy is to discretize the realization of the common noise entering the definition of an equilibrium in such a way that the equilibrium can take at most a finite number of outcomes as the realization of the common noise varies. This makesthesizeofthespaceofsolutionsmuchmorereasonable.Thesecondstepis topasstothelimitalongdiscretizedsolutions.Thesuccessofthisapproachcomes atthepriceofweakeningthenotionofequilibriuminthesensethattheequilibrium is only adapted to a filtration which is larger than the filtration generated by the commonnoise.ThisnotionofweaksolutionsistherationalefortheCompatibility ConditionintroducedinChapter1.Theconceptsofweakandstrongsolutionsare explained in Chapter 2. In analogy with strong solutions of stochastic differential equations,strongequilibriaarerequiredtobeadaptedtothefiltrationgeneratedby thecommonnoise.Naturally,weestablishaformofYamada-Watanabetheoryfor Nash equilibria. Quite expectedly, it says that weak solutions are strong provided that uniqueness holds in the strong sense. In analogy with the theory developed in the first volume, we prove that strong and weak equilibria may be represented by means of forward-backward systems of the conditional McKean-Vlasov type. Duetothepresenceofthecommonnoise,conditioningappearsintheformulation. To wit, we first develop, in the first section of the chapter, a theory of conditional propagation of chaos for particle systems with mean field interaction driven by a commonnoise. The construction of weak solutions is addressed in Chapter 3. There, we implement the aforementioned discretization approach. It consists in forcing the total number of realizations of an equilibrium to be finite. The passage to the limit along discretized solutions is achieved in the weak sense: we consider the asymptoticbehaviorofthejointlawoftheequilibriaandtheforwardandbackward processes characterizing the best response of the representative agent under the discretizedenvironment.Whiletightnessoftheforwardcomponentisinvestigated for the standard uniform topology on the space of continuous paths, tightness of thebackwardcomponentisestablishedfortheMeyer-Zhengtopologyonthespace ofright-continuouspathswithleftlimits.TheresultsoftheMeyer-Zhengtopology needed for this purpose are recalled in the first section of Chapter 3. In the last sections,weaddressthequestionofuniqueness.Likeformeanfieldgameswithout acommonnoise,meanfieldgameswithacommonnoiseareshowntohaveatmost