Almost Periodic Stochastic Processes

252 Pages·2011·3.47 MB·English

by  Paul H. Bezandry

Almost Periodic Stochastic Processes

Paul H. Bezandry • Toka Diagana

Almost Periodic
Stochastic Processes
Paul H. Bezandry Toka Diagana
Department of Mathematics Department of Mathematics
Howard University Howard University
2441 6th Street NW 2441 6th Street NW
20059 Washington District 20059 Washington District
of Columbia of Columbia
USA USA
[email protected] [email protected]

ISBN 978-1-4419-9475-2 e-ISBN 978-1-4 419-9476-9
D OI 10.1007/978-1-4419-9476-9
Springer New York Dordrecht Heidelberg London
L ibraryofCongressControlNumber:2011925256

Mathematics Subject Classification (2010): 34K50, 34K30, 35R60, 39A24, 39A50, 47D06, 60-XX,
60Axx, 65 J08
© Springer Science+Business Media, LLC 2011
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)
To ourfamilies

Acknowledgments
We would like to thank our wives and kids for support and encouragement. Their
patienceandencouragementhavemadeeverythingpossibleincludingthisbook.
WewishtothankmembersoftheHowardUniversity’sMathematicsDepartment
aswellastheCollegeofArts&Sciencesfortheirstrongsupportsincebothofus
joinedHowardUniversitytenyearsago.
We are grateful to Professors Terrence Mills and Alexander Pankov for proof-
readingalltheversionsofthisbook.Theircommentsandsuggestionshavesignifi-
cantlyimprovedthisbook.Oursincerestgratitudegoestobothofthem.
We are grateful to Springer for agreeing to publish our book. Additionally, we
wouldliketoexpressourdeepestgratitudetothereviewerforcarefulreadingofthe
bookandinsightfulcomments.
vii

Preface
This book analyzes almost periodic stochastic processes and their applications to
various stochastic differential equations, partial differential equations, and differ-
enceequations.Itisinpartasequeltheofauthors’recentwork[20,21,22,23,24,
55] on almost periodic stochastic difference and differential equations and has the
particularitytobeamongthefewbooksthatareentirelydevotedtoalmostperiodic
stochastic processes and their applications. The topics treated in it range from ex-
istence,uniqueness,boundedness,andstabilityofsolutionstostochasticdifference
anddifferentialequations.
Periodicity often appears in implicit ways in various natural phenomena. For
instance, this is the case when one studies the effects of fluctuating environments
on population dynamics. Though one can deliberately periodically fluctuate envi-
ronmental parameters in controlled laboratory experiments, fluctuations in nature
arehardlyperiodic.Almostperiodicityismorelikelytoaccuratelydescribenatural
fluctuations[63].Motivatedbythisobservation,wedecidedtowritethisbookthat
is devoted to the study of almost periodic (mild) solutions to stochastic difference
anddifferentialequations.Sincethebeginningofthecentury,thetheoryofalmost
periodicity has been developed in connection with problems related to differential
equations,dynamicalsystems,andotherareasofmathematics.Theclassicalbooks
of Bohr [32], Corduneanu [42], Fink [73], and Pankov [151] for instance gave a
nice presentation of the concept of almost periodic functions in the deterministic
settingaswellaspertinentresultsinthearea.Recently,therehasbeenanincreasing
interest in extending certain classical results to stochastic differential equations in
separable Hilbert spaces. This is due to the fact that almost all problems in a real
lifesituationtowhichmathematicalmodels areapplicablearebasicallystochastic
rather than deterministic. Nevertheless, the majority of mathematical methods are
basedondeterministicmodels.Forinstance,thetheoryofanalysisfrequentlyused
in deterministic models can often be utilized as a tool to obtain the solutions to
stochasticdifferentialequations.
The concept of almost periodicity for stochastic processes was first introduced
intheliteraturebySlutsky[166]attheendof1930s,whothenobtainedsomerea-
sonable sufficientconditions forsample pathsof astationary process to bealmost
ix