Sharp Large Deviation for the Energy of -Brownian Bridge

We study the sharp large deviation for the energy of -Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure. 1. Introduction We consider the following -Brownian bridge: where is a standard Brownian motion, , , and the constant . Let denote the probability distribution of the solution of (1). The -Brownian bridge is first used to study the arbitrage profit associated with a given future contract in the absence of transaction costs by Brennan and Schwartz [1]. -Brownian bridge is a time inhomogeneous diffusion process which has been studied by Barczy and Pap [2, 3], Jiang and Zhao [4], and Zhao and Liu [5]. They studied the central limit theorem and the large deviations for parameter estimators and hypothesis testing problem of -Brownian bridge. While the large deviation is not so helpful in some statistics problems since it only gives a logarithmic equivalent for the deviation probability, Bahadur and Ranga Rao [6] overcame this difficulty by the sharp large deviation principle for the empirical mean. Recently, the sharp large deviation principle is widely used in the study of Gaussian quadratic forms, Ornstein-Uhlenbeck model, and fractional Ornstein-Uhlenbeck (cf. Bercu and Rouault [7], Bercu et al. [8], and Bercu et al. [9, 10]). In this paper we consider the sharp large deviation principle (SLDP) of energy , where Our main results are the following. Theorem 1. Let be the process given by the stochastic differential equation (1). Then satisfies the large deviation principle with speed and good rate function defined by the following: where . Theorem 2. satisfies SLDP; that is, for any , there exists a sequence such that, for any , when approaches enough, where The coefficients may be explicitly computed as functions of the derivatives of and (defined in Lemma 3) at point . For example, is given by with , and . 2. Large Deviation for Energy Given , we first consider the following logarithmic moment generating function of ; that is, And let be the effective domain of . By the same method as in Zhao and Liu [5], we have the following lemma. Lemma 3. Let be the effective domain of the limit of ; then for all , one has with where and . Furthermore, the remainder satisfies Proof. By It？’s formula and Girsanov’s formula (see Jacob and Shiryaev [11]), for all and , Therefore, If , we can choose such that . Then where , and . Therefore, Proof of Theorem 1. From Lemma 3, we have and is steep; by the G？rtner-Ellis theorem (Dembo and Zeitouni [12]), satisfies the large deviation principle with speed and good