Some Large Deviations Results Based On Nonlinear Probabilities And The Applications In Insurance And Finance

Laxge deviation theory, which deals with limit problems that are different from central limit theory, is a very fruitful branch of limit theory of probability theory, and it is the precision of the law of large number theory, and also has importantapplications in mathematical statistic, analysis and physics. The theory of large deviations is traced to Khintchine (1929) Cramer (1938) and Chernoff (1952). Cram(?)r's theorem is about the large deviations of the sample mean of a sequence of independent identically distributed random variables. After this, Ventcel and Preidlin theory on small random perturbations of ordinary differential equations by an It(?) type stochastic term is taken up. Donsker and Varadhan developed large deviation problems for the empirical distributions of time homogeneous Markov chains, and Varadhan receives the Abel prize "for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation" in 2007.The studies of Prof. Peng and Prof. Chen in the field of backward stochastic differential equations have provided us a new environment. With the development of the theory of nonlinear expectation and nonlinear probability, the application of the large deviation theory in this field is a new challenge for us. Chapter 1: This Chapter is about the prior knowledge and related discussions.Through discussing a statistic problem, we give the definitions of large deviationand moderate deviation. In turn the large deviation principle of Varadhan [9] is given.Definition 1.1.2 (Large Deviation Principle(Varadhan [9]))Let I be a rate function on X. The sequence {ξ_n} is said to satisfy the large deviation principle on X with rate function I if the following two conditions hold.(a)Large deviation upper bound. For each closed subset F of X(b)Large deviation lower bound. For each open subset G of XAt the same time, we give the definitions of g-expectation and g-probability.Definition 1.2.1 Assume that (H1),(H2) and (H3) hold on g andξ∈L²(Ω,F,P). Let (y_s,z_s) be the solution of BSDEξ_g[ξ] is called the g-expectation of the random variableξ, defined byξ_g[ξ|F_t] is called the conditional g-expectation of the random variableξ, defined byDefinition 1.2.3 Assume that (H1),(H2) and (H3) hold on g. Given A∈F, the g-probability of A is defined by Chapter 2: This chapter is my main results. The first part, discusses a special nonlinear probability.Given two linear probabilities P, Q, which satisfy the large deviation principlewith rate functions I₁(x),I₂(x) respectively. We consider the classical mean problem.where for each i,ξ_i,i=1,2,…are independent, identically distributed (i.i.d) random variables.For the sake of convenience, we supposeWe define the new probabilityμ=P∧Q, and consider the large deviation principle about the new nonlinear probabilityμ.Proposition 2.1.1 Letμ=P∧Q, where P, Q are two probabilities and satisfy the large deviation principle with rate functions I₁(x),I₂(x) respectively, thenμsatisfies the large deviation with the rate function I(x), whereThe second part the discussion on the large deviations of g-expectation. Firstly, we give the definitions of mutually independence and identically distribution of randomvariances under g-probability.Definition 2.2.1 The random variables X₁,X₂,…,X_n are strongly, mutuallyindependent under g-probability P_g, ifand Definition 2.2.2 The random variables X₁,X₂,…,X_n are strongly, identicallydistributed under g-probability P_g, ifandFinally, we getProposition 2.2.4 (Cram(?)r Theorem under g-Expectation)Let X₁,X₂,…,X_n be a sequence of random variables, such that for each i, i,X_i,i=1,2,…n,…, are independent, identically distributed under g-probability P_μ, whereμ＞0. Assume thatThenwherei. e. the rate function isChapter 3: This chapter is about some applications and methods of large deviations in insurance and finance, we introduce the large deviation problems in the insurance model, the Cram(?)r-Lundberg estimate and the insurance-finance model.