Uncertain Calculus With Uncertain Renewal Process

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.