Stochastic Differential Systems: Proceedings of the 4th Bad Honnef Conference, June, 20–24, 1988

351 Pages·1989·5.456 MB·English

by  S. Albeverio, A. Hilbert (auth.), Norbert Christopeit, Kurt Helmes, Michael Kohlmann (eds.)

Lecture Notes ni
Control and
noitamrofnI Sciences
detidE yb amohT.M dna renyW.A
621
N. Christopeit, .K Helmes
.M Kohlmann (Editors)
Stochastic Differential Systems
Proceedings of the 4th daB Honnef Conference,
,enuJ 20-24, 8891
galreV-regnirpS
nilreB Heidelberg weN York
nodnoL siraP Tokyo Hong gnoK
Series Editors
M. Thoma • A. Wyner
Advisory Board
L. D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L. Massey • Ya Z. Tsypkin • A. .J Viterbi
Editors
Norbert Christopeit Michael Kohlmann
Institut fLir (~konometrie Fakultfit fLir Wirtschafts-
und Operations Research wissenschaften und Statistik
der Universitfit Bonn Universit&t Konstanz
Okonometrische Abteilung Postfach 5560
Adenauerallee 24-42 D-7750 Konstanz
D-5300 Bonn
Kurt Helmes
Department of Mathematics
University of Kentucky
Lexington, KY 40506
USA
ISBN 3-540-51299-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-51299-3 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging in Publication Data
Stochastic differential systems : proceedings of the 4th Bad Honnef conference,
June 20-24, 1988
/ N. Chdstopeit, K. Helmes, M. Kohlmann, eds.
(Lecture notes in control and information sciences ; 126)
Papers of the 4th Bad Honnef Conference on Stochastic Differential Systems.
ISBN 0-387-51299-3 (U.S.)
1. Stochastic systems - Congresses. 2. Differentiable dynamical systems - Congresses.
.I Christopeit, N. .1I Helmes, .K (Kurt) III. Kohlmann, M. (Michael)
.VI Bad Honnef Conference on Stochastic Differential Systems (4th : 1988) V. Series.
QA402.$846 1989
003-d¢20 89-11454
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PREFACE
This volume contains the major part of contributions to thc 4th Bad tIonnef Conference on
Stochastic Differential Systems held at Bad ttonnef, West Germany, June 20-24, 1988.Following the
tradition of the preceding Bad Honnef Conferences, the meeting was intended to highlight recent
advances in the areas of stochastic control and filter theory as well as stochastic analysis.
As sort of thematic "domains of attraction", special emphasis has been given to two rather
active fields of current rcsearch: the use of adaptive methods in stochastic systems analysis and the
theory of random fields. In view of the overwhelming flood of information accumulated in these two
areas in recent years, the most that could be hoped for at this conference was to offer a glimpse of
the status of research and to inspire interest and discussion in these fields. Several survey lectures
were intended to provide some introduction to the more mature parts of the theory for those less
acquainted with the subject, complemented by contributions that should give some taste of the
diversifying issues of current research both in thcory and practice. We leave it to the reader to
judge how well this goal has been achieved.
It is a privilege of the organizing committee to express its gratitude to the Deutsche Forschungs-
gemeinschaft, whose generous support made this conference possible. We are also indebted to the
members of the International Advisory Committee; assisting us with many valuable suggestions
concerning program and speakers they have a substantial share in the success of the conference.
Last, but not least, our thanks go to the staff of the Elly-ttSlterhoff-Stift for their kind hospitality
as wcll as to G. NSldeke and A. Schiitt for their skilfull job in data processing for the conference
and in preparing this volume.
Bonn, February 1989 .N ticpotsirhC
Helmes K.
M. Kohlmann
LIST OF PARTICIPANTS
S. Albeverio R.J. Chitashvili
Fakult//t und Institut Iiir Mathematik Dept. of Probability Theory and Math.
Ruhr-Universit/it Bochum Statistics
Postfaeh I0 21 48 Institute of Mathematics
4630 Bochum 1 Georgian Academy of Sciences
F.R.G. 150 a Plekhanov Ave.
380012 Tbilisi
U.S.S.R.
A.N. Al-Hussaini
Dept. of Statistics
University of Alberta N. Christopeit
Edmonton T6G 2G1 Institut filr Okonometrie und Operations Re-
Canada
search
Universit~it Bonn
Adanauerallee 24-42
S.A. Anulova
5300 Bonn 1
etutitsnI Control of secneicS
F.R.G.
Profsoyusnaya 65
117 806Moscow
U.S.S.R. M.II.A. Davis
Department of Electrical Engeneering
Imperial College of Science and Technology
A. Bensoussan
Exhibition Road
I.N.R.I.A.
London SW7 2BT
Domaine de Voluceau-Rocquencourt
Great Britain
B.P. 105
78135 Le Chesnay Cedcx
France T. E. Duncan
Dept. of Mathematics
University of Kansas
T. Bielecki
Lawrence, Kansas 66045
Institute of Mathematics
U.S.A.
Polish Academy of Scienccs
Sniadeckidl 8
00-950 Warszawa
R.J. Elliott
Poland Dept. of Statistics and Probability
University of Alberta
Edmonton, T6G2G1
K.J. Bierth
Canada
tutitsnI rd'i Angewandtc Mathematik
ti~tisrevinU Bonn
.rtsrelegeW 6 H.J. Engelbert
5300 Bonn 1
Sektion Mathematik
F.R.G. Friedrich-Schiller-Universit/it
6900 Jena
G.D.R.
P.E. Caines
Dept. of Electrical Engeneering
McGill University
L. Gerenc~er
3480 University Department of Electrical Engeneering
Montreal, PQ H3A 2A7
Me Gill University
Canada Montreal, Quebec, H3A 2A7
Canada
V
.J Glas K. Iwata
HochgrafenstraBc 65 Fakult~it und Institut f'dr Mathematik
7750 Konstanz Ruhr-Universit~it Bochum
F.R.G. Postfach 10 21 48
4630 Bochum 1
F.R.G.
B. Goldys
Institute of Mathematics
Polish Academy of Sciences A. Jakubowski
00-950 Warszawa Institut f'dr Mathematische Stochastik
Sniadeckich 8, P.O.Box 137 Universit~it GSttingen
Poland Lotzestr. 13
3400 G5ttingen
F.R.G.
G.L. Gomez
Abt. ffir Mathematik
Unlversit~.t Erlangen-Nilrnbcrg M. Jerschow
8920 Erlangen FB 6 Mathematik
F.R.G. Gesamthochschule Essen
Universit~itsstr. 2
4300 Essen
B. Grigelionis F.R.G.
Academy of Sciences of the Lithuanian SSR
Institute of Mathematics and Cybernetics
232600, Vilnius, 54 W. Kirsch
U.S.S.R. Fakultat un tutditsn I rd'f Mathematik
Ruhr-Universit£t Bochum
Postfach 10 21 48
A. lteinricher 4630 Bochum I
Dept. of Mathematics F.R.G.
University of Kentucky
Lexington, KY 40506
U.S.A. D. KShnlein
Institut f'dr Angewandte Mathematik
Universit~.t Bonn
K. Helmes Wegelerstr. 6
Dept. of Mathematics 5300 Bonn 1
University of Kentucky F.R.G.
Lexington, KY 40506
U.S.A.
M. Kohlmann
Fakult~it ffr Wirtschaftswissenschaften und
A. trebliH Statistik
Fakult£t un tutditsn I riif Mathematik Postfach 5560
Ruhr-Universit~t Bochum 7750 Konstanz 1
Postfach 10 21 48 F.R.G.
4630 Bochum 1
F.R.G.
T. Kottman~.
Institut fiir Okonometrie und Operations Re-
H. Hui search
tutitsnI fdr Angewandte Mathematik Universit~it Bonn
tgtisrevinU Bonn Adenauerallee 24-42
.rtsrelegeW 6 5300 Bonn 1
5300 Bonn 1 F.R.G.
F.R.G.
VI
P. KrSger .J Parisi
Mathematisches Institut Lehrstuhl flir Experimentalphysik II
Unlversitilt Erlangen-Niirnberg Universit~it Tiibingen
8521 Erlangen 7400 Tiibingen 1
F.R.G. F.R.G.
P.R. Kumar B. Pasik-Duncan
Decision and Control Laboratory Dept. of Mathematics
University of Illinois at Urbana-Champaign University of Kans~
1101 West Springfield Avenue Lawrence, Kansas 66045
Urbana, IL 61801 U.S.A.
U.S.A.
N.I. Portenko
H.J. Kushner Institute of Mathematics
Dept. of Applied Mathematics Ukrainian Academy of Sciences
Lefschetz Center for Dynamical Systems Repin Str. 3
Brown University Kiev
Providence, .I.R 02912 .R.S.S.U
U.S.A.
F. Russo
S. Kusuoka BIBOS
Research Institute for Mathematical Science Universit~it Bielefeld
Kyoto University Postfach 8640
606 Kyoto 4800 Bielefeld 1
Japan F.R.G.
T.L. Lai M. Sch~.l
Department of Statistics Institut filr Angewandte Mathematik
Sequoia Hall Universit~it Bonn
Stanford University Wegelerstr. 6
Stanford, CA 94305-4065 5300 Bonn 1
U.S.A. F.R.G.
P. Mandl Schauml6ffel K.-U.
Dept. of Probability and Mathematical Statis- ti~tisrevinU Bremen
tics tutitsnI rd-t Dynamische Systeme
Charles ytisrevinU eBartskehtoilbiB 1
Sokolovska 83 Postfach 44303 0
186 Prag 8 2800 Bremen 33
Czechoslovakia F.R.G.
S.P. Meyn K. Schilling
Department of Systems Engineering Hannoversche Sir. 68
Australian National University 3400 G6ttlngen
Canberra, ACT F.R.G.
Australia
S.E. Shreve
M. Musiela Dept. of Mathematics
School of Mathematics Carnegie-Mellon University
University of New South Wales ,hgrubsttiP Penn. 15213
P.O. Box 1 U.S.A.
Kensington, New SouthW ales
Australia
IIV
P. Spreij N. van Thu
Dept. of Economics Goetheinstitut
Free University P.O. Box 7161 7800 Freiburg i.Br.
1007 MC Amsterdam F.R.G.
sdnalrehteN
E. Wong
L. Stettner Department of Electrical Engineering
Institute of Mathematics and Computer Sciences
Polish Academy of Sclenccs University of California at Berkeley
Sniadeekich 8 Berkeley, CA 94720
00-950 Warszawa U.S.A.
Poland
K. Yamada
D. Surgailis Institute of Information Sciences and
Institute of Mathematics and Cybernetics Electronics
Academy of Sciences of the Lithuanian SSR University of Tsukuba
Akademijos 4 Tsukuba-shi, Ibaraki 305
232600, Vilnius Japan
,R.S.S.U
M. Zhiming
T.A. Toronjadze Fakuitgt und Institut i-fir Mathematik
Dept. of Probability Theory and Mathemati- Ruhr-Universitgt Bochum
cal Statistics Postfach 10 21 48
Institute of Mathematics 4630 Bochum 1
Georgian Academy of Sciences F.R.G.
150 a Plekhanov Ave.
Tbilisi 380012
U.S.S.R.
CONTENTS
S. ALBEVERIO, A. tIILBERT
Some results on Newton equation with an additional stochastic force ...................... 1
S. ALBEVERIO, M. RC)CKNER
On Dirichlet forms on topological vector spaces: existence and maximality ............... 41
S. ALBEVERIO, M. ZIIIMING
Nowhere Radon smooth measures, perturbation of Dirichlet forms and singular
quadratic forms ......................................................................... 32
A.N. AL-IIUSSAIN I
A generalization of Ito's formula ......................................................... 46
S.V. ANULOVA
General functional limit theorems for semimartingales ................................... 55
A. BENSOUSSAN
Nonlinear filtering for dynamic systems with singular perturbations ...................... 63
T. BIELECKI, B. GOLDYS
On recursive adaptive filtering: linear case ............................................... 72
R.J. CHITASttVILI
On the smooth fit boundary conditions in the optimal stopping problem
for semimartingales ..................................................................... 82
M.H.A. DAVIS, E.M. HEMERLY
Order determination and adaptive control of ARX models using the PLS criterion ........ 91
T.E. DUNCAN
Adaptive control of some partially observed linear stochastic systems ................... 201
R.J. ELLIOTT, M. KOItLMANN
The adjoint process in stochastic optimal control .......... . ............................ 115
R.J. ELLIOTT, M. KOHLMANN
Integration by parts and the MalIiavin calculus ......................................... 821
L.GERENCSER
Pathwize stability of random differential equations and the solution
of an adaptive control related problem ................................................. 140
G.L. GOMEZ
Some economic applications of stochastic calculus ...................................... 150
T. KOTTMANN
OLS-Estimation and rationality in model8 with forecasi~ feedback ....................... 261
IX
P. KROGER
Invariance "fo cones and comparison results for some classes of diffusion processes ........ 171
P.R. KUMAR
Performance and robustness in adaptive control of linear stochastic systems ............. 183
tI. KUSHNER
Singular perturbations for stochastic control ............................................ 196
T.L. LAI
Extended stochastic Lyapunov functions and recursive algorithms in
linear stochastic systems ................................................................ 206
A. LE BRETON, M. MUSIELA
Consistency sets of least squares estimates in stochastic regression models ............... 221
P. MANDL
Consistency of estimators in controlled systems ......................................... 227
S.P. MEYN, P.E. CAINES
Stochastic controllability and stochastic Lyapunov functions with applications
to adaptive and nonlinear systems ...................................................... 235
J. PARISI
A simple stochastic growth model for filamentary current structures
in semiconductor systems .............................................................. 258
B. PASIK-DUNCAN
Asymptotic normality of an estimator in a controlled investment model
with time-varying parameters .......................................................... 266
b. STETTNER
On invariant measures of filtering processes ............................................. 279
D. SURGAILIS, T. ARAK
Markov fields with discrete state space ................................................. 293
N. VAN THU
Strictly stationary processes with the linear prediction property ......................... 317
E. WONG
Multiparameter martingale and Markov process ........................................ 329
K. YAMADA
Limit process with the domain of attraction of a stable law ............................. 337
Some I-results on Newton Equation with
an Additional Stochastic Force
S. Albeverio#
A. Hilbert
Fakult/it fi'tr Mathema~ik, Ruhr-Universitiit-Bochum (FRG)
# BiBoS-Research Centre, SFB 237 Bochum-Essen-Dfisseldorf~ CERFIM Locarno (CH)
Abstract
In this report, based on joint work with E. Zehnder, we discuss stochastic perturbations of
classical Itamiltonitm systems by a white noise force. We give existence and uniqueness results
for file solutions of the equation of motion, allowing for forces growing stronger thml linearly at
infinity. Wc prove that Lcbcsguc mcasure in phase space is a a-finite inv~n'iant measure. Morcovcr
we give a Girsanov formula relating the solutions for a nonlinear force to those for a linear force.
1. Introduction
The study of stochastic perturbations of classical dynamical systems has becn developed in recent
years, however, the c~c of stochastic perturbations of classical Hmniltoniml systems has been
nntch less investigated on a mathcnmtical basis, despite its great iutcrcst in applications (celestial
mechanics, vibrations in nmchanical systems, wave propagation in solid state physics ...). One
reason for this is the fact that the structure of the classical flows themselves is much more com-
plicated. Orbits of very different long time bchaviour, in gcncral cannot be sep;n'atcd in finite
time intervals, stable and unstable bchaviom" being mixed, sec e.g. [At'], [Mo], and [Mo-Zc]. The
nature of the orbits depends on the dimension of tile system. For degrees of fl'ecdom exceeding
three the behaviour can vaz'y between periodic motion, mid Arnold diffusion. Often the difficult
mathematically rigorous invcstigat, ion of the longtime behaviour, has been replaced by heuristic
numcricM approaches, sec e.g. [Li-Lie].
In case tim perturbation is stoch~stic, hence typically non smooth, and undcr the restriction ofo ne
degree of freedom, Pot~cr [Po] (see also McKean {McK]) analysed nonlineea' oscillators perturbed
by a white noise force, described by the equations
where K is the deterministic force, • means time deriwttive, tb is white noine (the deriw~tive of
Brownian motion w(t)). Under assumptions on the fm'ce K(x) = -V'(x), V C Ct(/R) being
attra.cting towm'ds the origin, i.e. x - N(x) < ,O Potter proved the existence of global solutions
and results about recurrence and the invm'iance of Lebesgue measure baxd¢ under the flow given
in (1.1). Some of these results ht~ve been recently extended by Markus and Weerasinghe [MAW]
who also studied winding numbers associated with the solution process (x,v ) around the origin,
.]QGA[
see Mso