Taylor Approximations for Stochastic Partial Differential Equations

224 Pages·2011·1.39 MB·English

by  Arnulf Jentzen and Peter Kloeden

Taylor Approximations
for Stochastic Partial
Differential Equations
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Arnulf J EnTzEn
Princeton university
Princeton, new Jersey
PETEr E. Klo EDEn
Goethe university
f rankfurt am Main, Germany
Taylor Approximations
for Stochastic Partial
Differential Equations
Soci ETy for inDuSTri Al AnD APPli ED MAThEMATicS
Phil ADEl PhiA
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Library of Congress Cataloging-in-Publication Data
Jentzen, Arnulf.
Taylor approximations for stochastic partial differential equations / Arnulf Jentzen, Peter E.
Kloeden.
p. cm. -- (CBMS-NSF regional conference series in applied mathematics ; 83)
Includes bibliographical references and index.
ISBN 978-1-611972-00-9
1. Stochastic partial differential equations. 2. Approximation theory. I. Kloeden, Peter E.
II. Title.
QA274.25.J46 2011
515'.353--dc23
2011029546

is a registered trademark.
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Contents
Preface xi
ListofFigures xiii
1 Introduction 1
1.1 TaylorexpansionsforODEs . . . . . . . . . . . . . . . . . . . 1
1.1.1 TaylorschemesforODEs . . . . . . . . . . . . . 3
1.1.2 IntegralrepresentationofTaylorexpansions . . . 4
I RandomandStochasticOrdinaryDifferentialEquations 7
2 TaylorExpansionsandNumericalSchemesforRODEs 9
2.1 RODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 EquivalenceofRODEsandSODEs . . . . . . . . 10
2.1.2 SimplenumericalschemesforRODEs . . . . . . 12
2.2 Taylor-likeexpansionsforRODEs . . . . . . . . . . . . . . . 13
2.2.1 Multi-indexnotation . . . . . . . . . . . . . . . . 14
2.2.2 Taylorexpansionsofthevectorfield . . . . . . . 14
2.3 RODE–Taylorschemes . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Discretizationerror . . . . . . . . . . . . . . . . 17
2.3.2 ExamplesofRODE–Taylorschemes . . . . . . . 22
3 SODEs 25
3.1 ItôSODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Existence and uniqueness of strong solutions of
SODEs . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2 SimplenumericalschemesforSODEs . . . . . . 27
3.2 Itô–Taylorexpansions . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 IteratedapplicationoftheItôformula . . . . . . . 29
3.2.2 GeneralstochasticTaylorexpansions . . . . . . . 31
3.3 Itô–TaylornumericalschemesforSODEs . . . . . . . . . . . 32
vii
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viii Contents
3.4 Pathwiseconvergence . . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 NumericalschemesforRODEsappliedtoSODEs 36
3.5 Restrictivenessofthestandardassumptions. . . . . . . . . . . 37
3.5.1 CounterexamplesfortheEuler–Maruyamascheme 37
4 NumericalMethodsforSODEswithNonstandardAssumptions 43
4.1 SODEswithoutuniformlyboundedcoefficients . . . . . . . . 43
4.2 SODEsonrestrictedregions. . . . . . . . . . . . . . . . . . . 44
4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Anothertypeofweakconvergence . . . . . . . . . . . . . . . 47
II StochasticPartialDifferentialEquations 51
5 StochasticPartialDifferentialEquations 53
5.1 RandomandstochasticPDEs . . . . . . . . . . . . . . . . . . 53
5.1.1 MildsolutionsofSPDEs. . . . . . . . . . . . . . 54
5.2 Functionalanalyticalpreliminaries . . . . . . . . . . . . . . . 55
5.2.1 Hilbert–Schmidtandtrace-classoperators . . . . 55
5.2.2 Hilbertspacevaluedrandomvariables . . . . . . 56
5.2.3 Hilbertspacevaluedstochasticprocesses . . . . . 56
5.2.4 InfinitedimensionalWienerprocesses. . . . . . . 56
5.3 Settingandassumptions . . . . . . . . . . . . . . . . . . . . . 57
5.4 Existence,uniqueness,andregularityofsolutionsofSPDEs . . 58
5.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5.1 FinitedimensionalSODEs. . . . . . . . . . . . . 60
5.5.2 SecondorderSPDEs . . . . . . . . . . . . . . . . 60
5.5.3 FourthorderSPDEs . . . . . . . . . . . . . . . . 65
5.5.4 SPDEswithtime-dependentcoefficients . . . . . 67
6 NumericalMethodsforSPDEs 69
6.1 Anearlyresult . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Otherresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 TheexponentialEulerscheme . . . . . . . . . . . . . . . . . . 73
6.3.1 Convergence . . . . . . . . . . . . . . . . . . . . 74
6.3.2 Numericalresults . . . . . . . . . . . . . . . . . 75
6.3.3 Restrictivenessoftheassumptions . . . . . . . . 76
7 TaylorApproximationsforSPDEswithAdditiveNoise 79
7.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.1.1 Propertiesofthesolutions . . . . . . . . . . . . . 83
7.2 Autonomization . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.3.1 SemigroupgeneratedbytheLaplacian . . . . . . 91
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Contents ix
7.3.2 ThedriftasaNemytskiioperator . . . . . . . . . 92
7.3.3 Stochasticprocessasstochasticconvolution . . . 93
7.3.4 ConcreteexamplesofSPDEswithadditivenoise . 94
7.4 Taylorexpansions . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4.1 Integraloperators . . . . . . . . . . . . . . . . . 98
7.5 AbstractexamplesofTaylorexpansions . . . . . . . . . . . . 104
7.6 ExamplesofTaylorapproximations . . . . . . . . . . . . . . . 114
7.6.1 Space–timewhitenoise . . . . . . . . . . . . . . 114
7.6.2 TaylorapproximationsforanonlinearSPDE . . . 116
7.6.3 Smoothernoise . . . . . . . . . . . . . . . . . . 121
7.7 NumericalschemesfromTaylorexpansions . . . . . . . . . . 121
7.7.1 TheexponentialEulerscheme . . . . . . . . . . . 123
7.7.2 ARunge–KuttaschemeforSPDEs . . . . . . . . 124
8 TaylorApproximationsforSPDEswithMultiplicativeNoise 127
8.1 HeuristicderivationofTaylorexpansions . . . . . . . . . . . . 127
8.2 Settingandassumptions . . . . . . . . . . . . . . . . . . . . . 130
8.2.1 Stochasticheatequation . . . . . . . . . . . . . . 133
8.3 TaylorexpansionsforSPDEs . . . . . . . . . . . . . . . . . . 135
8.3.1 Integraloperators . . . . . . . . . . . . . . . . . 136
8.3.2 DerivationofsimpleTaylorexpansions . . . . . . 138
8.3.3 HigherorderTaylorexpansions . . . . . . . . . . 139
8.4 Stochastictreesandwoods . . . . . . . . . . . . . . . . . . . 140
8.4.1 Stochastictreesandwoods . . . . . . . . . . . . 140
8.4.2 Constructionofstochastictreesandwoods . . . . 142
8.4.3 Subtrees . . . . . . . . . . . . . . . . . . . . . . 145
8.4.4 Orderofstochastictreesandwoods . . . . . . . . 146
8.4.5 StochasticwoodsandTaylorexpansions . . . . . 147
8.5 ExamplesofTaylorapproximations . . . . . . . . . . . . . . . 149
8.5.1 AbstractexamplesofTaylorapproximations . . . 149
8.5.2 Applicationtothestochasticheatequation . . . . 151
8.5.3 FinitedimensionalSODEs. . . . . . . . . . . . . 152
8.6 NumericalschemesforSPDEs . . . . . . . . . . . . . . . . . 154
8.6.1 TheexponentialEulerscheme . . . . . . . . . . . 155
8.6.2 AninfinitedimensionalanalogueofMilstein’s
scheme . . . . . . . . . . . . . . . . . . . . . . . 156
8.6.3 Linear-implicitEulerandCrank–Nicolson
schemes . . . . . . . . . . . . . . . . . . . . . . 157
8.6.4 Globalandlocalconvergenceorders . . . . . . . 157
8.6.5 Numericalsimulations . . . . . . . . . . . . . . . 158
8.7 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.7.1 ProofofLemma8.3 . . . . . . . . . . . . . . . . 160
8.7.2 ProofofLemma8.5 . . . . . . . . . . . . . . . . 163
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x Contents
8.7.3 Someadditionallemmas . . . . . . . . . . . . . . 164
8.7.4 ProofofTheorem8.4 . . . . . . . . . . . . . . . 171
A RegularityEstimatesforSPDEs 173
A.1 Someusefulinequalities . . . . . . . . . . . . . . . . . . . . . 174
A.1.1 Minkowski’sintegralinequality . . . . . . . . . . 174
A.1.2 Burkholder–Davis–Gundy-typeinequalities . . . 175
A.2 Semigroupterm . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.3 Driftterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.4 Diffusionterm . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.5 Existenceanduniqueness . . . . . . . . . . . . . . . . . . . . 204
A.6 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Bibliography 209
Index 219
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Preface
The numerical approximation of stochastic partial differential equations
(SPDEs),specifically,stochasticevolutionequationsoftheparabolicorhyperbolic
type, encounters all of the difficulties that arise in the numerical solution of both
deterministicPDEsandfinitedimensionalstochasticordinarydifferentialequations
(SODEs)aswellasmanymoreduetotheinfinitedimensionalnatureofthedriving
noise processes. The state of development of numerical schemes for SPDEs com-
pareswiththatforSODEsintheearly1970s. Mostofthenumericalschemesthat
havebeenproposedtodatehavealoworderofconvergence,especiallyintermsof
anoverallcomputationaleffort,andonlyrecentlyhasitbeenshownhowtoconstruct
higherorderschemes.
ThebreakthroughforSODEsstartedwiththeMilsteinschemeandcontinued
with the systematic derivation of stochastic Taylor expansions and the numerical
schemesbasedonthem. ThesestochasticTaylorschemesarebasedonaniterated
applicationoftheItôformula. Thecrucialpointisthatthemultiplestochasticinte-
gralswhichtheycontainprovidemoreinformationaboutthenoiseprocesseswithin
discretization subintervals, and this allows an approximation of higher order to be
obtained. ThistheoryispresentedindetailinthemonographsKloeden&Platen[82]
andMilstein[95].
There is, however, no such Itô formula for the solutions of stochastic PDEs
inHilbertspacesorBanachspaces(seeChapter7formoredetails). Nevertheless,it
hasrecentlybeenshownthatTaylorexpansionsforthesolutionsofsuchequationscan
beconstructedbytakingadvantageofthemildformrepresentationofthesolutions.
Moreover,suchexpansionsarerobustwithrespecttothenoiseintheadditivenoise
case,i.e.,holdforothertypesofstochasticprocesseswithHöldercontinuouspaths
suchasfractionalBrownianmotion.
This book is based on recent work of the coauthors. Its style, contents, and
structure follow the series of lectures given by the second author, Peter Kloeden,
inAugust2010attheIllinoisInstituteofTechnologyinChicago. Themaindiffer-
encefromthelecturesistheexistenceanduniquenesstheoreminChapter5. Most
of that chapter and the entire appendix were written by the first coauthor, Arnulf
Jentzen.
xi
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