Asymptotic Analysis for Functional Stochastic Differential Equations

159 Pages·2016·1.129 MB·English

by  Jianhai Bao, George Yin, Chenggui Yuan

SPRINGER BRIEFS IN MATHEMATICS
Jianhai Bao
George Yin
Chenggui Yuan
Asymptotic Analysis
for Functional
Stochastic Differential
Equations
123
SpringerBriefs in Mathematics
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Michele Benzi
Palle Jorgensen
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Ping Zhang
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Jianhai Bao George Yin
(cid:129)
Chenggui Yuan
Asymptotic Analysis
for Functional Stochastic
Differential Equations
123
Jianhai Bao ChengguiYuan
Department ofMathematics Department Mathematics
Central SouthUniversity SwanseaUniversity
Changsha,Hunan Swansea
China UK
George Yin
Department ofMathematics
WayneState University
Detroit, MI
USA
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-46978-2 ISBN978-3-319-46979-9 (eBook)
DOI 10.1007/978-3-319-46979-9
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To my father Chengzhu Bao and my mother
Junhua Han
Jianhai Bao
To my wife Meimei Zhang
George Yin
To my wife Wangdong Yang
Chenggui Yuan
Preface and Introduction
This work focuses on dynamical systems that involve delays and random distur-
bances. The motivation of our study stems from a wide variety of systems in real
life in which random noise has to be taken into consideration and the effect of
delays cannot be avoided orignored. In fact, there are numerous sourcesofdelays
that we encounter in every-day life. For example, everyone has the experience of
delays in highway traffic, air traffic, and package delivery etc. Accompanying the
rapid progress in technology, digital computers, internet, and various hand-held
“smart” devices, delays become more and more prevalent in physical, cyber-
physical, and biological systems. In the new era, delays are ubiquitous in wired or
wireless communications, queueing, and networked control problems.
In this work, we concentrate on such systems that are described by functional
stochasticdifferentialequations.Afunctionalstochasticdifferentialequationisone
whose state space depends on the past. Accompanying the extensive literature on
functional differential equations for deterministic systems (see, e.g., the mono-
graph [62]), the study of such stochastic systems was initiated in 1964 by Itô and
Nisio[69].Forregularityandstochasticstabilityofsolutionprocesses,andMarkov
trajectories plus the infinitesimal generator for segment processes, we refer to the
monographs [96] and [106], respectively. Because their importance, significant
efforts have also been devoted to the study of delay differential systems and more
generally functional differential systems, which are exemplified by chemical pro-
cesses, biological systems, and communication systems. It is observed that delays
may have detrimental impact on stability and performance of dynamical systems.
Their presence adds substantial difficulties in analysis of long-term behavior of
systems and stability. Facing the challenges, we need to have a thorough under-
standing on systems with delays and treat such systems with great care.
vii
viii PrefaceandIntroduction
Motivational Examples
There are many examples and numerous applications involving functional differ-
ential equations with noise. Among the early works on controlled stochastic dif-
ferential delay equations is [85], whereas early treatment of stability of stochastic
delay equations can be found in [82]. In recent years, emerging applications in
financial engineering have drawn much attention; see [2, 4, 7, 75] and references
therein. Although Black-Scholes formula is one of the most important results in
finance,thefeasibilityofthemodelhasbeenquestionedbecauseoftheassumption
of constant appreciation and volatility rates. To treat more realistic situation, effort
has also been directed to the development of dynamical models that take into
consideration the influence of past history.
Thisbriefisdevotedtothestudyoflargetimebehaviorofstochasticfunctional
differential equations. As a motivational example, we consider the following con-
trolled functional stochastic differential equations in an infinite horizon, which is a
modification ofthat considered in[84,Chapter 5]. Suppose that XðtÞisa diffusion
process living in Rn and that X is the associated memory segment process X ¼
t t
fXðtþθÞ:(cid:2)τ(cid:3)θ(cid:3)0gwithτ[0.LetU bethesetofcontrols,andbð(cid:4)Þandσð(cid:4)Þ
beappropriatefunctions(moreprecisenotationwillbegiveninlaterchapters),and
WðtÞ bea standardBrownianmotion.Consider thefollowingfunctional stochastic
differential equation
dXðtÞ¼bðX;uðXÞÞdtþσðXÞdWðtÞ:
t t t
We wish to minimize a long run average cost functional given by
(cid:2)Z (cid:3)
1 T
lim Eu kðX;uðXÞÞdt :
T!1T 0 t t
Theideaoftheergodiccostproblemistoreplacetheinstantaneousmeasureby
thatoftheergodicmeasuresothattheexpectationbecomesanaveragewithrespect
totheergodicmeasureleadingtoasubstantialreductionofcomplexity.Suchstudy
isofgreatutilityinmanyapplications.However,tobeabletocarryoutthedesired
task, one needs to make sure that under suitable conditions, there is an ergodic
measure for such systems. This is one of the main objectives of the current brief.
As a second example, consider an adaptive control problem of controlled dif-
fusion with past dependence treated in [46]. For a fixed τ[0, let C¼
Cð½(cid:2)τ;0(cid:5);RnÞ be the space of continuous functions. Consider a controlled process
defined by
(cid:4)
dXðtÞ¼ff ðXðtÞÞþf ðX;uðXÞ;α(cid:6)ÞgdtþgðXðtÞÞdWðtÞ
1 2 t t
Xð0Þ¼ξ2C;
PrefaceandIntroduction ix
whereXðtÞ2Rn,α(cid:6) isanunknownparameterlivinginRd,andWð(cid:4)Þisastandard
Rn-valued Brownian motion, uð(cid:4)Þ2U (cid:7)Rm with U being a compact set and
X ¼fXðtþθÞ:(cid:2)τ(cid:3)θ(cid:3)0g. The functions f ð(cid:4)Þ and gð(cid:4)Þ satisfy global Lipschitz
t 1
condition. The diffusion is assumed to be non-degenerate in that gðxÞg(cid:6)ðxÞ(cid:8)cI
n(cid:9)n
for some c[0 and all x2Rn with g(cid:6) denoting the transpose of g. The function
f ð(cid:4)Þ is assumed to be bounded and Borel measurable on C. The objective is to
2
minimize the long-run average cost
Z
1 T
Jðu;ξ;α(cid:6)Þ¼limsupTEξ;u;α(cid:6) kðXs;uðXsÞÞds;
T!1 0
where kð(cid:4)Þ:C(cid:9)U7!R is a bounded and continuous function. In [46], it was
shown that the unknown parameter α(cid:6) can be estimated by a biased maximum
likelihoodestimatorandacertaintyequivalentcontrolcanbeobtained.Thecontrol
so designed yields an almost self-optimizing adaptive control. Again, in this pro-
cess, the ergodicity plays an essential role.
Recently, there are much interests in treating the so-called consensus formation
for networked systems and multi-agent systems. Related published works in phy-
sics, computer science, control engineering, ecology, biology, social sciences, as
wellasinthejournalNatureamongothers,showthegrowinginterestsfromawide
rangeofcommunities.Thegoalistoachieveacommonobjectivesuchasposition,
speed, load distribution, etc. for a group of members often referred to as mobile
agents. Such models have been successfully used since late 1980s in computer
graphics, physics, control engineering, and social network modeling, to describe
collective behavior such as flocking, schooling, autonomous vehicles, and other
groupbehavioramongothers.Adiscrete-timemodelofautonomousagents,which
can be viewed as points or particles, all moving in the same plane with the same
speed but with different headings was proposed in [132], in which each agent
updatesitsheadingusingalocalrulebasedontheaverageheadingsofitsownand
itsneighbors.Thismodelturnsouttoberelatedtoaformulationintroducedearlier
for simulating animation offlocking and schooling behaviors in [120]. A version
of the scheme taking into consideration of time delay with communication latency
and possibly random-switching topology may be written as a stochastic approxi-
mation type algorithm of the following form
xnþ1 ¼xnþμMðαnÞxn(cid:2)bd=μcþμG^ðxn(cid:2)bd=μc;αn;ζn(cid:2)bd=μc;~ζnÞ;
with the initial segment x for k ¼(cid:2)bd=μc;...;0 being arbitrary. In the above, x
k n
is in a multi-dimensional Euclidean space, μ[0 is the stepsize of consensus
control algorithm, G^ is an appropriate function, d[0 is a constant and bd=μc
denotes the integer part of d=μ representing the time delays, α is a discrete-time
n
MarkovchainwithstatespaceM¼f1;...;m gandtransitionmatrixPε ¼IþεQ
0
(ε[0 is a small parameter, I is the identity matrix, and Q is a generator of a
continuous-timeMarkovchain),MðiÞisthegeneratorofacontinuous-timeMarkov
x PrefaceandIntroduction
chain for each i2M, d[0 is a constant with bd=μc representing the delay, and
fζ g and f~ζ g are sequences representing measurement or observation noise.
n n
Assumingthatμ¼OðεÞandtakingacontinuous-timeinterpolationxεðtÞ¼x and
n
αε ¼α for t2½nε;ðnþ1ÞεÞ. Then xεð(cid:4)Þ belongs to a function space that consists
n
offunctions that are right continuous with left limits endowed with the Skorohod
topology. Under appropriate conditions, it can be shown that the noises are aver-
aged out, xεð(cid:4)Þ has a weak limit characterized by the switched pure delay equation
x_ðtÞ¼MðαðtÞÞxðt(cid:2)dÞ, where αð(cid:4)Þ is a continuous-time Markov chain [the weak
limitofαεð(cid:4)Þ].Furthermore,itcanbedemonstratedthatxεð(cid:4)þtεÞ(withtε !1as
ε!0) converges weakly to θ, a vector with all components being the same (i.e.,
consensus is reached). With more effort, one can analyze the corresponding esti-
mation error sequence fx (cid:2)θg and show that a suitable scaling leads to a
n
stochastic differential delay equation with Markov switching of the form
dzðtÞ¼MðαðtÞÞzðt(cid:2)dÞdtþGðαðtÞÞdWðtÞ;
foranappropriatefunctionGð(cid:4)ÞandastandardmultidimensionalBrownianmotion
Wð(cid:4)Þ; see [148] for more details together with many references therein.
Purposes of This Brief
This brief is mainly for research purposes. It is written for probabilists, applied
mathematicians, engineers, and scientists who need to use delay systems in their
work. Selected topics from the brief can also be used in a graduate level topics
course in probability and stochastic processes.
The format of the SpringerBriefs in Mathematics gives us an excellent oppor-
tunity to focus on the long-term behavior of the functional stochastic differential
equations. This very focused approach enables us to emphasize the central theme;
our study encompasses ergodicity. Although there are many excellent treatises on
stochastic differential delay equations and functional stochastic differential equa-
tions, short monographs devoted to ergodicity of functional stochastic differential
equationsseemtobescarcetodate.Itwouldcertainlybewelcomedtohaveawork
collect a number of long-run properties about functional stochastic systems.
Nevertheless,becauseoftheformatoftheBriefs,wearenotabletomakethisbook
acomprehensivetreatmentoffunctionalstochasticdifferentialequations.Infact,we
are not even able to include the vast literature in a short book like this.
PrefaceandIntroduction xi
Outline of the Book
Thisbook isorganizedasfollows. Chapter 1 isdevoted toergodicity offunctional
stochastic differential equations and Chapter 2 focuses on ergodicity without dis-
sipative conditions. Chapter 3 ascertains rates of convergence of Euler–Maruyama
procedures. Chapter 4 obtains large deviations estimates for neutral functional
stochastic differential equations with jumps. Chapter 5 gives an application to an
interest model in an infinite horizon. Finally, to make the brief relatively
self-contained,twoappendicesaregivenattheendofthebooktocollectanumber
of results on existence and uniqueness of solutions offunctional stochastic differ-
ential equations, Markov properties, as well as certain technical results such as
variation of constants formulas.
Acknowledgements
Without the help and encouragement of many people, this book project would not
have been completed. We would like to use this opportunity to thank many col-
leaguesandfriendshelpingusinbringingthisbriefintobeing.Inparticular,Mu-Fa
Chen,XianpingGuo,ZhentingHou,JunhaoHu,NielsJacob,JunpingLi,Zaiming
Liu, Xuerong Mao, Jinghai Shao, Aubrey Truman, Feng-Yu Wang, Le Yi Wang,
FukeWu,Jiang-LunWu,andQingZhangworkedwithusonresearchproblemsfor
systemswithdelays,functionalstochasticdifferentialequations,andrelatedissues.
We thank them for their help, encouragement, and inspiration.
We thank the reviewers for the constructive comments and suggestions. Our
thanks also go to Donna Chernyk and Springer professionals for their help and
assistance in finalizing the book. The research of J. Bao was supported in part by
the National Natural Science Foundation of China under grant No. 11401592, and
the research of C. Yuan was supported in part by the EPSRC and NERC. During
thepastyears,theresearchofG.YinwassupportedinpartbytheNationalScience
Foundation, the Air Force Office of Scientific Research, and the Army Research
Office,underdifferentresearchprojectsfordifferentresearchtopicsduringdifferent
times. The supports of the funding agencies are greatly appreciated.
Changsha, China Jianhai Bao
Detroit, USA George Yin
Swansea, UK Chenggui Yuan
August 2016