Some Results Of Large Deviation In Nonlinear Cases And Application In Finance

In probability theory, large deviation theory concerns the asymp-totic behavior of remote tails of sequences of probability distributions, so large deviation theory is a very fruitful branch of limit theory in probability theory, and it deals with limit problems which are dif-ferent from the central limit theorem and the law of large numbers. Generally speaking, large deviation theory is the precision of the law of large numbers, thus it is one of the most effective ways to gather information out of a probabilistic model. Some basic ideas of the large deviation theory can be tracked back to Khintchine, Cramer and Cher-noff, although a clear unified formal definition was introduced in 1966 by Varadhan who received the Abel prize for his fundamental contri-butions to probability theory and in particular for creating a unified theory of large deviation in 2007. Cramer's theorem is about the large deviations of the sample mean of a sequence of independent identically distributed random variables and Schilder's theorem gives an estimate for the probability that a (scaled-down) sample path of Brownian mo-tion will stray far from the mean path (which is constant with value 0). After this, Freidlin-Wentzell's theorem which generalizes Schilder's theorem for standard Brownian motion gives an estimate for the prob-ability that a (scaled-down) sample path of an Ito diffusion will stray far from the mean path. Donsker and Varadhan developed large de-viation problems for the empirical distributions of time homogeneous Markov chains.Large deviation theory provides a good method to calculus the probability of rare event, which will have great impact once it happens although its probability may be very small. Usually the probability of rare event derived from large deviation method can be expressed as a solution of variational problem. Therefore, large deviation theory is a powerful tool to resolve many problems that it is different to resolve by other methods and it has important applications in mathematical statistic, statistical mechanic, quantum mechanic, information theory and risk management.Regretfully, so far the classic large deviation theory is confined to the linear case such as linear probability. linear expectation and so on. However, with the development of the theory of nonlinear expecta-tion and nonlinear probability, it is necessary to study large deviation theory in the nonlinear case. For example, in risk pricing, in which the traditional mathematical expectation plays an important role, the price functions don't satisfy the linear property, which cause that the price of the sum of two risks is usually smaller than the sum of the price of the two risks, so the traditional mathematical expectation al-ways results in the problem which puzzled the economists many years such as the Allais paradox and the Ellesberg paradox in utility theory. Thus, with the purpose of more extensive applications of large devia- tion theory in nonlinear field, this paper presents some results of large deviation in the nonlinear cases.This paper will be organized as follow.Section 1:In subsection 1.1, we first give the definition of large de-viation and other related prior knowledge through a statistic problem and introduce the large deviation principle of Varadhan and Laplace principle which statements the equivalent description of the large de-viation principle. Next we introduce the preliminaries of SDE theory and state that the solution of the SDE with small noise satisfies the large deviation principle.In section 1.2, we introduce the the knowledge of forward-backward stochastic differential equation and related g-expectation, and finally give the existence and uniqueness theorem of the solution of FBSDE and an estimation about the solution.Section 2:In subsection 2.1, we introduce the preliminaries of capacity.In subsection 2.2. we give the definitions of the nonlinear expecta-tions and probabilities - the upper expectation and the lower expecta-tion as well as the upper probability and the lower probability which usually are called capacity. Furthermore, we study some properties of the upper probability and the lower probability. Then we get the central limit theorem under these capacities from which we obtain the large deviation principle for these capacities.Section 3:In subsection 3.1, we mainly prove the convergence of the solution of FBSDE with small noise.In subsection 3.2, we first introduce a variational representation. by which one can derive the standard large deviation principle for small noise diffusions as a new method. Inspired by this work, we use a new method to prove the large deviation principle of the solution of FBSDE with small noise applying this variational representation.In section 3.3, we introduce the knowledge of financial risk mea-surement, give an application of the large deviation principle of the solution of FBSDE with small noise in this field.