Error calculus for finance and physics: The language of Dirichlet forms

244 Pages·2003·0.987 MB·English

by  Nicolas Bouleau

de Gruyter Expositions in Mathematics 37
Editors
O.H.Kegel, Albert-Ludwigs-Universität, Freiburg
V.P.Maslov, Academy of Sciences, Moscow
W.D.Neumann, Columbia University, New York
R.O.Wells, Jr., Rice University, Houston
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3 TheStefanProblem,A.M.Meirmanov
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5 TheRiemannZeta-Function,A.A.Karatsuba,S.M.Voronin
6 ContactGeometryandLinearDifferentialEquations,V.E.Nazaikinskii,V.E.Shatalov,
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22 AnIntroductiontoLorentzSurfaces,T.Weinstein
23 LecturesinRealGeometry,F.Broglia(Ed.)
24 EvolutionEquationsandLagrangianCoordinates,A.M.Meirmanov,V.V.Pukhnachov,
S.I.Shmarev
25 CharacterTheoryofFiniteGroups,B.Huppert
26 PositivityinLieTheory:OpenProblems,J.Hilgert,J.D.Lawson,K.-H.Neeb,E.B.Vinberg
(Eds.)
27 AlgebraintheStone-CˇechCompactification,N.Hindman,D.Strauss
28 HolomorphyandConvexityinLieTheory,K.-H.Neeb
29 Monoids,ActsandCategories,M.Kilp,U.Knauer,A.V.Mikhalev
30 RelativeHomologicalAlgebra,EdgarE.Enochs,OvertounM.G.Jenda
31 NonlinearWaveEquationsPerturbedbyViscousTerms,ViktorP.Maslov,PetrP.Mosolov
32 ConformalGeometryofDiscreteGroupsandManifolds,BorisN.Apanasov
33 CompositionsofQuadraticForms,DanielB.Shapiro
34 ExtensionofHolomorphicFunctions,MarekJarnicki,PeterPflug
35 LoopsinGroupTheoryandLieTheory,Pe´terT.Nagy,KarlStrambach
36 AutomaticSequences,FriedrichvonHaeseler
Error Calculus
for Finance and Physics:
The Language of
Dirichlet Forms
by
Nicolas Bouleau
≥
Walter de Gruyter · Berlin · New York
Author
Nicolas Bouleau
E´cole Nationale des Ponts et Chausse´es
6 avenue Blaise Pascal
77455 Marne-La-Valle´e cedex 2
France
e-mail: [e-mail]
Mathematics Subject Classification 2000:
65-02; 65Cxx, 91B28, 65Z05, 31C25, 60H07, 49Q12, 60J65, 31-02, 65G99, 60U20,
60H35, 47D07, 82B31, 37M25
Key words:
error, sensitivity, Dirichlet form, Malliavin calculus, bias, Monte Carlo, Wiener space,
Poisson space, finance, pricing, portfolio, hedging, oscillator.
EP Printedonacid-freepaperwhichfallswithintheguidelines
oftheANSItoensurepermanenceanddurability.
LibraryofCongress(cid:18)Cataloging-in-PublicationData
Bouleau,Nicolas.
Error calculus for finance and physics : the language of Dirichlet
forms/byNicolasBouleau.
p. cm(cid:18)(DeGruyterexpositionsinmathematics;37)
Includesbibliographicalreferencesandindex.
ISBN3-11-018036-7(alk.paper)
1.Erroranalysis(Mathematics) 2.Dirichletforms. 3.Random
variables. I.Title. II.Series.
QA275.B682003
511(cid:25).43(cid:18)dc22 2003062668
ISBN3-11-018036-7
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Preface
ToGustaveChoquet
Ourprimaryobjectivehereinisnottodeterminehowapproximatecalculationsintro-
duceerrorsintosituationswithaccuratehypotheses,butinsteadtostudyhowrigorous
calculationstransmiterrorsduetoinaccurateparametersorhypotheses. Unlikequan-
titiesrepresentedbyentirenumbers,thecontinuousquantitiesgeneratedfromphysics,
economicsorengineeringsciences,asrepresentedbyoneorseveralrealnumbers,are
compromisedbyerrors. Thechoiceofarelevantmathematicallanguageforspeaking
abouterrorsandtheirpropagationisanoldtopicandonethathasincitedalargevariety
of works. Without retracing the whole history of these investigations, we can draw
themainlinesofthepresentinquiry.
Thefirstapproachistorepresenttheerrorsasrandomvariables. Thissimpleidea
offers the great advantage of using only the language of probability theory, whose
power has now been proved in many fields. This approach allows considering error
biases and correlations and applying statistical tools to guess the laws followed by
errors. Yet this approach also presents some drawbacks. First, the description is
too rich, for the error on a scalar quantity needs to be described by knowledge of
a probability law, i.e. in the case of a density, knowledge of an arbitrary function
(andjointlawswiththeotherrandomquantitiesofthemodel). Bydefinitionhowever,
errorsarepoorlyknownandtheprobabilitymeasureofanerrorisveryseldomknown.
Moreover, in practical cases when using this method, engineers represent errors by
meansofGaussianrandomvariables,whichmeansdescribingthembyonlytheirbias
andvariance. Thiswayhastheunavoidabledisadvantageofbeingincompatiblewith
nonlinearcalculations. Secondly,thisapproachmakesthestudyoferrortransmission
extremely complex in practice since determining images of probability measures is
theoreticallyobvious,butpracticallydifficult.
Thesecondapproachistorepresenterrorsasinfinitelysmallquantities. Thisof
course does not prevent errors from being more or less significant and from being
comparedinsize. Theerrorsareactuallysmallbutnotinfinitelysmall;thisapproach
therefore is an approximate representation, yet does present the very significant ad-
vantage of enabling errors to be calculated thanks to differential calculus which is a
veryefficienttoolinboththefinitedimensionandinfinitedimensionwithderivatives
inthesenseofFréchetorGâteaux.
Ifweapplyclassicaldifferentialcalculus,i.e.formulaeofthetype
dF(x,y)=F(cid:16)(x,y)dx+F(cid:16)(x,y)dy
1 2
vi Preface
wehavelostalloftherandomcharacteroftheerrors;correlationoferrorsnolonger
has any meaning. Furthermore, by nonlinear mapping, the first-order differential
calculusapplies: typicallyifx = ϕ(s,t)andy = ψ(s,t),thendx = ϕ(cid:16) ds +ϕ(cid:16) dt
1 2
anddy =ψ(cid:16) ds+ψ(cid:16) dt,and
1 2
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
dF ϕ(s,t),ψ(s,t) = F(cid:16)ϕ(cid:16) +F(cid:16)ψ(cid:16) ds+ F(cid:16)ϕ(cid:16) +F(cid:16)ψ(cid:16) dt.
1 1 2 1 1 2 2 2
In the case of Brownian motion however and, more generally, of continuous semi-
martingales, Itô calculus displays a second-order differential calculus. Similarly, it
is indeed simple to see that error biases (see Chapter I, Section 1) involve second
derivativesintheirtransmissionbynonlinearfunctions.
The objective of this book is to display that errors may be thought of as germs
ofItôprocesses. Wepropose,forthispurpose,introducingthelanguageofDirichlet
formsforitstremendousmathematicaladvantages,aswillbeexplainedinthisbook.
In particular, this language allows error calculus for infinite dimensional models, as
mostoftenappearinphysicsorinstochasticanalysis.
Deterministicsensitivity
Deterministic analysis: Interval
approaches derivationwithrespectto calculus
theparametersofthemodel
Errorcalculususing
Probabilistic Dirichletforms Probability
approaches firstordercalculus second order cal- theory
only dealing with culus with vari-
variances ancesandbiases
Infinitesimalerrors Finiteerrors
The approach we adopt herein is therefore intermediate: the errors are infinitely
small,buttheircalculusdoesnotobeyclassicaldifferentialcalculusandinvolvesthe
first and second derivatives. Although infinitely small, the errors have biases and
variances(andcovariances). ThisaspectwillbeintuitivelyexplainedinChapterI.
Theabovetabledisplaysthevariousapproachesforerrorcalculations. Itwillbe
commented on in ChapterV, Section 1.2. Among the advantages of Dirichlet forms
(whichactuallylimitItôprocessestosymmetricMarkovianprocesses)letusempha-
sizeheretheirclosedcharacter(cf.ChaptersIIandIII).Thisfeatureplaysasimilar
role in this theory to that of σ-additivity in probability theory. It yields a powerful
extensiontoolinanysituationwherethemathematicalobjectsthroughwhichwecom-
putetheerrorsareonlyknownaslimitofsimplerobjects(finite-dimensionalobjects).
Preface vii
ThistextstemsfromapostgraduatecoursetaughtattheParis6andParis1Uni-
versities and supposes as prerequisite a preliminary training in probability theory.
Textbookreferencesaregiveninthebibliographyattheendofeachchapter.
Acknowledgements. I express my gratitude to mathematicians, physicists and fi-
nance practitioners who have reacted to versions of the manuscript or to lectures
on error calculus by fruitful comments and discussions. Namely Francis Hirsch,
PaulMalliavin,GabrielMokobodzki,SüleymanÜstünel,DominiqueLépingle,Jean-
Michel Lasry,Arnaud Pecker, Guillaume Bernis, Monique Jeanblanc-Picqué, Denis
Talay,MoniquePontier,NicoleElKaroui,Jean-FrançoisDelmas,ChristopheChorro,
FrançoisChevoirandMichelBauer. Mystudentshavealsotobethankedfortheirsur-
prisereactionsandquestions. Imustconfessthatduringthelastyearsofelaboration
ofthetext,themostusefuldiscussionsoccurredfrompeople,colleaguesandstudents,
who had difficulties understanding the new language. This apparent paradox is due
to the fact that the matter of the book is emerging and did not yet reach a definitive
form. Forthesamereasonisthereaderaskedtoforgivetheremainingobscurities.
Paris,October2003 NicolasBouleau

Contents
Preface v
I Intuitiveintroductiontoerrorstructures 1
1 Errormagnitude 1
2 Descriptionofsmallerrorsbytheirbiasesandvariances 2
3 Intuitivenotionoferrorstructure 8
4 Howtoproceedwithanerrorcalculation 10
5 Application: PartialintegrationforaMarkovchain 12
Appendix. Historicalcomment: Thebenefitofrandomizingphysical
ornaturalquantities 14
BibliographyforChapterI 16
II Strongly-continuoussemigroupsandDirichletforms 17
1 Strongly-continuouscontractionsemigroupsonaBanachspace 17
2 TheOrnstein–UhlenbecksemigrouponRandtheassociated
Dirichletform 20
Appendix. DeterminationofDfortheOrnstein–Uhlenbecksemigroup 28
BibliographyforChapterII 31
III Errorstructures 32
1 Maindefinitionandinitialexamples 32
2 Performingcalculationsinerrorstructures 37
3 Lipschitzfunctionalcalculusandexistenceofdensities 41
4 Closabilityofpre-structuresandotherexamples 44
BibliographyforChapterIII 50
IV Imagesandproductsoferrorstructures 51
1 Images 51
2 Finiteproducts 56
3 Infiniteproducts 59
Appendix. Commentsonprojectivelimits 65
BibliographyforChapterIV 66
x Contents
V Sensitivityanalysisanderrorcalculus 67
1 Simpleexamplesandcomments 67
2 Thegradientandthesharp 78
3 Integrationbypartsformulae 81
4 SensitivityofthesolutionofanODEtoafunctionalcoefficient 82
5 Substructuresandprojections 88
BibliographyforChapterV 92
VI Errorstructuresonfundamentalspacesspace 93
1 ErrorstructuresontheMonteCarlospace 93
2 ErrorstructuresontheWienerspace 101
3 ErrorstructuresonthePoissonspace 122
BibliographyforChapterVI 135
VII Applicationtofinancialmodels 137
1 Instantaneouserrorstructureofafinancialasset 137
2 Fromaninstantaneouserrorstructuretoapricingmodel 143
3 ErrorcalculationsontheBlack–Scholesmodel 155
4 Errorcalculationsforadiffusionmodel 165
BibliographyforChapterVII 185
VIII Applicationsinthefieldofphysics 187
1 Drawinganellipse(exercise) 187
2 Repeatedsamples: Discussion 190
3 CalculationoflengthsusingtheCauchy–Favardmethod(exercise) 195
4 Temperatureequilibriumofahomogeneoussolid(exercise) 197
5 Nonlinearoscillatorsubjecttothermalinteraction:
TheGrüneisenparameter 201
6 Naturalerrorstructuresondynamicsystems 219
BibliographyforChapterVIII 229
Index 231