Some Large Deviation Results Under Nonlinear Expectations And Their Applications In Risk Models

In probability theory,the theory of large deviation is a branch with great achievements in recent years.It focuses on the asymptotic behavior of tail events of probability distribution series,which belongs to a part of limit theory in probability theory.Its basic ideas can be traced back to Khintchine(1929)[1],Cramer(1938)[2]and Chernoff(1952)[3].The initial problem is about the convergence of random sums of independent identically distributed random variables with mean value of 0 and variance of 1.The convergence results can be obtained by the law of large numbers and the central limit theorem,but we are concerned with the more precise convergence problem.We are concerned about the rate of convergence and whether we can give the expression of the corresponding rate function.Developed from such a series of problems,the theory of large deviation has become more perfect nowadays.(Ref.S.R.S Varadhan(1984)[68],D.W.Strook(1984)[69],Deuschel J.D.?Stroock(1989)[65],A.Bucklew(1990)[40],A.Dembo,O.Zeitouni(1998)[4]etc.).It is precisely because the large deviation studies how small the probability of rare events is,and can give the convergence rate function of such small probabili-ty events.Therefore,under the realistic background,the theory large deviation is widely studied and applied.Nowadays,one of the main applications of large de-viation theory is risk theory.Risk itself is a small probability event.Risk is the basis of insurance.Insurance uses the principle of leverage to counteract unknown risks by continuously collecting premiums.Risk theory is based on probability and statistics.It mainly describes the loss risk and operation risk in insurance operation quantitatively,establishes relevant models,studies the nature of these risk models,carries out the most effective risk analysis and risk control for insurance operation in reality,and provides guarantee and technical support for them.From the classi-cal risk model,we can see that the claim amount process can be expressed as the sum of random variables.In this way,we can apply a powerful tool,the theory of large deviation,to solve the limit convergence problem of the claim amount process.So far,many scholars have discussed on this aspect.For example,the application of classical precise large deviations in heavy-tailed distributions begins with the literature Asmussen.S.and Kluppelberg.C.(1996)[82].Scholars gradually apply it to various risk models and various stochastic processes,and get corresponding large deviation results.For example,the large deviation principle of Levy process is given in literature DE ACOSTA.A.(1994)[93].Embrechts.P.,Kluppelberg.C.and Mikosch.T.(1997)[36]gives the large deviation principle of classical Poisson risk process.Claudio Macci(2005)[83]gives the large deviation principle of compound Markov renewal process.Ref.[59][92][91][74][90]etc.There are many applications description.Since the thirteenth century,different forms of financial crises have begun to take place globally.The global financial crisis broke out in 2007,which once again revealed the complexity of the financial system.Therefore,it is of far-reaching theo-retical and technical significance to develop this quantitative analysis of uncertainty risk.Nonlinear expectation theory is a kind of science that helps us to study uncer-tainty.Since 2005,Academician Shige Peng has gradually established and perfected the theory of nonlinear expectation.Large deviation theory is precision of the law of large number,with the continuous improvement of the non-linear expectation and the non-linear probability theory,how to express it and how to apply it,this poses a new challenge to us.This is also the starting point of this paper.In fact,compared with the classical large deviation theory,the research results on large deviation the-ory and its related applications under nonlinear expectations are poor.At present,under sublinear expectations,the upper bound of Cramer theorem is derived from Feng Hu(2010)[9].Gao Fuqing(2010)[5]applies the polishing estimation method and the standard sub-additive method to obtain the large deviation principle for stochastic differential equations driven by G-Brownian motion.Zengjing Chen and Jie Xiong(2012)[8]obtained the large deviation principle of diffusion process under sublinear expectations.Under the new probability framework of nonlinear expec-tations,this paper continues to find suitable methods to study the large deviation theory and its application in financial risk model.The main structure and contents of this paper are introduced below.In Chapter 1,the background and current situation of this study are elaborated,and the main contents of this paper are briefly described.In Chapter 2,we first,introduce some definitions and properties of sublinear expectations and some results about large deviations under sublinear expectations.Next,the upper bound of large deviations in compact set is proved by applying the finite covering theorem.Then the exponential tightness in large deviation is applied to obtain the upper bound of large deviation for any closed set.We call it Gartner-Ellis theorem under sublinear expectations.This theorem is a generalization of Cramer theorem,which considers the large deviations of the sum of independent random variables.Finally,the classical insurance model is introduced,and the main conclusions of this chapter are applied to the classical insurance model to obtain a class of large deviation upper bounds for the compound Poisson process.More important,the concrete expression of its rate function is obtained.In this way,according to the different distribution of claim amount,concrete results can be calculated.This is a major innovation of this paper.This application has not been done so far,which opens up a new prospect for the application of large deviation theory in risk models under sublinear expectations.In Chapter 3,we give the asymptotic relation of large deviations between the counting process and its inverse process under sublinear expectations.Notice that there is a very important equivalence between the counting process and its inverse process,?Tn?t? and only if {N(t)>n}.By proving that both processes satis-fy Gartner-Ellis conditions under sublinear expectations,we can obtain that both N(t)and {Tn} satisfy the large deviation principle,and the rate functions of both processes are also related.As an application of this result,we also give an expres-sion of the corresponding rate function for the inverse process of G-Poisson process under sublinear expectations.This expression depends only on the bounds of in-tensity for G-Poisson process,which is easier to achieve in practice.This is a good application of the result of this chapter.This is the first study of large deviations in other risk models under sublinear expectations.Furthermore,we can also obtain the large deviation results of the renewal process under the sublinear expectations and the expression of its rate function.This is also a small breakthrough in the large deviation of stochastic processes under sublinear expectations.In Chapter 4,we consider a risk model with delay claims.By using Gartner-Ellis theorem under sublinear expectation and its solution to the large deviation problem of compound Poisson process in classical risk model,the large deviation problem of the new model is solved by exponential equivalence,and the expression of the specific rate function is obtained,which is very meaningful for the actual insurance operation and monitoring.This is the first study of large deviations in other risk models under sublinear expectations.In Chapter 5,we note that G-Poisson processes also play an important role in risk models under sublinear expectations.Combined with the important application of Chapter 2 above,i.e.the large deviation result representation of the correspond-ing compound G-Poisson process,we further generalize the compound Poisson risk model and obtain two new risk models.Under the generalized compound Poisson risk model,the upper bound of large deviation for its total claim process is also derived.In addition,the reinsurance risk model is also discussed,two kinds of rein-surance payment methods are given in the same model.Under the reinsurance risk model of this double insurance,the expression of the large deviation rate function is also obtained.This is the main content of this paper.Observing carefully,this paper mainly focuses on the large deviation of the sum of independent random variables under sublinear expectation.After obtaining the corresponding Gaxtner-Ellis theorem un-der sublinear expectations,the application of this theorem to compound Poisson processes is decisive.This has laid the foundation for the large deviations on var-ious types of insurance risk model under sublinear expectations.In addition,the asymptotic relationship between the counting process and its inverse process under sublinear expectations is also proved by Gartner-Ellis condition.It can be said that these aspects are brand new.I believe that this paper will play a role in attracting more and more good results of large deviations will be obtained.