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Probability Theory and Stochastic Modelling 92 Hiroshi Kunita Stochastic Flows and Jump-Diffusions Probability Theory and Stochastic Modelling Volume 92 Editors-in-chief PeterW.Glynn,Stanford,CA,USA AndreasE.Kyprianou,Bath,UK YvesLeJan,Orsay,France AdvisoryBoard SørenAsmussen,Aarhus,Denmark MartinHairer,Coventry,UK PeterJagers,Gothenburg,Sweden IoannisKaratzas,NewYork,NY,USA FrankP.Kelly,Cambridge,UK BerntØksendal,Oslo,Norway GeorgePapanicolaou,Stanford,CA,USA EtiennePardoux,Marseille,France EdwinPerkins,Vancouver,Canada HalilMeteSoner,Zürich,Switzerland TheProbabilityTheoryandStochasticModellingseriesisamergerandcontin- uationofSpringer’stwowellestablishedseriesStochasticModellingandApplied ProbabilityandProbabilityandItsApplicationsseries.Itpublishesresearchmono- graphsthatmakeasignificantcontributiontoprobabilitytheoryoranapplications domain in which advanced probability methods are fundamental. Books in this seriesareexpectedtofollowrigorousmathematicalstandards,whilealsodisplaying the expository quality necessary to make them useful and accessible to advanced studentsaswellasresearchers.Theseriescoversallaspectsofmodernprobability theoryincluding (cid:129) Gaussianprocesses (cid:129) Markovprocesses (cid:129) Randomfields,pointprocessesandrandomsets (cid:129) Randommatrices (cid:129) Statisticalmechanicsandrandommedia (cid:129) Stochasticanalysis aswellasapplicationsthatinclude(butarenotrestrictedto): (cid:129) Branchingprocessesandothermodelsofpopulationgrowth (cid:129) Communicationsandprocessingnetworks (cid:129) Computationalmethodsinprobabilityandstochasticprocesses,includingsimu- lation (cid:129) Geneticsandotherstochasticmodelsinbiologyandthelifesciences (cid:129) Informationtheory,signalprocessing,andimagesynthesis (cid:129) Mathematicaleconomicsandfinance (cid:129) Statisticalmethods(e.g.empiricalprocesses,MCMC) (cid:129) Statisticsforstochasticprocesses (cid:129) Stochasticcontrol (cid:129) Stochasticmodelsinoperationsresearchandstochasticoptimization (cid:129) Stochasticmodelsinthephysicalsciences Moreinformationaboutthisseriesathttp://www.springer.com/series/13205 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 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Thisbookisdedicatedtomyfamily 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 martingales 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)