Research On Some Limit Theorems And Programming Principles Under Uncertainty Theory

In real situations,there is a class of uncertain information described by human language.When the sample size is large enough,probability theory provides a tool to deal with such information.However,sometimes few or no samples are available.Consequently,uncertainty theory was pioneered by Professor Liu Baoding to model human uncertainty information more reasonably.Except for human uncertainty,financial uncertainty is another source of indeterminacy.With the rapid development of science and technology,the financial risks we face are becoming more and more complex and dynamic,the traditional Choquet expectation is no longer applicable to deal with this type of financial risk.For this purpose,sublinear expectation theory was established by Academician Peng Shige and is becoming a robust analytical and computational tool for studying the dynamic financial risk problems under Knight uncertainty.For a long time,there is a class of complex systems that contain not only random variables with sublinear characteristics but also uncertain variables,such as investment activities in incomplete financial markets influenced by government macro-regulation,redundant design of logistics systems,etc.In order to deal with such complex systems,reference[8]proposed a U-S chance theory based on uncertainty theory and sublinear expectation theory.This paper can be divided into two parts.In part 1,the law of large numbers,the central limit theorem,and the law of the iterated logarithm for Bernoulli uncertain sequences are proved within the framework of uncertainty theory.In part 2,uncertain random programming is proposed within the framework of the U-S chance theory.To obtain the above results,this thesis first proposes several new notions such as weakly dependent,Bernoulli uncertain sequence,and continuity of uncertain measure.Then the limit theorems of uncertain variables are established by exploring the relationship between uncertain measure and probability measure.Secondly,under the framework of U-S chance theory,the operational law of uncertain random variables is proposed,and a new way of defining the expectation of uncertain random variables is given based on the sublinear expectation and Choquet integral,respectively.In addition,the properties and algorithms of expectations are studied.Finally,this thesis provides four uncertain random programming models and applies them to investment issues in an incomplete financial market and a system reliability design problem.