Equations involving malliavin calculus operators : applications and numerical approximation

139 Pages·2017·2.876 MB·English

by  Levajkovic, Tijana, Mena, Hermann

SPRINGER BRIEFS IN MATHEMATICS
Tijana Levajković
Hermann Mena
Equations
Involving Malliavin
Calculus Operators
Applications
and Numerical
Approximation
123
SpringerBriefs in Mathematics
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ć
Tijana Levajkovi Hermann Mena
(cid:129)
Equations Involving
Malliavin Calculus Operators
Applications and Numerical Approximation
123
TijanaLevajković
Department ofMathematics
University of Innsbruck
Innsbruck
Austria
Hermann Mena
Department ofMathematics
Yachay Tech
SanMiguel deUrcuquí
Ecuador
and
Department ofMathematics
University of Innsbruck
Innsbruck
Austria
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-65677-9 ISBN978-3-319-65678-6 (eBook)
DOI 10.1007/978-3-319-65678-6
LibraryofCongressControlNumber:2017949124
Mathematics Subject Classification (2010): 60G20, 60G22, 65J10, 60H07, 60H40, 60H10, 60H15,
60H20,60H30,60H35,35R60,34H05,93E20,47D06,46N30,49N10,93C20
©TheAuthor(s)2017
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To Branko and Zvezdana
Tijana Levajković
To Marco and Juanita
Hermann Mena
Preface
This book provides a comprehensive and unified introduction to stochastic differ-
ential equations involving Malliavin calculus operators. Particularly, it contains
results on generalized stochastic processes, linear quadratic optimization and
specific applications that appear in the literature as a series of papers ranging from
classical Hilbert spaces to numerical approximations in a white noise analysis
setting. The intended audience are researchers and graduate students interested in
stochastic partial differential equations and related fields. This book is
self-containedforreadersfamiliarwithwhitenoiseanalysisandMalliavincalculus.
A major contribution of this book is the development of generalized Malliavin
calculus in the framework of white noise analysis, based on chaos expansion
representationofstochasticprocessesanditsapplicationforsolvingseveralclasses
ofstochastic differential equationswith singular data, i.e., singular coefficients and
singular initial conditions, involving the main operators of Malliavin calculus.
Thisbookisdividedintofourchapters.Thefirst,entitledWhiteNoiseAnalysis
and Chaos Expansions, includes notation and provides the reader with the theo-
retical background needed tounderstand thesubsequent chapters.Inparticular,we
introduce spaces of random variables and stochastic processes, and consider pro-
cesses that have finite variance on classical and fractional Gaussian white noise
probability spaces. We also present processes with infinite variance, particularly
Kondratiev stochastic distributions. We introduce the Wick and ordinary multi-
plication of the processes and state where these operations are well defined.
In Chap. 2, Generalized Operators of Malliavin Calculus, the Malliavin
derivative operator D, the Skorokhod integral d and the Ornstein–Uhlenbeck
operatorRareintroducedintermsofchaosexpansions.Themainpropertiesofthe
operators,whichareknownintheliteratureforthesquareintegrableprocesses,are
proven using the chaos expansion approach and extended for generalized and test
stochastic processes. Moreover, we discuss fractional versions of these operators.
Chapter 3, Equations Involving Malliavin Calculus Operators, is devoted to the
studyofseveraltypesofstochasticdifferential equationsthatinvolvetheoperators
ofMalliavincalculus,introducedinthepreviouschapter.Inparticular,wedescribe
the range of the operators D, d and R.
vii
viii Preface
In Chap. 4, we present applications of the chaos expansion method in optimal
control and stochastic partial differential equations. In particular, we consider the
stochasticlinearquadraticoptimalcontrolproblemwherethestateequationisgiven
byastochasticdifferentialequationoftheIto-Skorokhod typewithdifferent forms
of noise disturbances, operator differential algebraic equations arising in fluid
dynamics, stationary equations and fractional versions of the studied equations.
Moreover, we provide a numerical framework based on chaos expansions and
perform numerical simulations.
We would like to express our gratitude to our institutions, University of
Innsbruck (Austria) and Yachay Tech (Ecuador) for giving us the opportunity to
work in great environments. H. Mena thanks the support of the Austrian Science
Foundation(FWF)–projectid:P27926.Specialthanksgotoallourco-authorsand
colleagues, who contributed greatly in making this project enjoyable and success-
ful. Last but not least, we would like to thank our families for their love and
support.
Innsbruck, Austria Tijana Levajković
August 2017 Hermann Mena
Contents
1 White Noise Analysis and Chaos Expansions .. .... .... ..... .... 1
1.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 1
1.2 Deterministic Background... .... .... .... .... .... ..... .... 2
1.3 Spaces of Random Variables .... .... .... .... .... ..... .... 4
1.4 Stochastic Processes.. ..... .... .... .... .... .... ..... .... 18
1.5 Operators .. .... .... ..... .... .... .... .... .... ..... .... 35
References.. .... .... .... ..... .... .... .... .... .... ..... .... 39
2 Generalized Operators of Malliavin Calculus.. .... .... ..... .... 43
2.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 43
2.2 The Malliavin derivative.... .... .... .... .... .... ..... .... 44
2.3 The Skorokhod Integral .... .... .... .... .... .... ..... .... 50
2.4 The Ornstein-Uhlenbeck Operator .... .... .... .... ..... .... 54
2.5 Properties of the Operators of Malliavin Calculus .... ..... .... 59
2.6 Fractional Operators of the Malliavin Calculus... .... ..... .... 70
References.. .... .... .... ..... .... .... .... .... .... ..... .... 72
3 Equations Involving Mallivin Calculus Operators .. .... ..... .... 75
3.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 75
3.2 Equations with the Ornstein-Uhlenbeck Operator. .... ..... .... 76
3.3 First Order Equation with the Malliavin Derivative Operator. .... 79
3.4 Nonhomogeneous Equation with the Malliavin Derivative
Operator... .... .... ..... .... .... .... .... .... ..... .... 84
3.5 Wick-Type Equations Involving the Malliavin Derivative ... .... 85
3.6 Integral Equation .... ..... .... .... .... .... .... ..... .... 92
References.. .... .... .... ..... .... .... .... .... .... ..... .... 95
4 Applications and Numerical Approximation... .... .... ..... .... 97
4.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 97
4.2 A Stochastic Optimal Control Problem in Infinite
Dimensions. .... .... ..... .... .... .... .... .... ..... .... 99
ix
x Contents
4.3 Operator Differential Algebraic Equations .. .... .... ..... .... 114
4.4 Stationary Equations.. ..... .... .... .... .... .... ..... .... 117
4.5 A Fractional Optimal Control Problem. .... .... .... ..... .... 119
4.6 Numerical Approximation... .... .... .... .... .... ..... .... 121
References.. .... .... .... ..... .... .... .... .... .... ..... .... 130
Chapter 1
White Noise Analysis and Chaos Expansions
Abstract Intheframeworkofwhitenoiseanalysis,randomvariablesandstochastic
processescanberepresentedintermsofFourierseriesinaHilbertspaceorthogonal
basis,namelyintheirchaosexpansionforms.Webrieflysummarizebasicconcepts
and notations of white noise analysis, characterize different classes of stochastic
processes(test,squareintegrableandgeneralizedstochasticprocesses)intermsof
their chaos expansion representations and review the main properties of the Wick
calculusandstochasticintegration.
1.1 Introduction
White noise analysis, introduced by Hida in [9] and further developed by many
authors[11, 26],asadisciplineofinfinitedimensionalanalysis,hasfoundapplica-
tionsinsolvingstochasticdifferentialequations(SDEs)andthusinthemodelingof
stochastic dynamical phenomena arising in physics, economy, biology [6, 27, 29,
34,37].Inthiscontext,whitenoiseanalysiswasproposedasaninfinitedimensional
analogueoftheSchwartztheoryofdeterministicgeneralizedfunctions.
Stochastic processes withinfinite variance, e.g. the whitenoise process, appear
in many cases as solutions of SDEs. The Hida spaces and the Kondratiev spaces
[9, 11] have been introduced as the stochastic analogues of the Schwartz space of
tempereddistributionsinordertoprovideastricttheoreticalmeaningforthiskindof
processes.Thespacesoftestprocessescontainhighlyregularprocesseswhichallow
onetodetecttheactionofgeneralizedprocesses.Thechaosexpansionofastochastic
processprovidesaseriesdecompositionofasquareintegrableprocessinaHilbert
spaceorthogonalbasisbuiltuponaclassofspecialfunctions,Hermitepolynomials
andfunctions,intheframeworkofwhitenoiseanalysis.Inordertobuildspacesof
stochastictestandgeneralizedfunctions,onehastouseseriesdecompositionsvia
orthogonalfunctionsasabasis,withcertainweightsequences.Here,wefollowthe
classicalHidaapproach[9],whichsuggeststostartwithanuclearspaceG andits
dual G(cid:2), such that G ⊆ L2(R) ⊆ G(cid:2), and then take the basic probability space to
be Ω = G(cid:2) endowed with the Borel sigma algebra of the weak topology and an
appropriateprobabilitymeasure P.
©TheAuthor(s)2017 1
T.Levajkovic´andH.Mena,EquationsInvolvingMalliavinCalculusOperators,
SpringerBriefsinMathematics,DOI10.1007/978-3-319-65678-6_1