Table Of ContentProbability 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
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AndreasE.Kyprianou,Bath,UK
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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)
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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
