Table Of ContentProbability Theory and Stochastic Modelling 85
Xiaoying Han
Peter E. Kloeden
Random Ordinary
Differential
Equations and
Their Numerical
Solution
Probability Theory and Stochastic Modelling
Volume 85
Editors-in-chief
Søren Asmussen, Aarhus, Denmark
Peter W. Glynn, Stanford, CA, USA
Yves Le Jan, Orsay, France
Advisory Board
Martin Hairer, Coventry, UK
Peter Jagers, Gothenburg, Sweden
Ioannis Karatzas, New York, NY, USA
Frank P. Kelly, Cambridge, UK
Andreas Kyprianou, Bath, UK
Bernt Øksendal, Oslo, Norway
George Papanicolaou, Stanford, CA, USA
Etienne Pardoux, Marseille, France
Edwin Perkins, Vancouver, BC, Canada
Halil Mete Soner, Zürich, Switzerland
The Probability Theory and Stochastic Modelling series is a merger and
continuation of Springer’s two well established series Stochastic Modelling and
Applied Probability and Probability and Its Applications series. It publishes
researchmonographsthatmakeasignificantcontributiontoprobabilitytheoryoran
applications domain in which advanced probability methods are fundamental.
Books in this series are expected to follow rigorous mathematical standards, while
alsodisplayingtheexpositoryqualitynecessarytomakethemusefulandaccessible
toadvancedstudentsaswellasresearchers.Theseriescoversallaspectsofmodern
probability theory including
(cid:129) Gaussian processes
(cid:129) Markov processes
(cid:129) Random fields, point processes and random sets
(cid:129) Random matrices
(cid:129) Statistical mechanics and random media
(cid:129) Stochastic analysis
as well as applications that include (but are not restricted to):
(cid:129) Branching processes and other models of population growth
(cid:129) Communications and processing networks
(cid:129) Computational methods in probability and stochastic processes, including
simulation
(cid:129) Genetics and other stochastic models in biology and the life sciences
(cid:129) Information theory, signal processing, and image synthesis
(cid:129) Mathematical economics and finance
(cid:129) Statistical methods (e.g. empirical processes, MCMC)
(cid:129) Statistics for stochastic processes
(cid:129) Stochastic control
(cid:129) Stochastic models in operations research and stochastic optimization
(cid:129) Stochastic models in the physical sciences
More information about this series at http://www.springer.com/series/13205
Xiaoying Han Peter E. Kloeden
(cid:129)
Random Ordinary
Differential Equations
and Their Numerical Solution
123
XiaoyingHan PeterE. Kloeden
Auburn,AL Schoolof Mathematics andStatistics
USA HuazhongUniversity of Science
andTechnology
Wuhan, Hubei
China
ISSN 2199-3130 ISSN 2199-3149 (electronic)
Probability Theoryand Stochastic Modelling
ISBN978-981-10-6264-3 ISBN978-981-10-6265-0 (eBook)
DOI 10.1007/978-981-10-6265-0
LibraryofCongressControlNumber:2017951158
Mathematics Subject Classification (2010): 37H10, 60H10, 60H35, 34F05, 37H70, 60H30, 65LC30,
65L05,65L06,65L20,92-08
©SpringerNatureSingaporePteLtd.2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore
For my parents
—Xiaoying Han
Für Karin
—Peter E. Kloeden
Preface
Randomordinarydifferentialequations(RODEs)areordinarydifferentialequations
(ODEs) that include a stochastic process in their vector field. They seem to have
had a shadow existence to Itô stochastic differential equations (SODEs), but have
been around for as long as if not longer and have many important applications. In
particular, RODEs play a fundamental role in the theory of random dynamical
systems.
In the older engineering and physics literature, a simpler kind of RODE is
investigatedwiththevectorfieldbeingchosenrandomlyratherthandependingona
stochasticprocess.SuchRODEsarestillofinterestintheuncertaintyquantification
community.Theyareaspecialcaseofthoseconsideredinthisbook,butwillnotbe
treated separately.
RODEs, unlike SODEs, can be analysed pathwise with deterministic calculus,
but require further treatment beyond that of classical ODE theory. Specifically,
since the driving process in a RODE has at most Hölder continuous sample paths,
the solution sample paths are continuously differentiable, but the sample paths
of the derivative are no more than Hölder continuous in time. The resulting vector
fieldafterinsertionofthedrivingstochasticprocessisatmostHöldercontinuousin
time, no matter how smooth the vector field is in its original variables. Thus, the
solutionsofRODEsdonothavesufficientsmoothnesstohaveTaylorexpansionsin
the usual sense.
Taylorexpansionsareaverybasictoolinnumericalanalysis.Theyallowoneto
derive one-step numerical schemes for ODEs of arbitrary high order. In practice,
such Taylor schemes are rarely implemented, but are used instead as a theoretical
comparison for determining the convergence orders of other schemes that have
been derived by more heuristic methods. This theory does not apply directly to
RODEs or SODEs. Although classical numerical schemes for ODEs can be used
pathwise for RODEs, they rarely attain their traditional order.
The situation with RODEs issimilarto that for Itô SODEs, the sample paths of
which are just Hölder continuous and not even differentiable. For Itô SODEs,
stochasticTaylorexpansionscanbederivedusinganiteratedapplicationoftheItô
formula, the stochastic chain rule, in integral form. These stochastic Taylor
vii
viii Preface
expansions were the starting point for the derivation of consistent higher order
numerical schemes for SODEs. A similar approach will be used in this book for
RODEs.
Amajor motivation for writingthis bookistomake more widely knownrecent
resultsonthederivationofhigherordernumericalschemesforRODEs.Anotheris
tomakeRODEsthemselvesandthecloselyassociatedtheoryofrandomdynamical
systems better known too. A particular motivation of personal interest to us is to
show how RODEs are being used in the biological sciences, where non-Gaussian
and bounded noise are often more realistic than the conventionally used Itô
calculus.
Auburn, USA Xiaoying Han
Wuhan, China Peter E. Kloeden
March 2017
Acknowledgements
Both authors were substantially supported by the Thousand Talent Program for
Foreign Experts of the Chinese Central Government and the Huazhong University
of Science and Technology.
We thank Pamela Burrage, Denise Chang, Jiayin Jin, Thomas Lorenz, Andreas
Neuenkirch, Yuanling Niu, Andreas Rößler, Florian Rupp, Peng Wang, Xiaojie
Wang and Fuke Wu for carefully reading parts of the manuscript and for their
usefulfeedback.Inaddition,wethankFritzColonius,LarsGrüne,ThomasLorenz
and Thomas Wanner for their assistance with the literature. The valuable insights
andsuggestionsoftheanonymousreviewersofanearlymanuscripthavebeenvery
helpful to us and are deeply appreciated.
We are particularly grateful to Yusuke Asai, who was working towards his
doctorateonthenumericalanalysisofRODEsattheGoetheUniversityinFrankfurt
am Main, as we were writing this book. He read various earlier versions of the
manuscriptwithgreatcareandprovideduswithmanyofthefiguresandsimulation
results as well as very useful comments.
ix
Guide to Reading This Book
Wesuggestthatallreaders firstreadChap.1whereweintroducerandomordinary
differential equations (RODEs) and briefly indicate various numerical methods for
them that have appeared in the literature in the past.
Readersinterestedintheirapplications,inparticularinabiologicalcontext,can
obtainanoverviewbyreadingthefirstpagesofeachchapterinPartIVatanytime.
The later parts of these chapters apply the numerical methods and various results
from the theory of random dynamical systems that are developed in the book.
Those readers who are mainly interested in numerical methods could skip
Chap. 4 on random dynamical systems and Chap. 5 on numerical dynamics or
return to them later when the material in them is needed, e.g. as in some of the
applications in Part IV.
ThenumericalschemesforRODEsinthisbookarederivedsystematicallyusing
various kinds of generalised Taylor expansions in integral form, depending on the
structureoftheRODEsorthenatureofthenoiseinthem.Theseareintroducedand
investigated inPart II.This isessential reading for those who plan tocontribute to
further mathematical developments of the subject. Numerical schemes based on
them are presented in Part III, with the notation and Taylor approximations from
Part II briefly restated for the readers’ convenience.
Itisnotessential toreadthechaptersinthesepartsinalinearfashion.Areader
whoisinterestedonlyinaparticulartypeofRODEordrivingnoisecouldjustread
thecorrespondingchaptersinthePartsIIandIII.Nevertheless,werecommendthat
readers skim first over all chapters, here and elsewhere, without focusing on
technicaldetails,toobtainanoverviewofthecontentsandtoseehowthedifferent
aspects are interrelated.
xi
