【作者】 姚凯；

【导师】 刘宝碇；

【作者基本信息】 清华大学 ， 数学， 2013， 博士

【摘要】 不确定理论是处理人的信度的一个数学系统.不确定过程用来描述随时间变化的不确定现象,本质上它是一列不确定变量.有一类轨道不连续的不确定过程称为不确定更新过程,它用来描述一类变化过程不连续的不确定现象.在不确定更新过程的基础上,本论文提出了不确定交错更新过程用来描述交替工作和休息的不确定系统,计算了它的平均工作率的不确定分布,并证明了基本更新定理,完善了不确定更新理论.基于不确定更新过程,本论文建立了一套积分理论,用来处理不确定过程关于不确定更新过程的积分和微分.同时,本论文证明了这种积分满足关于被积项的线性性和关于积分区间的可加性,并给出了不确定更新过程的积分理论的基本定理.不确定更新过程的积分理论扩展了不确定积分理论的研究内容,为研究带跳不确定微分方程奠定了基础.为了描述不连续变化的不确定现象所服从的变化规律,本论文提出了由不确定更新过程驱动的微分方程,即带跳不确定微分方程.本论文给出了两类带跳不确定微分方程的解析解,并证明了带跳不确定微分方程的解的存在唯一性定理.此外,本论文还定义了带跳不确定微分方程在不确定测度意义下的稳定性,给出了带跳不确定微分方程稳定的充分条件.本论文为研究不确定环境下带跳的金融市场及最优控制等提供了理论依据,使不确定微分方程能够更好的指导实践.本文的创新点主要有：·定义了不确定交错更新过程,给出了平均工作率的不确定分布,证明了交错更新定理；·建立了不确定过程关于不确定更新过程的积分理论,研究了该积分的一些数学性质,并给出了它的基本定理；·定义了带跳不确定微分方程,给出了两类方程的解析解,并证明了方程的解的存在唯一性和稳定性定理.更多还原

【Abstract】 Uncertainty theory is a branch of axiomatic mathematics to deal with human’sbelief degree. Uncertain process, aiming to describe the evolution of uncertain phe-nomena, is essentially a sequence of uncertain variables. Uncertain renewal process isa type of sample-discontinuous uncertain process, and it is used to model discontinu-ously varying uncertain phenomena. Based on uncertain renewal process, this thesisproposes a new type of uncertain process called uncertain alternating renewal processto describe an uncertain system which works and rests alternately. The uncertainty dis-tribution of the average working rate is given, and the alternating renewal theorem isproved. As a result, the uncertain renewal theory is expanded.Based on uncertain renewal process, this thesis builds a new theory of uncertaincalculus to deal with the integral and diferential of an uncertain process with respectto uncertain renewal process. The integral is proved to meet with the linearity on theintegrand and the additivity on the bounds. In addition, the fundamental theorem of un-certain calculus with uncertain renewal process is verified, which gives the diferentialof a function of uncertain process with respect to uncertain renewal process. Uncertaincalculus with uncertain renewal process, which extends the area of uncertain calculustheory, is the basis to study uncertain diferential equation with jumps.In order to describe the rule that a discontinuously varying uncertain phenomenonobeys, this thesis proposes a type of diferential equation driven by uncertain renewalprocess, i.e., uncertain diferential equation with jumps. It gives analytic solutions fortwo types of uncertain diferential equations with jumps. Besides, it gives an existenceand uniqueness theorem for the proposed diferential equation. In addition, this thesisproposes a definition of stability in the sense of uncertain measure for uncertain difer-ential equation with jumps, and gives a sufcient condition for the diferential equationbeing stable. These results provide a theoretical basis for further research in many ar-eas such as uncertain financial market and uncertain optimal control with jumps. As a result, uncertain diferential equation will do a better job in practice.The contributions of this thesis are:It proposes a definition of uncertain alternating renewal process, and gives anuncertainty distribution of average working rate. Besides, it proves an alternatingrenewal theorem.It builds uncertain calculus with respect to uncertain renewal process. Someproperties of the integral are investigated, and the fundamental theorem is veri-fied.It introduces a concept of uncertain diferential equation with jumps, and givesanalytic solutions for two types of the proposed diferential equations. In ad-dition, it gives sufcient conditions for an uncertain diferential equation withjumps to have a unique solution and to be stable.更多还原