Stochastic Differential Equations: With Applications to Physics and Engineering

414 Pages·1991·47.7 MB·English

by  Kazimierz Sobczyk (auth.)

Stochastic DifferentialEquations
Mathematics and Its Applications (EastEuropean Series)
MaDagingEditor:
M.HAZEWINKEL
CentrelorMathematicsandComputerScience.Amsterdam.TheNetherlands
EditorialBoard:
A.BIALYNICKI-BIRULA,Institute01Mathematics.WarsawUniversity,Poland
H.KURKE,HumboldtUniversity,Berlin,OD.R.
J.KURZWER.,MathematicsInstitute.Academy01Sciences.Prague,Czechoslovakia
L.LEINDLER,Bo/yaiInstitute,Szeged,Hungary
L.LOVASZ,Bo/yaiInstitute,Szeged,Hungary
D.S.MITRINOVIC,UniversityofBelgrade.Yugoslavia
S.ROLEWIcz,PolishAcademyofSciences.Warsaw, Poland
BL.H.SENDOV,BulgarianAcademyolSciences,Sofia,Bulgaria
LT.TODOROV,BulgarianAcademy01Sciences.Sofia.Bulgaria
H.1RIEBEL,Universityoflena,OD.R.
Volume40
Stochastic
Differential Equations
With Applications to Physics and Engineering
by
Kazimierz Sobczyk
Institute ofF undamental Technological Research,
Polish Academy of Sciences,
Warsaw, Poland
~
"
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
lJbnry of Congras Calaloging-in-Publication Data
Sobczvk, Klzl.l.rz.
Stocha.tic diff.rential equations : with applications to
physics and engineering I by Klzl.lerz Sobczyk.
p. c •• -- (Mlth'tltles Ind Its Ippllcltlons. Elst European
aerl.s)
Inc I ud.s Index.
Blbllogrlphy: p.
ISBN 978-1-4020-0345-5 ISBN 978-94-011-3712-6 (eBook)
DOI 10.1007/978-94-011-3712-6
1. Stochastic dlffsrentlll equltlons. I. Title. II. Series:
Matheeltlcs Ind Its Ippllcltlons (Kluwer ACldeele Publishers). Elst
Europ.an .erl •••
QA274.2!.S55 1990
519.2--dc20 89-15287
PriIfIed on acld-free paper
AD Riabts Reserved
o
1991 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1991
Softcover reprint of the hardcover 1st edition 1991
No put of 1be ma1erial protected by this copyright notice may be reproduced or
1IIilized in any form or by any means, electronic or mechanical,
iDcludiDg pbotocOpyiDg, recording or by any infonnation storage and
retrieval system, without written permission from the copyright owner.
To
mywife Anna
andchildren
Joanna, Pawel, Jacek, Marcin
SERIESEDITOR'SPREFACE
'Etmoi, ..~silavaitsuCO.llUlJaltenrevc:nir, One acMcc matbcmatica bu JaIdcred the
jen'yseraispointaBe.' humanrac:c. ItbuputCOIDIDOD_ beet
JulesVerne -wb'a.citbdoup, 0Jl !be~IbcII_t
to!bedustycauialcrIabc&d'diMardod__
Theseriesisdivergent; thc:reforcwemaybe
abletodosomethingwithit. I!.ticT.Bc:I1
O.Hcavisidc
Mathematicsisa toolforthought. Ahighlynecessarytoolinaworldwhen:bothfeedbackandnon
linearitiesabound. Similarly. allkindsofpartsofmathematicsserveas toolsforotherpartsandfor
othersciences.
Applyinga simplerewritingrule to the quote on the right above one finds such statcmaltsas:
'One service topology has rendered mathematical physics ...•; 'One servicelogichasrendered c0m
puterscience...'; 'Oneservicecategorytheoryhasrenderedmathematics...'.Allarguably true. And
allstatementsobtainablethiswayformpartoftheraisond'etreofthisseries.
This series, Mathematics and Its Applications. started in 19n. Now that over one hundred
volumeshaveappeareditseemsopportunetoreexamineitsscope.AtthetimeI wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However. the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branc:hes. It
also happens, quite often in fact, that branches which were thought to be completely.
disparatearesuddenly seen to be related. Further. thekind and levelofsophistication
of mathematics applied in various sciences has changed drastically in recent yean:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometryinteractswithphysics; the Minkowskylemma:codingtheory and thestructure
of water meet one another in packing and covering theory; quantum dads, aystal
arc
defects and mathematical programming profit from homotopy theory; Lie algebras
relevanttofiltering; andpredictionandelectricalengineeringcanuseSteinspaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics'. 'CFD·. 'completelyintegrablesystems', 'chaos, synergeticsand large-scale
order', which are almost impossible to fit into the existingclassificationschemes. They
drawuponwidelydifferentsectionsofmathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more dort is
neededand soare books that canhelp mathematiciansand scientistsdo so. Accordingly MIA will
continuetotry tomakesuchbooksavailable.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interactionareas one should add stringtheory when: Riemann surfaces, algebraicgeometry. modu
lar functions. knots, quantum field theory. Kac-Moody algebras, monstrous moonshine(and more)
allcometogether. And to theexamplesofthingswhichcanbeusefullyappliedletmeaddthetopic
'finite geometry'; a combination of words which sounds like it might not eval exist, let alone be
applicable. Andyetitisbeingapplied: tostatisticsviadesigns, to radar/sonardetectionarrays(via
finiteprojectiveplanes). and to busconnectionsofVLSIchips(vis,diffaaK'lesets). 1'hcrcseemsto
be no partof(so-calledpure) mathematics thatis notinimmediatedangerofbeingapplied. And,
accordingly. the applied mathematician needs to be aware of much more. Besides analysis and
numerics. the traditionalworkhorses, he may need allkinds ofcombinatorics, algebra, probability.
andsoon.
In addition, the applied scientist needs to cope inaeasingly with the nonlinear world and the
vii
viii Series Editor'sPreface
extra mathematical sophistication that this requires. For that is where the rewards are. linear
modds are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linearworld thatinfinitesimalinputs may resultin macroscopicoutputs(orviceversa). Toappreci
atewhat I amhintingat: ifelectronicswerelinearwewouldhaveno fun with transistorsand com
puters;wewouldhaveno]V;infactyouwouldnotbereadingtheselines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration,p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no tdling where all this is leading
fortunately.
Thus theoriginal scopeof theseries, whichforvarious(sound) reasons now comprisesfive sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), stillapplies. It has beenenlargeda bitto includebooks treatingofthe toolsfrom onesubdis
ciplinewhichareusedinothers.Thus theseriesstillaimsatbooksdealingwith:
- a central concept which plays an important role in several different mathematical and/or
scientificspecializationareas;
- new applicationsoftheresultsandideasfrom oneareaofscientificendeavourintoanother;
- influenceswhich the results, problems and concepts ofone field of enquiry have, and have had,
on thedevelopmentofanother.
Many phenomena in nature have a stochastic component or, for a variety of reasons, can be well
modelledin terms ofequations with stochasticaspects. So the topic ofstochasticdifferentialequa
tions should bea flourishingonein mathematics, with many applications. And soitis, and, indeed
the applications range from oil platforms in the north sea (stochastic mechanics) to biology and
from physics to computernetworks. The topicis also, mathematicallyspeaking, a deep and beauti
ful one.
There are several good books on stochastic differential equations. But a book that combines a
thorough, sdfcontained treatment of the topic with actual real life applications; a book that also
really tellsthereaderwhatall thisbeautifultheory isgoodfor and, moreover, discusseshow todeal
with these things numerically for simulationpurposes, is rare. Thisis sucha bookand itisa pleas
uretowelcomeitin thisseries.
Theshortestpathbetweentwo truthsin the Neva lend boob, for no ooe eva returns
real domain plWC5 through the complea them; the only books I have in my library
domain. arebooksthatotherfolkhavelentme.
J.Hadamard Anatole France
La physique ne nous donne pas seuIement Thefunction ofan eapertisnot tobemore
I'occasionde Rsoudre des probllmes ... eIIe right thanolbapeople,but tobewrongfor
nousfaitpressentirIasolution. moresophisticatedreasons.
H.Poincare DavidButler
Amsterdam, December 1990 MichielHazewinkel
CONTENTS
Series Editor's Preface vii
Preface ix
INTRODUCTION:
ORIGIN OF STOCHASTIC DIFFERENTIAL EQUATIONS 1
I. STOCHASTIC PROCESSES - SHORT RESUME 6
1. INTRODUCTORY REMARKS 6
2. PROBABILITY AND RANDOM VARIABLES 7
2.1. Basic concepts 7
2.2. Some probability distributions 11
2.3. Convergenceofsequences ofrandom variables 13
2.4. Entropy and information ofrandom variables 16
3. STOCHASTIC PROCESSES - BASIC CONCEPTS 18
4. GAUSSIAN PROCESSES 24
5. STATIONARY PROCESSES 26
6. MARKOV PROCESSES 29
6.1. Basic definitions 29
6.2. Diffusion processes 32
6.3. Methods ofsolving the Kolmogorov equation 41
6.4. Vector diffusion processes 46
7. PROCESSES WITH INDEPENDENT INCREMENTS; .47
WIENER PROCESS AND POISSON PROCESS
7.1. Definition and general properties 47
7.2. Wiener process .49
7.3. Poisson process 52
7.4. Processes related to Poisson process 54
8. POINT STOCHASTIC PROCESSES 56
9. MARTINGALES 58
10. GENERALIZED STOCHASTIC PROCESSES;
WHITE NOISE 60
x Contents
11. PROCESSES WITH VALUES IN HILBERT SPACE 64
12. STOCHASTIC OPERATORS 67
EXAMPLES 68
II. STOCHASTIC CALCULUS: PRINCIPLES AND RESULTS 82
13. INTRODUCTORY REMARKS 82
14. PROCESSES OF SECOND ORDER; MEAN SQUARE ANALYSIS 82
14.1. Preliminaries 82
14.2. Mean-square continuity 84
14.3. Mean-square differentiation 85
14.4. Mean-square stochastic integrals 89
14.5. Orthogonal expansions 93
14.6. Transformationsofsecond-order stochastic processes 95
14.7. Mean-square ergodicity 96
15. ANALYTICAL PROPERTIES OF SAMPLE FUNCTIONS 98
15.1. Sample function integration , 98
15.2. Sample function continuity 100
15.3. Sample function differentiation 102
15.4. Relation to second-order properties 102
16. ITO STOCHASTIC INTEGRAL 106
17. STOCHASTIC DIFFERENTIALS. ITO FORMULA 113
18. COUNTING STOCHASTIC INTEGRAL 119
19. GENERALIZATIONS 124
EXAMPLES 125
III. STOCHASTIC DIFFERENTIAL EQUATIONS:
BASIC THEORY 137
20. INTRODUCTORY REMARKS 137
21. REGULAR STOCHASTIC DIFFERENTIAL EqUATIONS 139
21.1. Mean-square theory 139
21.2. Sample function solutions 141
21.3. Analysis viastochastic operators 143
21.4. Asymptotic analysis 147
21.5. Stationary solutions 154