【作者】 刘微；

【导师】 张勇；

【作者基本信息】 吉林大学 ， 概率论与数理统计， 2022， 博士

【摘要】 随着科学社会的发展,为了克服金融统计学、数理经济学、风险度量和金融中超套期保值等方面的不确定性因素的影响,对非线性概率/期望理论的研究成为近来热门研究方向.而次线性期望就是一个较为一般却具有重要地位的非线性期望,本文主要研究次线性期望空间下线性过程的几个极限定理,在推广前人理论成果的同时,进一步完善和丰富次线性期望理论成果,希望能将其更加合理、广泛的运用到实际当中.首先,本文在第一章绪论部分简要介绍了次线性期望理论的发展进程和研究现状,然后介绍了次线性期望理论基础知识.其次,本文根据Peng框架(次线性期望理论)下的分布、独立性等新型概念,将严平稳序列的定义从经典概率空间推广到次线性期望空间中,进而重新定义了次线性期望下的线性过程.第二章研究次线性期望空间下由独立同分布随机变量序列生成的线性过程的中心极限定理和不变原理.第三章主要研究由次线性期望诱导的容度意义下平稳独立序列生成的线性过程的重对数律.第四章主要研究由一族概率测度取上确界所定义的上概率空间下平稳序列生成的线性过程的大偏差原理.最后总结了本文所做的主要工作及取得的成果,同时为后续的研究工作提出了一些问题和设想.更多还原

【Abstract】 In the 1930s,Kolmogorov proposed the axiomatic system of probability theory by means of measure theory,which laid the foundation for the development of modern probability theory.Since then,probability theory has developed rapidly and has been widely used in many fields,such as mathematical statistics,stochastic control,information science and economics.With the development of science and society,the application of classical limit theory is limited by the linear additive condition to some extent,so it can not reasonably explain and predict many uncertain phenomena.This makes the nonlinear probability/expectation theory deal with the problem of model uncertainty.For example,the advantages of financial statistics,mathematical economics,risk measurement and super hedging in finance are becoming increasingly prominent.Driven by these uncertainty modeling,many scholars at home and abroad have set off an upsurge of research on various nonlinear probability/expectation limit theories,which also stimulates our interest in further exploration and research.In the development of nonlinear probability/expectation theory,Choquet-capacity,Choquet-expectation,sub-linear expectation,upper and lower probabilities,upper and lower expectations and so on are representative and influential.In 1954,Choquet(1954)proposed the concept of capacity.Many important conclusions of classical probability theory have been generalized in the framework of tolerance(c.f.Wasserman and Kadane(1990),Marinacci and montrucchio(2004),Chen and kulberger(2006),etc.)Peng(2006,2008a,2009)introduced a more generalized sub-linear expectation theory,and used the sub-linear expectation space(Ω,×,E)to replace the probability space(Ω,F,P).As we know,a classical probability space can comprehensively and systematically obtain all the important conclusions in probability theory,probability statistics theory and stochastic analysis.In the sub-linear expectation space,the distribution,independence,associate and stationarity of random variables are all understood in a new way.Peng(2010,2019b),combined with the theory of partial differential equations,proposed the concepts and properties of G-normal distribution,maximal distribution,G-Brownian motion,G-ito stochastic calculus,G-martingale,etc.,and established a relatively complete theoretical system.Denis,Hu and Peng(2011),Peng(2007,2008a,2010)et al.proved the law of large numbers under sub-linear expectation,the central limit theorem,the stochastic integral theory driven by nonlinear Brownian motion and the corresponding stochastic analysis theory.It is a substantial extension of the classical Ito stochastic analysis theory,and its results are still valid under the original linear expectation space.Corresponding to the nonlinear expectation theory is the nonlinear probability(capacity)theory.The classical expectation and probabilitiy are uniquely determined by each other,but there is no one-to-one correspondence between them under the nonlinear expectation.The nonlinear probability can be uniquely determined by a given nonlinear expectation,but the reverse is not true.See Section 1 of Chapter 2 for detailed explanation.Therefore,the nonlinear expectation theory and the nonlinear probability theory are two related but different theoretical systems,and it becomes very necessary to study the nonlinear probability theory(c.f.Maccheroni and Marinacci(2005),Hu(2010),Chen and Hu(2014),Zhang(2021a),et al.Different from the study of single fixed probability in classical probability theory,many scholars have studied the limit theory of proba.bility on the definition of upper certainty of a family of probability measures(c.f.Gao and Xu(2010),Chen,Hu and Li(2013),Tan and Zong(2010),etc.).Interestingly,the upper probability is also a capacity and is inextricably related to the sub-linear expectation.For details,please refer to Section 1 of Chapter 3.Inspired by the theoretical achievements of the scholars above,this paper based on the relationship between the capacity,upper and lower expectation,upper and lower probability and sub-linear expectation.The aim is to further study the limit theorem of linear process under sub-linear expectation,popularize the results of predecessors,and further improve the results of sub-linear expectation theory,hoping to apply it more reasonably and widely in practice.Firstly,according to the new concepts of distribution and independence in Peng’s framework,we give the definition of strictly stationary sequence under sub-linear expectation.We further redefine the linear process under sub-linear expectations.Secondly,some results of this paper are given.In chapter 2,we study the central limit theorem and invariance principle of linear processes under sub-linear expectation spaces.Firstly,a key lemma is obtained by using linear process partial sum decomposition,Kronecker’s lemma,and Rosenthal’s inequality under sub-linear expectations.Then we use the properties of bounded Lipschitz function,sub-linear expected self-control and the improved Peng-central limit theorem by Zhang(2016b)And Donsker’s invariance principle of Zhang(2015)were obtained the central limit theorem and invariance principle for linear processes generated by independent identically distributed random variables under sub-linear expectations,This not only enriches the results of the limit theory of sub-linea.r expectation,but also extends the classical Kolmogorov’s central limit theorem and invariance principle.In chapter 3,we mainly study the law of iterated logarithm of linear processes induced by sub-linear expectation under capacity.First,a key lemma in the sense of sub-linear expectation are obtained by truncating the random variable,the relation between sub-linear expectation and Choquet-expectation,the definition of Choquetexpectation,the countable additivity of capacity and the exponential inequality under sub-linear expectation.Then,using the decomposition of the partial sum of linear processes,the transformation of Choquet-expectation and integral,and the Borel-Cantelli lemma,it is proved that the part and the tail of the linear process go to zero with respect to capacity.Finally,using the law of iterated logarithm of Zhang(2016b),we obtain the main result of this chapter:the law of iterated logarithm for the linear process generated by a stationary sequence under sub-linear expectation.The result enriches the theory of tolerance(non-additive probability)limit and is also a natural extjension of the law of iterated logarithm under the classical additive probability.In chapter 4,we study the large deviation principle for linear processes under upper probability space defined by a family of probability measures.Two preliminary lemmas are obtained by using the monotone convergence theorem under sub-linear expectations and H?lder inequality.Then by controlling the upper bound of Cramer functional,using the decomposition of the partial sum of linear processes,upper(lower)semicontinuous function properties,stability,And the upper bound lemma of Cramer by Gao and Jiang(2010)is used to obtain the upper bound of large deviation for stationary linear processes under sub-linear expectation.By using the triangle inequality,the capacity property,the Chebyshev’s inequality under sub-linear expectation and the large deviation principle of Gao and Xu(2011),the lower bound of the large deviation of the linear process of stationary sequence generation under sub-linear expectation is obtained.This part of the result is a generalization of the large deviation of upper probability.It also enriches the limit theory of upper probability space.At the end of the article,we summarize the main work and achievements of this paper,and puts forward some questions and assumptions for the follow-up research work.更多还原