Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation

264 Pages·2013·2.07 MB·English

by  Carl Graham

Stochastic Modelling
StochasticMechanics
RandomMedia and Applied Probability
SignalProcessing
andImageSynthesis (Formerly:
MathematicalEconomicsandFinance ApplicationsofMathematics)
StochasticOptimization
StochasticControl 68
StochasticModelsinLifeSciences
Editedby P.W. Glynn
Y.LeJan
AdvisoryBoard M. Hairer
I.Karatzas
F.P. Kelly
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Forfurthervolumes:
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Carl Graham (cid:2) Denis Talay
Stochastic
Simulation and
Monte Carlo
Methods
Mathematical Foundations
of Stochastic Simulation
CarlGraham DenisTalay
CentredeMathématiquesAppliquées INRIA
ÉcolePolytechnique,CNRS SophiaAntipolis,France
Palaiseau,France
ISSN0172-4568 StochasticModellingandAppliedProbability
ISBN978-3-642-39362-4 ISBN978-3-642-39363-1(eBook)
DOI10.1007/978-3-642-39363-1
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013945076
MathematicsSubjectClassification: 60H10,65U05,65C05,60J30,60E7,65R20
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Preface
Theextraordinaryincreaseofcomputercapacitynowencouragesphysicists,biolo-
gists, economists, and engineers to model and simulate numerically tremendously
complex phenomena, in order to answer scientific questions, industrial needs, and
societalrequirementsforriskevaluationandcontrol.Therefore,expertsinvarious
fields aim to solve incredibly complex high dimensional systems of partial differ-
ential equations (PDE), couplings of local stochastic dynamics and deterministic
macroscopicequations,etc.
Stochasticapproachesappeartobeuseful,andsometimesmandatory,inthefol-
lowingtwocontexts.First,onecannotexpectthatverycomplexphenomenaleadto
perfectlycalibratedmathematicalmodels,oreventoperfectmathematicalmodels,
so that uncertainties or stochastic components are involved in the equations. Sec-
ond, stochastic numerical methods allow one to solve deterministic problems, of
whichthehighdimensionorsingularitiesrenderclassicaldeterministicmethodsof
resolution intractable or inaccurate, provided that the solutions can be represented
intermsofprobabilitydistributionsofrandomvariablesorstochasticprocesses.
ThecombinationofstochasticanalysisandPDEtheoryarenecessaryto:
• obtainstochasticrepresentationsofsolutionsofdeterministicPDE,
• constructeffectivestochasticnumericalmethods,
• obtainpreciseerrorestimatesintermsofthenumericalparametersofthesemeth-
ods,undertheconstraintthattheytakeintoaccountthecriticalsituationswhere
thestochasticnumericalmethodsareusedandtheobjectivesoftheirusers.
This monograph aims to introduce the reader to these difficult issues in a self-
contained way. We particularly emphasize the essential role played by martingale
theoryinallthetheoreticalandnumericalaspectsoftheseissues.Wealsohavede-
votedasubstantialpartofthebooktotheconstructionofsimulationalgorithms:the
readers who are mainly interested in numerical issues may skip the mathematical
proofsandconcentrateonthealgorithmsandtheconvergencerateestimates.
Compared to other textbooks, this monograph presents the specificity of devel-
oping the mathematical tools which are necessary to construct effective stochastic
v
vi Preface
simulationmethodsandobtainaccurateerrorestimates,andofreunitinginanon-
classicalwayseveraladvancedtheoreticaltopics,someofwhichwenowlist.
We focus on non-asymptotic error estimates for Monte Carlo methods, on the
backwardmartingaletechniquetoprovetheStrongLawofLargeNumbers,andon
elementarynotionsonlogarithmicSobolevinequalitiestoprovebasicconcentration
inequalities.Weplacegreatemphasisonthepathwiseconstructionandsimulation
ofPoissonprocesses,discretespaceMarkovprocesses,andsolutionsofItôstochas-
ticdifferentialequations.
We use the notions of infinitesimal generators and stochastic flows to estab-
lishstochasticrepresentationsofparabolicpartialdifferentialequationsorintegro-
differentialequations.Weintensivelyusethesestochasticrepresentationsofevolu-
tion equations to prove optimal error estimates for stochastic simulation methods.
AsexplainedinChap.1,stochasticnumericalmethodsareusedtocomputequan-
tities expressedin terms of the probabilitydistributionof stochasticprocesses; we
therefore essentially consider numerical errors in the weak sense rather than in an
Lp-normorinapathwisesensewhichonlyprovidecrudeinformationontheaccu-
racyofpracticalsimulations.
Wepresentvariancereductiontechniquesinnontrivialsituations,whichleadsus
touseoptimizedGirsanovtransformationsandtointroducethereadertostochastic
optimizationprocedures.
Forfurtherinformationonthecontentsofthefirsttwochapters,thereaderisad-
visedtoconsultthehugeliteraturewhichconcernsCentralLimitTheorems,Edge-
worthexpansions,LargeDeviationsPrinciples,concentrationinequalities,andsim-
ulationalgorithmsforfinite-dimensionalrandomvariables.See,e.g.,Devroye[10],
Feller[15,16],Petrov[43],Shiryayev[44],andreferencestherein.
Thefirstbookonthediscretizationofstochasticdifferentialequationsisdueto
Milstein [37]. Several other books have been published on this topic with a rather
different point of view to ours: most of them focus on particular applications or
variousdiscretizationmethodswhereas,asalreadyemphasized,weconcentrateon
the mathematical methodologies which allow one to get sharp convergence rates.
Therefore we encourage the reader to consult the selected references below, ref-
erences therein, and other useful references, to get further algorithmic or applied
informationonstochasticsimulations.
Fortimedependentmodels,MonteCarlomethodsarederivedfromthesimula-
tionofMarkovprocesses,possiblydiscretized.Inthiscontext,e.g.,Asmussenand
Glynn [4] treat the mathematics of queueing theory and some related areas, with
an emphasis on stationary regimes. Glasserman [20] focuses on numerical meth-
ods for financial models. Kloeden and Platen [28] present an extended catalog of
variantsoftheEulerandMilsteindiscretizationschemesforstochasticdifferential
equations. Lapeyre, Pardoux and Sentis [32] present an overview of applications
of Monte Carlo simulations of stochastic processes. Milstein and Tretyakov [38]
notably study discretization methods for stochastic differential systems with sym-
plectic structure, Hamiltonian systems, and small noise systems, layer simulation
methodsandrandomwalksimulationsforstochasticsystemswithboundarycondi-
tions.TheCIMEvolume[21]developedtheweakconvergenceofdiscretizedpro-
Preface vii
cesses,withastrongemphasisonstochasticinteractingparticlesystemsrelatedto
non-linearpartialdifferentialequations.
WeherehavenottackledsuchimportanttopicsasMalliavincalculustechniques
togetoptimalconvergenceratesanddevelopvariancereductionmethods,longtime
simulationsofergodicMarkovprocessesandthestochasticapproximationoftheir
invariantprobabilitydistributions,approximationmethodsforreflectedandstopped
diffusionprocesses,simulationofLévyprocessesanddiscretizationofLévydriven
stochastic differential equations, quantization simulation techniques, or exact sim-
ulationmethods.Thesesubjectsneedmathematicaltoolswhicharebeyondtheob-
jectivesofthisfirstvolume,andwewilladdresstheminaforthcomingvolume.
OurothervolumeswillconcernthestochasticsimulationmethodsforPartialDif-
ferentialEquationswithnon-smoothcoefficientsandthestochasticparticlemethods
fortheanalysisandthenumericalresolutionofnon-linearPartialDifferentialEqua-
tions.
We hope that Master or Ph.D. students with notions on stochastic calculus and
researchers interested in the mathematical or numerical aspects of stochastic sim-
ulations will find this series of monographs useful, in order to be introduced into
some advanced topics in Probability theory, to improve the accuracy and the con-
fidence intervals of their simulations, or to acquire some fundamental knowledge
beforereadingadvancedresearchpapersonnumericalprobability.
Palaiseau,France CarlGraham
Sophia-Antipolis,France DenisTalay
February2012
Acknowledgements
We warmly thank Benjamin Jourdain for his very valuable comments on the first
draftofthisbook,andheandCarolineHillairetforhavingtaughtitscontentswith
usattheÉcolePolytechnique(Palaiseau,France).
ix
Contents
PartI PrinciplesofMonteCarloMethods
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 WhyUseProbabilisticModelsandSimulations? . . . . . . . . . . 3
1.1.1 WhatAretheReasonsforProbabilisticModels? . . . . . . 4
1.1.2 WhatAretheObjectivesofRandomSimulations? . . . . . 6
1.2 OrganizationoftheMonograph . . . . . . . . . . . . . . . . . . . 9
2 StrongLawofLargeNumbersandMonteCarloMethods . . . . . . 13
2.1 StrongLawofLargeNumbers,ExamplesofMonteCarloMethods 13
2.1.1 StrongLawofLargeNumbers,AlmostSureConvergence . 13
2.1.2 Buffon’sNeedle . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 NeutronTransportSimulations . . . . . . . . . . . . . . . 15
2.1.4 StochasticNumericalMethodsforPartialDifferential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 SimulationAlgorithmsforSimpleProbabilityDistributions . . . . 18
2.2.1 UniformDistributions . . . . . . . . . . . . . . . . . . . . 19
2.2.2 DiscreteDistributions . . . . . . . . . . . . . . . . . . . . 20
2.2.3 GaussianDistributions . . . . . . . . . . . . . . . . . . . . 21
2.2.4 CumulativeDistributionFunctionInversion,Exponential
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.5 RejectionMethod . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Discrete-TimeMartingales,ProofoftheSLLN . . . . . . . . . . . 25
2.3.1 RemindersonConditionalExpectation . . . . . . . . . . . 25
2.3.2 MartingalesandSub-martingales,BackwardMartingales . 27
2.3.3 ProofoftheStrongLawofLargeNumbers . . . . . . . . . 30
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Non-asymptoticErrorEstimatesforMonteCarloMethods . . . . . 37
3.1 ConvergenceinLawandCharacteristicFunctions . . . . . . . . . 37
3.2 CentralLimitTheorem . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 AsymptoticConfidenceIntervals . . . . . . . . . . . . . . 41
3.3 Berry–Esseen’sTheorem . . . . . . . . . . . . . . . . . . . . . . 42
xi