Springer Uncertainty Research Kai Yao Uncertain Differential Equations Springer Uncertainty Research Springer Uncertainty Research Springer Uncertainty Research is a book series that seeks to publish high quality monographs, texts, and edited volumes on a wide range of topics in both funda- mental and applied research of uncertainty. New publications are always solicited. This book series provides rapid publication with aworld-widedistribution. Editor-in-Chief Baoding Liu Department of Mathematical Sciences Tsinghua University Beijing 100084, China http://orsc.edu.cn/liu Email: [email protected] Executive Editor-in-Chief Kai Yao School of Economics and Management University of Chinese Academy of Sciences Beijing 100190, China http://orsc.edu.cn/yao Email: [email protected] More information about this series at http://www.springer.com/series/13425 Kai Yao Uncertain Differential Equations 123 KaiYao Schoolof Economics andManagement UniversityofChineseAcademyofSciences Beijing China ISSN 2199-3807 ISSN 2199-3815 (electronic) SpringerUncertainty Research ISBN978-3-662-52727-6 ISBN978-3-662-52729-0 (eBook) DOI 10.1007/978-3-662-52729-0 LibraryofCongressControlNumber:2016941292 ©Springer-VerlagBerlinHeidelberg2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbHBerlinHeidelberg To my parents Yuesheng Yao Xiuying Zhang Preface Uncertainty theoryisabranch of mathematicsfor modelingbelief degrees.Within the framework of uncertainty theory, uncertain variable is used to represent quantitieswithuncertainty,anduncertainprocessisusedtomodeltheevolutionof uncertain quantities. Uncertain differential equation is a type of differential equa- tions involving uncertain processes. Since it was proposed in 2008, uncertain dif- ferentialequationhasbeensubsequentlystudiedbymanyresearchers.Sofar,ithas become the main tool to deal with dynamic uncertain systems. Uncertain Variable Uncertain measure is used to quantify the belief degree that an uncertain event is supposedtooccur,anduncertainvariableisusedtorepresentquantitieswithhuman uncertainty. Chapter 2 is devoted to uncertain measure, uncertain variable, uncer- taintydistribution,inverseuncertaintydistribution,operationallaw,expectedvalue, and variance. Uncertain Process Uncertain process is essentially a sequence of uncertain variables indexed by the time. Chapter 3 introduces some basic concepts about an uncertain process, including uncertainty distribution, extreme value, and time integral. vii viii Preface Contour Process Contourprocessisatypeofuncertainprocesseswithsomespecialstructuressothat its main properties are determined by a spectrum of its sample paths. Solutions of uncertain differential equations are the most frequently used contour processes. Chapter 4 is devoted to such processes and proves the set of contour processes is closed under the extreme value operator, time integral operator, and monotone function operator. Uncertain Calculus Uncertain calculus deals with the differentiation and integration of uncertain pro- cesses. Chapter 5 introduces the Liu process, the Liu integral, the fundamental theorem, and integration by parts. Uncertain Differential Equation Uncertaindifferentialequationisatypeofdifferentialequationsinvolvinguncertain processes.Chapter6isdevotedtotheuncertaindifferentialequationsdrivenbythe Liu processes. It discusses some analytic methods and numerical methods for solving uncertain differential equations. In addition, the existence and uniqueness theorem,andstabilitytheoremsonthesolutionofanuncertaindifferentialequation are also covered. For application, it introduces two stock models and derives their option pricing formulas as well. Uncertain Calculus with Renewal Process Renewal process is a type of discontinuous uncertain processes, which is used to record the number of renewals of an uncertain system. Chapter 7 is devoted to uncertain calculus with respect to renewal process. It introduces the renewal pro- cess,theYaointegralandtheYaoprocess,includingthefundamentaltheoremand integration by parts. Preface ix Uncertain Differential Equation with Jumps Uncertain differential equation with jumps is essentially a type of differential equationsdrivenbyboththeLiuprocessesandtherenewalprocesses.Chapter8is devoted to uncertain differential equation with jumps, including the existence and uniqueness,andstabilityofitssolution.Italsointroducesastockmodelwithjumps and derives its option pricing formulas for application purpose. Multi-Dimensional Uncertain Differential Equation Multi-dimensional uncertain differential equation is a system of uncertain differ- ential equations. Chapter 9 introduces multi-dimensional Liu process, multi-dimensional uncertain calculus, and multi-dimensional uncertain differential equation. High-Order Uncertain Differential Equation High-order uncertain differential equation is a type of differential equations involving the high-order derivatives of uncertain processes. Chapter 10 is devoted tohigh-orderuncertaindifferentialequationsdrivenbytheLiuprocesses.Itgivesa numerical method for solving high-order uncertain differential equations. In addi- tion,theexistenceanduniquenesstheoremonthesolutionofahigh-orderuncertain differential equation is also covered. Uncertainty Theory Online If you would like to read more papers related to uncertain differential equations, please visit the Web site at http://orsc.edu.cn/online. Purpose Thepurposeofthisbookwastoprovideatoolforhandlingdynamicsystemswith humanuncertainty.Thebookissuitableforresearchers,engineers, andstudentsin the field of mathematics, information science, operations research, industrial engineering, economics, finance, and management science. x Preface Acknowledgment ThisworkwassupportedinpartbyNationalNaturalScience FoundationofChina (GrantNo.61403360).IwouldliketoexpressmysinceregratitudetoProf.Baoding Liu of Tsinghua University for his rigorous supervision. My sincere thanks also go to Prof. Jinwu Gao of Renmin University of China, Prof. Xiaowei Chen of NankaiUniveristy,Prof.RuiqingZhaoofTianjinUniversity,Prof.YuanguoZhuof Nanjing University of Science and Technology, and Prof. Jin Peng of Huanggang NormalUniversity.Iamalsodeeplygratefultomywife,MeixiaWang,forherlove and support. Beijing Kai Yao February 2016