Optional Processes ; Theory and Applications (Chapman and Hall/CRC Financial Mathematics Series)

393 Pages·2020·2.705 MB·English

by  Mohamed Abdelghani , Alejandro Melnikov

Optional Processes
CHAPMAN & HALL/CRC
Financial Mathematics Series
Aims and scope:
The field of financial mathematics forms an ever-expanding slice of the financial sector. This series aims to capture new
developments and summarize what is known over the whole spectrum of this field. It will include a broad range of textbooks,
reference works and handbooks that are meant to appeal to both academics and practitioners. The inclusion of numerical code and
concrete real-world examples is highly encouraged.
Series Editors
M.A.H. Dempster
Centre for Financial Research
Department of Pure Mathematics and Statistics
University of Cambridge
Dilip B. Madan
Robert H. Smith School of Business
University of Maryland
Rama Cont
Department of Mathematics
Imperial College
Interest Rate Modeling
Theory and Practice, 2nd Edition
Lixin Wu
Metamodeling for Variable Annuities
Guojun Gan and Emiliano A. Valdez
Modeling Fixed Income Securities and Interest Rate Options
Robert A. Jarrow
Financial Modelling in Commodity Markets
Viviana Fanelli
Introductory Mathematical Analysis for Quantitative Finance
Daniele Ritelli, Giulia Spaletta
Handbook of Financial Risk Management
Thierry Roncalli
Optional Processes: Theory and Applications
Mohamed Abdelghani, Alexander Melnikov
For more information about this series please visit: https://www.crcpress.com/Chapman-and-HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH
Optional Processes
Theory and Applications
Mohamed Abdelghani
Morgan Stanley, New York, USA
Alexander Melnikov
University of Alberta
Firsteditionpublished2020
byCRCPress
6000BrokenSoundParkwayNW,Suite300,BocaRaton,FL33487-2742
andbyCRCPress
2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN
(cid:13)c 2020Taylor&FrancisGroup,LLC
CRCPressisanimprintofTaylor&FrancisGroup,LLC
Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers
haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologizetocopy-
right holders if permission to publish in this form has not been obtained. If any copyright material has not been
acknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint.
ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,transmitted,or
utilizedinanyformbyanyelectronic,mechanical,orothermeans,nowknownorhereafterinvented,includingpho-
tocopying,microfilming,andrecording,orinanyinformationstorageorretrievalsystem,withoutwrittenpermission
fromthepublishers.
Forpermissiontophotocopyorusematerialelectronicallyfromthiswork,accesswww.copyright.comorcontactthe
Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that
[email protected]
Trademark Notice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyfor
identificationandexplanationwithoutintenttoinfringe.
ISBN:9781138337268(hbk)
ISBN:9780429442490(ebk)
TypesetinCMR
byNovaTechsetPrivateLimited,Bengaluru&Chennai,India
Contents
Preface ix
Introduction xi
1 Spaces, Laws and Limits 1
1.1 Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Measurable Spaces, Random Variables and Laws . . . . . . . . . . . . . . . . . . . 3
1.2.1 Measurable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Measurable functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Atoms and separable fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 The case of real-valued random variables. . . . . . . . . . . . . . . . . . . . 5
1.2.5 Monotone class theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.6 Probability and expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.6.1 Convergence of random variables . . . . . . . . . . . . . . . . . . . 9
1.2.6.2 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.6.3 Uniform integrability . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.6.4 Completion of probability spaces . . . . . . . . . . . . . . . . . . . 11
1.2.6.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.6.6 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Analytic Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Paving and analytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Separable sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Lusin and Souslin spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.3.1 Souslin-Lusin Theorem . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Capacities and Choquet’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4.1 Constructing capacities . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.5 Theorem of cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Stochastic Processes 29
2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1 Time law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 Canonical process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Processes on Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Adapted processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2 Progressive measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Paths Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Processes on dense sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
v
vi Contents
2.3.2 Upcrossings and downcrossings . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.3 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.3.1 Doob’s separability theorems . . . . . . . . . . . . . . . . . . . . . 45
2.3.4 Progressive processes of random sets . . . . . . . . . . . . . . . . . . . . . . 47
2.3.5 Almost equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.5.1 Pseudo-Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Random Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.4.1 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.2 Basic properties of stopping times . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.3 Stochastic intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4.4 Optional and predictable σ-fields . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4.5 Predictable stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.6 Classification of stopping times . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.7 Quasi-left-continuous filtrations . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4.8 Optional and predictable cross-sections . . . . . . . . . . . . . . . . . . . . 80
2.5 Optional and Predictable Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3 Martingales 89
3.1 Discrete Parameter Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.1.2 Right and left closed supermartingales . . . . . . . . . . . . . . . . . . . . . 91
3.1.3 Doob’s stopping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.3.1 Extension to unbounded stopping times . . . . . . . . . . . . . . . 94
3.1.4 Fundamental inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.4.1 Maximal lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1.4.2 Domination in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1.4.3 Martingales upcrossings and downcrossings . . . . . . . . . . . . . 100
3.1.5 Convergence and decomposition theorems . . . . . . . . . . . . . . . . . . . 102
3.1.5.1 Almost sure convergence of supermartingales . . . . . . . . . . . . 103
3.1.5.2 Uniform integrability and martingale convergence . . . . . . . . . 104
3.1.5.3 Riesz decompositions of supermartingales . . . . . . . . . . . . . . 105
3.1.5.4 Krickeberg decomposition of martingales . . . . . . . . . . . . . . 107
3.1.6 Some applications of convergence theorems . . . . . . . . . . . . . . . . . . 108
3.2 Continuous Parameter Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1 Supermartingales on countable sets . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1.1 Fundamental inequalities . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.1.2 Existence of right and left limits . . . . . . . . . . . . . . . . . . . 111
3.2.2 Right-continuous supermartingale . . . . . . . . . . . . . . . . . . . . . . . 114
3.2.3 Projections theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.4 Decomposition of supermartingales . . . . . . . . . . . . . . . . . . . . . . . 127
3.2.4.1 Functional analytic decomposition theorem . . . . . . . . . . . . . 127
3.2.4.2 Extension to non-positive functionals . . . . . . . . . . . . . . . . 131
3.2.4.3 Decomposition of positive supermartingale of class D . . . . . . . 132
3.2.4.4 The general case of Doob decomposition . . . . . . . . . . . . . . 136
Contents vii
4 Strong Supermartingales 139
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2 Projection Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.3 Special Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.4 Mertens Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.5 Snell Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5 Optional Martingales 159
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Increasing and Finite Variation Processes . . . . . . . . . . . . . . . . . . . . . . . 166
5.3.1 Integration with respect to increasing and finite variation processes . . . . . 167
5.3.2 Dual projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.4 Decomposition Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.4.1 Decomposition of elementary processes . . . . . . . . . . . . . . . . . . . . . 172
5.4.2 Decomposition of optional martingales . . . . . . . . . . . . . . . . . . . . . 177
5.5 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.5.1 Predictable and optional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.5.2 Kunita-Watanabe inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.6 Optional Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.6.1 Integral with respect to square integrable martingales . . . . . . . . . . . . 189
5.6.2 Integral with respect to martingales with integrable variation . . . . . . . . 192
5.6.3 Integration with respect to local optional martingales . . . . . . . . . . . . 194
6 Optional Supermartingales Decomposition 201
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Riesz Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3 Doob-Meyer-Galchuk Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.3.1 Decomposition of DL class . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7 Calculus of Optional Semimartingales 215
7.1 Integral with Respect to Optional Semimartingales . . . . . . . . . . . . . . . . . 215
7.2 Formula for Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.3 Stochastic Integrals of Random Measures . . . . . . . . . . . . . . . . . . . . . . . 223
7.4 Semimartingales and Their Characteristics . . . . . . . . . . . . . . . . . . . . . . 229
7.4.1 Canonical representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.4.2 Component representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.5 Uniform Doob-Meyer Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.5.1 Supporting lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8 Optional Stochastic Equations 243
8.1 Linear Equations, Exponentials and Logarithms . . . . . . . . . . . . . . . . . . . 243
8.1.1 Stochastic exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.1.2 Stochastic logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.1.3 Nonhomogeneous linear equation . . . . . . . . . . . . . . . . . . . . . . . . 249
8.1.4 Gronwall lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.2 Existence and Uniqueness of Solutions of Optional Stochastic Equations . . . . . . 252
viii Contents
8.2.1 Stochastic equation with monotonicity condition . . . . . . . . . . . . . . . 253
8.2.2 Existence and uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . 255
8.2.2.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.2.2.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
8.2.3 Remarks and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.3 Comparison of Solutions of Optional Stochastic Equations . . . . . . . . . . . . . 269
8.3.1 Comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.3.2 Remarks and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 Optional Financial Markets 281
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
9.2 Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
9.3 Martingale Deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.3.1 The case of stochastic exponentials . . . . . . . . . . . . . . . . . . . . . . . 284
9.3.2 The case of stochastic logarithms . . . . . . . . . . . . . . . . . . . . . . . . 288
9.4 Pricing and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.5 Absence of Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.6 Examples of Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.6.1 Ladlag jumps diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.6.1.1 Computing a local martingale deflator . . . . . . . . . . . . . . . . 298
9.6.1.2 Pricing of a European call option . . . . . . . . . . . . . . . . . . 299
9.6.1.3 Hedging of a European call option . . . . . . . . . . . . . . . . . . 302
9.6.2 Basket of stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
9.6.3 Defaultable bond and a stock . . . . . . . . . . . . . . . . . . . . . . . . . . 304
10 Defaultable Markets on Unusual Space 307
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.2 Optional Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
10.3 Defaultable Cash-Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
10.3.1 Portfolio with default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
10.4 Probability of Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.5 Valuation of Defaultable Cash-Flow and Examples . . . . . . . . . . . . . . . . . . 317
11 Filtering of Optional Semimartingales 325
11.1 The Filtering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
11.2 The Usual Case of Optimal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 326
11.2.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
11.2.2 Martingales’ integral representation . . . . . . . . . . . . . . . . . . . . . . 335
11.2.3 Filtering of cadlag semimartingales . . . . . . . . . . . . . . . . . . . . . . . 348
11.3 The Unusual case of Optimal Filtering . . . . . . . . . . . . . . . . . . . . . . . . 358
11.3.1 Filtering on unusual stochastic basis . . . . . . . . . . . . . . . . . . . . . . 358
11.3.2 Filtering on mixed stochastic basis . . . . . . . . . . . . . . . . . . . . . . . 360
11.4 Filtering in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Bibliography 367
Index 377
Preface
This book deals with the subject of stochastic processes on unusual probability spaces and their
applicationstomathematicalfinance,stochasticdifferentialequationsandfilteringtheory.Unusual
probability spaces are probability spaces where the information σ-algebras are neither right- nor
left,neithercontinuousnorcomplete.Thetermunusualbelongstothefamousexpertinthegeneral
theoryofstochasticprocesses,C.Dellacherie,whoinitiatedthestudyofunusualprobabilityspaces
about 50 years ago. These spaces allow for the existence of a richer class of stochastic processes,
such as optional martingales and semimartingales.
FurtherdevelopmentsatthatperiodweredonebyLepingle,Horowitz,LenglartandGaltchouk.
Inthesepublications,amodernversionofstochasticanalysiswascreatedunderthe“unusualcondi-
tions”.Intheusualtheory,right-continuousandleftlimits(r.c.l.l.)semimartingalesaremeasurable
with respect to an optional σ-algebra on the product of sample space and time, generated by all
right-continuous processes adapted to a right-continuous and complete filtration. However, in the
unusual case they are not, and it is necessary to assume that they are optional processes so as to
provide the existence of their regular modifications which admit finite right limits and left limits
(r.l.l.l.). The existence of such initial theory calls for its further development as well as for further
applications in today’s times with new research challenges.
Besides the pure theoretical motivation for studying optional processes, we should note that
optional processes can be used in different areas of applications. In particular, one can provide a
strong motivation for this research that comes from the modeling of electricity spot prices, the
complexity of which lies in the modeling of severe and frequent price spikes. In this case, optional
processes allow for a more accurate representation of the spot price dynamics compare to cadlag
processes.
Therefore, in this book we present a comprehensive treatment of the stochastic calculus of
optional processes on unusual probability spaces and its applications. We begin with a foundation
chapterontheanalyticbasisofoptionalprocesses,thenthereareseveralchaptersonthestochastic
calculusofmartingalesandsemimartingales.Wewillcovermanytopicssuchaslinearstochasticdif-
ferentialequationswithrespecttooptionalsemimartingales;solutionstothenonhomogeneouslinear
stochasticdifferentialequation;Gronwalllemma;existenceanduniquenessofsolutionsofstochastic
equationsofoptionalsemimartingalesundermonotonicityconditions;andtheoremsoncomparison
of solutions of stochastic equations of optional semimartingale under Yamada conditions.
Furthermore, a financial market model based on optional semimartingales is presented and
methods for finding local martingale deflators are also given. Arbitrage pricing and hedging of
contingent claims in these markets are treated. A new theory of defaultable markets on unusual
probability spaces is presented. Also, several examples of financial applications are given: a laglad
jump diffusion model and a portfolio of a defaultable bond and a stock.
The book also contains a version of the uniform Doob-Meyer decomposition of optional semi-
martingales.Itsfundamentalroleinmathematicalfinanceiswellestablished,duetoitsapplication
ix