Table Of ContentProbability Theory and Stochastic Modelling 92
Hiroshi Kunita
Stochastic Flows
and
Jump-Diffusions
Probability Theory and Stochastic Modelling
Volume 92
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Hiroshi Kunita
Stochastic Flows
and Jump-Diffusions
123
HiroshiKunita
KyushuUniversity(emeritus)
Fukuoka,Japan
ISSN2199-3130 ISSN2199-3149 (electronic)
ProbabilityTheoryandStochasticModelling
ISBN978-981-13-3800-7 ISBN978-981-13-3801-4 (eBook)
https://doi.org/10.1007/978-981-13-3801-4
LibraryofCongressControlNumber:2019930037
MathematicsSubjectClassification:60H05,60H07,60H30,35K08,35K10,58J05
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Preface
TheWienerprocessandthePoissonrandommeasurearefundamentaltothestudy
of stochastic processes; the former describes a continuous random evolution, and
the latter describes a random phenomenon that occurs at a random time. It was
showninthe1940sthatanyLévyprocess(processwithindependentincrements)is
representedbyaWienerprocessandaPoissonrandommeasure,calledtheLévy–
Itôrepresentation.Further,Itôdefinedastochasticdifferentialequation(SDE)based
onaWienerprocess.HedefinedalsoanequationbasedonaWienerprocessanda
Poissonrandommeasure.Inthismonograph,wewishtopresentamoderntreatment
ofSDEanddiffusionandjump-diffusionprocesses.Inthefirstpart,wewillshow
that solutions of SDE will define stochastic flows of diffeomorphisms. Then, we
discuss the relation between stochastic flows and heat equations. Finally, we will
investigatefundamentalsolutionsoftheseheatequations(heatkernels),throughthe
studyoftheMalliavincalculus.
It seems to be traditional that diffusion processes and jump processes are
discussed separately. For the study of the diffusion process, theory of partial
differentialequationsisoftenused,andthisfacthasattractedalotofattention.On
theotherhand,thestudyofjumpprocesseshasnotdevelopedrapidly.Onereason
might be that for the study of jump processes, we could not find effective tools in
analysis such as the partial differential equation in diffusion processes. However,
recently, the Malliavin calculus for Poisson random measure has been developed,
andwecanapplyittosomeinterestingproblemsofjumpprocesses.
A purpose of this monograph is that we present these two theories simultane-
ously. In each chapter, we start from continuous processes and then proceed to
processes with jumps. In the first half of the monograph, we present the theory
of diffusion processes and jump-diffusion processes on Euclidean spaces based
on SDEs. The basic tools are Itô’s stochastic calculus. In Chap.3, we show that
solutions of these SDEs define stochastic flows of diffeomorphisms. Then in
Chap.4,relationsbetweenadiffusion(orjump-diffusion)andaheatequation(ora
heatequationassociatedwithintegro-differentialequation)willbestudiedthrough
propertiesofstochasticflow.
vii
viii Preface
In the latter half of the monograph, we will study the Malliavin calculus on
the Wiener space and that on the space of Poisson random measure. These two
typesofcalculusarequitedifferentindetail,buttheyhavesomeinterestingthings
in common. These will be discussed in Chap.5. Then in Chap.6, we will apply
the Malliavin calculus to diffusions and jump-diffusions determined by SDEs. We
will obtain smooth densities for transition functions of various types of diffusions
andjump-diffusions.Further,weshowthatthesedensityfunctionsarefundamental
solutionsforvarioustypesofheatequationsandbackwardheatequations;thus,we
construct fundamental solutions for heat equations and backward heat equations,
independent of the theory of partial differential equations. Finally, SDEs on
subdomains of a Euclidean space and those on manifolds will be discussed at the
endofChaps.6and7.
Acknowledgements Most part of this book was written when the author was working on
Malliavin calculus for jump processes jointly with Masaaki Tsuchiya and Yasushi Ishikawa in
2014–2017.Discussionswiththemhelpedmegreatlytomakeandrectifysomecomplicatednorm
estimations,whichcannotbeavoidedforgettingthesmoothdensity.ThecontentsofSects.5.5,
5.6,and5.7overlapwiththejointworkwiththem[46].Further,IshikawareadChap.7carefully
andgavemeusefuladvice.Iwishtoexpressmythankstothemboth.Iwouldalsoliketothank
the anonymous referees and a reviewer, who gave me valuable advices for improving the draft
manuscript.Finally,itismypleasuretothankMasayukiNakamura,editoratSpringer,whohelped
megreatlytowardthepublicationofthisbook.
Fukuoka,Japan HiroshiKunita
Introduction
Stochastic differential equations (SDEs) based on Wiener processes have been
studied extensively, after Itô’s work in the 1940s. One purpose was to construct
a diffusion process satisfying the Kolmogorov equation. Results may be found in
monographsofStroock–Varadhan[109],Ikeda–Watanabe[41],Oksendal[90],and
Karatzas–Shreve[55].Later,thegeometricpropertyofsolutionswasstudied.Itwas
shownthatsolutionsofanSDEbasedonaWienerprocessdefineastochasticflow
ofdiffeomorphisms[59].
In 1978, Malliavin [77] introduced an infinite-dimensional differential calculus
on a Wiener space. The theory had an interesting application to solutions of
SDEs based on the Wiener process. He applied the theory to the regularity of the
heat kernel for hypo-elliptic differential operators. Then, the Malliavin calculus
developed rapidly. Contributions were made by Bismut [9], Kusuoka–Stroock
[69,71],Ikeda–Watanabe[40,41],Watanabe[116,117],andmanyothers.
At the same period, the Malliavin calculus for jump processes was studied in
parallel(seeBismut[10],Bichteler–Gravereau–Jacod[7],andLeandre[74]).Later,
Picard[92]proposedanotherapproachtotheMalliavincalculusforjumpprocesses.
InsteadoftheWienerspace,hedevelopedthetheoryonthespaceofPoissonrandom
measure and got a smooth density for the law of a “nondegenerate” jump Markov
process.Then,Ishikawa–Kunita[45]combinedthesetwotheoriesandgotasmooth
densityforthelawofanondegeneratejump-diffusion.Thus,theMalliavincalculus
canbeappliedtoalargeclassofSDEs.
Inthismonograph,wewillstudytwotypesofSDEsdefinedonEuclideanspace
and manifolds. One is a continuous SDE based on a Wiener process and smooth
coefficients. We will define the SDE by means of Fisk–Stratonovich symmetric
integrals, since its solution has nice geometric properties. The other is an SDE
with jumps based on the Wiener process and the Poisson random measure, where
coefficientsforthecontinuouspartaresmoothvectorfieldsandcoefficientsforjump
parts are diffeomorphic maps. These two SDEs are our basic objects of study. We
wanttoshowthatbothoftheseSDEsgeneratestochasticflowsofdiffeomorphisms
andthesestochasticflowsdefinediffusionprocessesandjump-diffusionprocesses.
In the course of the argument, we will often consider backward processes, i.e.,
ix
x Introduction
stochastic processes describing the backward time evolution. It will be shown
that inverse maps of a stochastic flow (called the inverse flow) define a backward
Markov process. Further, we show that the dual process is a backward Markov
process and it can be defined directly by the inverse flow through an exponential
transformation.Ineachchapter,wewillstartwithtopicsrelatedtoWienerprocesses
andthenproceedtothoserelatedtoPoissonrandommeasures.Investigatingthese
two subjects together, we can understand both the Wiener processes and Poisson
randommeasuresmorestrongly.
Chapters1and2arepreliminaries.InChap.1,weproposeamethodofstudying
the smooth density of a given distribution, through its characteristic function
(Fouriertransform);itwillbeappliedtothedensityproblemofinfinitelydivisible
distributions. Then, we introduce some basic stochastic processes and backward
stochastic processes. These include Wiener processes, Poisson random measures,
martingales,andMarkovprocesses.InChap.2,wediscussstochasticintegrals.Itô
integrals and Fisk–Stratonovich symmetric integrals based on continuous martin-
gales and Wiener processes are defined, and Itô’s formulas are presented. Then,
we define stochastic integrals based on (compensated) Poisson random measures.
Further, we will give Lp-estimates of these stochastic integrals; these estimates
will be used in Chaps.3 and 6 for checking that stochastic flows have some nice
properties.Thebackwardstochasticintegralswillalsobediscussed.
InChap.3,wewillrevisitSDEsandstochasticflows,whichwerediscussedby
theauthor[59,60].AcontinuousSDEond-dimensionalEuclideanspaceRd based
onad(cid:2)-dimensionalWienerprocess(W1,...,Wd(cid:2))isgivenby
t t
(cid:2)d(cid:2)
dX = V (X ,t)◦dWk +V (X ,t)dt, (1)
t k t t 0 t
k=1
where ◦dWk denotes the symmetric integral based on the Wiener process Wk. If
t t
coefficientsV (x,t),k =0,...,d areofC∞,1-class,itisknownthatthefamilyof
k b
solutions {Xx,s,s < t} of the SDE, starting from x at time s, have a modification
t
{Φ (x),s <t},whichiscontinuousins,t,x andsatisfies(a)Φ :Rd →Rd are
s,t s,t
C∞-diffeomorphisms,(b)Φ = Φ ◦Φ foranys < t <ualmostsurely,and
s,u t,u s,t
(c) Φ and Φ are independent. {Φ } is called a continuous stochastic flow of
s,t t,u s,t
diffeomorphismsdefinedbytheSDE.
A similar problem was studied for SDE based on the Wiener process and the
Poissonrandommeasure.LetN(dtdz)beaPoissonrandommeasureonthespace
U = [0,T]×(Rd(cid:2) \{0})withtheintensitymeasuren(dtdz) = dtν(dz),whereν
isaLévymeasurehavinga“weakdrift.”WeconsideranSDEdrivenbyaWiener
processandPoissonrandommeasure:
(cid:2)d(cid:2) (cid:3)
dXt = Vk(Xt,t)◦dWtk +V0(Xt,t)dt + (φt,z(Xt−)−Xt−)N(dtdz),
k=1 |z|>0+
(2)
