Table Of ContentStochastic Evolution Systems
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POLY VA NOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Volume 35
Stochastic
Evolution Systems
Linear Theory and Applications
to Non-linear Filtering
by
B. L. Rozovskii
Former/y: Institute ofA dvanced Studies for Chemistry
Managers and Engineers,
Moscow, U.S.S.R.
Nowat: Department of Mathematics,
The University of North Carolina at Char/otte,
Charlotte, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data
~czcvsk 11. 8. L. (Bor~$ L 'IIOV1Ch)
[t:voiîufS'0nnye stokhastichesk~e 51stemy. Erlg1ishl
StOChast1c evo'ution $yste~s , 11near theory and appl1catlons to
non-linear f; ;terirg Oy B.L. Rczcvskil.
I
~. cm. -- (~athematlcs ana its appl1catl~ns (Soviet series»
Translatj~~ of ~vol~fSionr.ye stc.hasticheskle slstemy.
lncludes l~dex.
ISBN 978-94-010-5703-5 ISBN 978-94-011-3830-7 (eBook)
DOI 10.1007/978-94-011-3830-7
1. StochastlC part'a1 dlfferentlal e~Uations. 1. Tltle.
II. Serles. Mathematlcs ana 'ts applicatlo~S (Kluwer AcademiC
P~bl1shers). Soviet ser'Ss.
OA27Q.25.R6913 1990
519.2--dc19 88-39995
ISBN 978-94-010-5703-5
Printed on acid-free pap er
Translated [rom the Russian by A. Yarkho
This is the translation of the original work
3BOJIIOUl10HHbIE CTOXACTI14ECKI1E CI1CTEMbI
n. ....a u ftOJNI •• ap ........ Cft'l'llCTllh •
CII)"Ialtq,a:llpo&lCCCOa
Published bu Nauka Publishers, Moscow, © 1983.
AlI Rights Reserved
© 1990 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1990
Softcover reprint ofthe hardcover Ist edition 1990
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TO MY MOTHER
SERIES EDITOR'S PREFACE
'Et moi, "'J si j'avait su comment en revcnir, One seMcc mathematics has rendered the
je n'y semis point aile.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shclf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I 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 branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
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 e1fort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
viii SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
mOOds are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no 1V; in fact you would not be reading these lines.
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 te1ling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- infiuences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
As the author writes, up to fairly recently an Ito stochastic differential equation was almost invari
ably an ordinary stochastic differential equation. Since then (the mid seventies) the situation has
changed drastically and fields like signal processing, magnetohydrodynamics, relativistic quantum
mechanics, population dynamics, and genetics, oceanology, and others gave rise to numerous sto
chastic partial differential equations, mainly linear evolution equations du = A (t, u)dt + B (t, u)dw,
where w is Wiener noise and where the drift, A (t,u), and diffusion, B(t,u), are operators, usually
unbounded .. In addition such diffusion equations have turned out to be important in the theory of
(nonlinear) nonstochastic second-order partial differential equations of elliptic or parabolic type;
see, for instance, KIylov's book on this topic (also published in this series).
Already in 1983 the basic decision was made to translate the book. Also already at that time it
was clear that an additional chapter should be included. Indeed the author absolutely refused to
allow a straightforward translation. Writing the additional material took longer than anticipated
partly because the author moved in this period from the USSR to the USA and mostly because the
author wanted to do a thorough job worthy of his reputation and talents. The result is well worth
waiting for and the delay has had the advantage that the developments of the so-called Malliavin
calculus (stochastic variational calculus) and its applications could be thoroughly included.
The shortcst path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
I'o cc:asion de resoudre des problemes ... eIle right than other people, but to be wrong for
nous fait prcssentir Ia solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, 7 October 1990 Michiel Hazewinkel
PREFACE
TO THE ENGLISH EDITION
The subject of this book is linear stochastic partial differential equations and
their applications to the theory of diffusion processes and non-linear filtering.
Until recently, the term "stochastic differential equation" did not need any
specifications - in 99 cases out of 100 it was applied to ordinary stochastic
differential equations. Their theory started to develop at the beginning of the
1940s, based on Ito's stochastic calculus [50], [51], and now forms one of the most
beautiful and fruitful branches of the theory of stochastic processes, [36], [52], [49],
[63], [90], [132].
In the middle of the 1970s, however, the situation changed: in various
branches of knowledge (primarily, in physics, biology, and control theory) a vast
number of models were found that could be described by stochastic evolution
partial differential equations. Such models were used, for example to describe a
free (boson) field in relativistic quantum mechanics, a hydromagnetic dynamo
process in cosmology, diffraction in random-heterogeneous media in statistical
physics, the dynamics of populations for models with a geographical structure in
population genetics, etc.
The emergence of this new type of equation was simultaneously stimulated by
ix
x PREFACE
the inner needs of the theory of differential equations. Such equations were
effectively used to study parabolic and elliptic second-order equations in infinite
dimensional spaces.
An especially powerful impetus to the development of the theory of evolution
stochastic partial differential equations was given by the problem of non-linear
filtering of diffusion processes.
The filtering problem (estimation of the "signal" by observing it when it is
mixed with a "noise") is one of classical problem in the statistics of stochastic
processes. It also belongs to a rare type of purely engineering problems that have
a precise mathematical formulation and allows for a mathematically rigorous
solution.
The first remarkable results in connection with filtering of stationary processes
were obtained by A.N. Kolmogorov [60] and N. Wiener [146]. After the paper by
R. Kalman and R. Bucy [54] was published, the 1960s and 1970s witnessed a rapid
development of filtering theory for systems whose dynamics could be described by
Ito's stochastic differential equations. The results were first summed up by R. Sh.
Liptser and A.N. Shiryayev [90] and G. Kallianpur [53].
One of the key results of the modern non-linear filtering theory states that the
solution of the filtering problem for the processes described by Ito's ordinary
stochastic equations is equivalent to the solution of an equation commonly called
the filtering equation. The filtering equation is a typical example of an evolution
PREFACE xi
stochastic partial differential equation. An equation of this type can be regarded
as an "ordinary" Ito equation
du(t) = A(t, u(t»dt + B(t, u(t» dW(t)
for the process u(t) taking values in a function space X. The coefficients A, B (of
"drift" and "diffusion") in this equation are operators (unbounded, as a rule), and
W(t) is a "white noise" taking values in a function space. Such an equation may
be regarded as a system (an infinite one, if the space X is infinite) of one
dimensional Ito's equations. Below we shall call equations (systems of equations)
of this type stochastic evolution systems.
The theory of stochastic evolution systems is quite a young branch of science
but it has nevertheless generated many interesting and important results, much
more than it would be reasonable to include in one book. Books [5], [8], [144],
[81], [19], [17], [105], [141], etc. contain sections devoted to the theory.
In the present monograph, the author has set himself the following program:
(1) to cover the general theory of linear stochastic evolution systems (LSES)
with unbounded drift and diffusion operators:
(2) to contstruct the theory of Ito's second-order parabolic equations;
(3) to investigate, on the basis of the latter, the filtering problem and related
issues (interpolation, extrapolation) for processes whose trajectories can be
described by Ito's ordinary equations.
The first point of the program is dealt with in Chapters 2 and 3, the second
