Uncertain Calculus With Finite Variation Processes

An uncertain process is essentially a sequence of uncertain variables indexed bytime or space. Uncertain calculus is a branch of mathematics which deals with integra-tion and diferentiation of functions of uncertain processes. There is an important typeof uncertain process whose sample paths are functions with finite variation, knownas the finite variation process (including canonical process, renewal process and sta-tionary independent increment process). In this thesis, the goal is to study uncertaincalculus based on the finite variation process. We use the uncertain integral to inte-grate an uncertain process with respect to the finite variation process. The uncertainintegral has the properties of continuity and linearity. After giving the definition ofintegrability for uncertain process, the sufcient conditions for existence of uncertainintegral are proposed. In the framework of uncertain integral, the fundamental theoremof diferentiation of function of uncertain processes is proved. In addition, the integra-tion by parts formula and the formula for change of variables are derived. In order tobetter study the properties of uncertain calculus, the uncertain integral is extended tohigher dimensional space, and the high-dimensional case of the fundamental theoremis proved.Based on the uncertain calculus, we study the diferential equation driven by finitevariation uncertain processes whose solution is also an uncertain process. Using thefundamental theorem in uncertain calculus, some special uncertain diferential equa-tions are solved. As diferential equations, most of the equations are very difcult tosolve. In order to better use uncertain diferential equation to solve practical problems,existence and uniqueness theorems for uncertain diferential equations are proved. Inour practice, stability of the uncertain diferential equation is often considered, and thusthe stability concept for uncertain diferential equation is introduced and conditions forstability of linear uncertain diferential equations are proposed. Our new contributionsin this thesis are: The definition of the uncertain integral of an uncertain process with respect toa finite variation uncertain process is introduced. Sufcient condition for suchintegrability of an uncertain process is given and its mathematical properties isstudied;The definition of diferentiability about an uncertain process with respect to afinite variation uncertain process is introduced. The fundamental theorem todeal with diferentiation is proved. This uncertain integral is extended to higherdimensional spaces;The uncertain diferential equation driven by finite variation processes is intro-duced. Some methods to solve the analytic solution are given. Besides, exis-tence and uniqueness theorems for this type of uncertain diferential equationsare proved. Finally, the definition of the stability and the stability conditions areproposed.