Limit Theory Of Probability And Probability Inequalities Under Sublinear Expectation

Driven by need of financial risk and insurance, professor Shige Peng of Shandong university, who is also one academician of the Chinese Academy of Sciences, had raised the notion of sublinear expectation creatively, and gave complete axiom system. The axiom system of sublinear expectation make up deficiency of classical probability space in the application of financial field. With the developing of many years, theory of sublinear expectation has been accepted by most of the probability of scholars, and being getting noticed by more and more experts from other fields. The theory of sublinear expectation is now studied deeper and wider.LDP theory is one of very important branch of theory of advanced probability, its main intention is to characterizing the limit of probability of type of exponent, LDP is much more precise than Law of Large Numbers. With the developing of last decades, LDP becomes a very hot research direction of probability.It can be said that convergence is loyal friend of limit of probability, without convergence, the research on limit of probability would lose meaning. Different type of convergence depicts different property the limit of probability. Convergence is a powerful tool in probability especially in limit of probability. Convergence in case of sublinear expectation is also a very important tool.Probability inequality is also a very pivotal role in theory of probability. It’s even more importance than quality in some case. With correctly using probability inequality in proving study, we can get our result more convenient.In this paper, the main target is to explore the theory of probability limit (large deviation theory and convergence of r.v. sequence); and the theory of probability inequalities in the case of sublinear expectation.In the part of the theory of probability, we prove "Varadhan Integral Lemma" and "Bryc’s Inverse Varadhan Lemma " in the case of sublinear expectation. Besides, in the first part of chapter 5, we give definition of uniformly integrable in case of sublinear expectation and prove the theorem for determination of uniformly integrable in the same case. In the second part we gaive three notations of convergence(q.s., in choquet,  convergence), then we also give the proof of derivation of the relationship for q.s. convergence, uniformly integrable and in probability convergence. In the part of probability inequalities, we prove 2 probability inequalities in the case of sublinear expectation.