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Probability Theory and Stochastic Modelling 90 Umut Çetin Albina Danilova Dynamic Markov Bridges and Market Microstructure Theory and Applications Probability Theory and Stochastic Modelling Volume 90 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 Umut Çetin (cid:129) Albina Danilova Dynamic Markov Bridges and Market Microstructure Theory and Applications 123 UmutÇetin AlbinaDanilova DepartmentofStatistics DepartmentofMathematics LondonSchoolofEconomics LondonSchoolofEconomics London,UK London,UK ISSN2199-3130 ISSN2199-3149 (electronic) ProbabilityTheoryandStochasticModelling ISBN978-1-4939-8833-4 ISBN978-1-4939-8835-8 (eBook) https://doi.org/10.1007/978-1-4939-8835-8 LibraryofCongressControlNumber:2018953309 Mathematics Subject Classification (2010): 60J60, 91B44, 60H20 (primary), 60G35, 91G80, 60F05 (secondary) ©SpringerScience+BusinessMedia,LLC,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerScience+BusinessMedia,LLC, partofSpringerNature. Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A. Toourfamilies Emel,ChristianandAlice. Preface Duringthecourseofourresearchonequilibriummodelsofasymmetricinformation in market microstructure theory, we have realised that one needed to apply techniquesfromdifferentbranchesofstochasticanalysistotreatthesemodelswith mathematical rigour. However, these subfields of stochastic analysis—to the best of our knowledge—are not presented in a single volume. This book intends to addressthisissueandprovidesoneconciseaccountofallfundamentaltheorythat isnecessaryforstudyingsuchequilibriummodels. Equilibrium in these models can be viewed as an outcome of a game among asymmetricallyinformedagents.Thelessinformedagentsinthesegamesendeav- our to infer the information possessed by the agents with superior information. Thisobviouslynecessitatesagoodunderstandingofthestochasticfilteringtheory. On the other hand, the equilibrium strategy of an agent with superior information is to drive a commonly observed process to a given random variable without distorting the unconditional law of the process. Construction of such strategy turns out to be closely linked to the conditioning of Markov processes on their terminal value. Moreover, this construction needs to be admissible and adapted to the agent’s filtration, which brings us to the study of stochastic differential equations(SDEs)representingMarkovbridges.Therefore,anadequateknowledge ofstochasticfiltering,MarkovbridgesandSDEsisessentialforathoroughanalysis ofasymmetricinformationmodels. The aim of this book is to build this knowledge. Although there are many excellent texts covering various aspects of the aforementioned three fields, the standardassumptionsintheseliteratureareoftentoorestrictivetobeappliedinthe context of asymmetric information models. Driven by this need from applications weextendalotofresultsknownintheliterature.Therefore,thisbookcanalsobe viewedasacomplementarytexttothestandardliterature.Proofsofstatementsthat alreadyexistintheliteratureareoftenomittedandaprecisereferenceisgiven. This book assumes the reader has some knowledge of stochastic calculus and martingale theory in continuous time. Although familiarity with SDEs will make its reading more enjoyable, no prior knowledge on this subject is necessary. The vii viii Preface exposition is largely self-contained, which allows it to be used as a graduate textbookonequilibriummodelsofinsidertrading. Thematerialpresentedhereisdividedintotwoparts.PartIdevelopsthemath- ematicalfoundationsforSDEs,staticanddynamicMarkovbridges,andstochastic filtering. Equilibrium models of insider trading and their analysis constitute the contentsofPartII. In Chap.1 we present preliminaries of the theory of Markov process including thestrongMarkovpropertyandtherightcontinuityofthefiltrationsandintroduce Fellerprocesses.Naturallyinthischapterweselecttheresultsthatarenecessaryfor thedevelopmentofthetheoryofMarkovbridges. As proofs of the results presented in Chap.1 will remain unaltered under an assumption of path continuity, we have refrained from assuming path regularity in that chapter. However, we will confine ourselves to diffusion processes for the restofthebooksincethetheoryofSDErepresentationforgeneraljump-diffusion bridgesisyettobedeveloped. Chapter 2 is devoted to stochastic differential equations and their relation with thelocalmartingaleproblem.InparticularstandardresultsonthesolutionsofSDEs andcomparisonofone-dimensionalSDEshavebeenextendedtoaccommodatethe exploding nature of the coefficients that are inherent to the SDEs associated with bridges. Chapter 3 is an overview of stochastic filtering theory. Kushner–Stratonovich equation for the conditional density of the unobserved signal is introduced and uniqueness of its solution is proved using a suitable filtered martingale problem pioneeredbyKurtzandOcone[84]. UsingthetheorypresentedinChaps.1and2wedeveloptheSDErepresentation of Markov processes that are conditioned to have a prespecified distribution μ at a given time T in Chap.4. Two types of conditioning are considered: weak conditioningreferstothecasewhenμisabsolutelycontinuouswithrespecttothe original law at time T whereas strong conditioning corresponds to the case when μ is a Dirac mass. We also discuss the relationship between such bridges and the enlargementoffiltrationstheory. ThebridgesconstructedinChap.4arecalledstaticsincetheirfinalbridgepoint isgiveninadvance.Chapter5considersanextensionofthistheorywhenthefinal point is not known in advance but is revealed over time via the observation of a given process. To verify that the law of these dynamic bridges coincides with the lawoftheoriginalMarkovprocesswhenconsideredintheirownfiltration,weuse techniquesfromChap.3. Part II is concerned with the applications of the theory in Part I and starts with Chap.6,whichprovidesthedescriptionoftheKyle–Backmodelofinsidertrading astheunderlyingframeworkforthestudyofequilibriuminthechaptersthatfollow. Chapter6alsocontainsaproofinageneralsettingofthe‘folkresult’thatonecan limit insider’s trading strategies to absolutely continuous ones. Chapter 7 presents an equilibrium in this framework when the inside information is dynamic in the absenceofdefaultrisk.Italsoshowsthatequilibriumisnotuniqueinthisfamilyof models.Chapter8studiestheimpactofdefaultriskintheequilibriumoutcome. Preface ix The book grew out of a series of paper with our long-term collaborator and colleague Luciano Campi, who has also read large portions of the first draft and suggestedmanycorrectionsandimprovements forwhichwearegrateful.Wealso thank Christoph Czichowsky and Michail Zervos for their suggestion on various parts of the manuscript. This book was discussed at the Financial Mathematics ReadingGroupattheLSEandwearegratefultoitsparticipantsfortheirinput. London,UK UmutÇetin October2017 AlbinaDanilova Contents PartI Theory 1 MarkovProcesses............................................................ 3 1.1 MarkovProperty........................................................ 3 1.2 TransitionFunctions.................................................... 5 1.3 MeasuresInducedbyaMarkovProcess .............................. 8 1.4 FellerProcesses......................................................... 11 1.4.1 PotentialOperators............................................. 11 1.4.2 DefinitionandContinuityProperties.......................... 14 1.4.3 StrongMarkovPropertyandRightContinuityofFields..... 18 1.5 Notes.................................................................... 21 2 StochasticDifferentialEquationsandMartingaleProblems ........... 23 2.1 InfinitesimalGenerators................................................ 23 2.2 LocalMartingaleProblem ............................................. 27 2.3 StochasticDifferentialEquations...................................... 40 2.3.1 LocalMartingaleProblemConnection........................ 40 2.3.2 ExistenceandUniquenessofSolutions....................... 44 2.3.3 TheOne-DimensionalCase ................................... 57 2.4 Notes.................................................................... 61 3 StochasticFiltering .......................................................... 63 3.1 GeneralEquationsfortheFilteringofMarkovProcesses............ 63 3.2 Kushner–StratonovichEquation:ExistenceandUniqueness ........ 72 3.3 Notes.................................................................... 78 4 StaticMarkovBridgesandEnlargementofFiltrations ................. 81 4.1 StaticMarkovBridges ................................................. 81 4.1.1 WeakConditioning............................................. 84 4.1.2 StrongConditioning ........................................... 90 4.2 ConnectionwiththeInitialEnlargementofFiltrations............... 111 4.3 Notes.................................................................... 117 xi