Sharp Large Deviation for the Energy of α-Brownian Bridge

where W is a standard Brownian motion, t ∈ [0, T), T ∈ (0,∞), and the constant α > 1/2. Let P α denote the probability distribution of the solution {X t , t ∈ [0, T)} of (1). The α-Brownian bridge is first used to study the arbitrage profit associatedwith a given future contract in the absence of transaction costs by Brennan and Schwartz [1]. α-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 α-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 OrnsteinUhlenbeck (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 S t , where


Introduction
We consider the following -Brownian bridge: where is a standard Brownian motion, ∈ [0, ), ∈ (0, ∞), and the constant > 1/2. Let denote the probability distribution of the solution { , ∈ [0, )} 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.

(5)
The coefficients , may be explicitly computed as functions of the derivatives of and (defined in Lemma 3) at point . For example, ,1 is given by with = ( ) ( ), and ℎ = ( ) ( ).

Large Deviation for Energy
Given > 1/2, 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.

Sharp Large Deviation for Energy
For > 1/(2 − 1), let Then where E is the expectation after the change of measure For , one gets the following.

Lemma 6.
For all > 1/(2 − 1), the distribution of under converges to (0, 1) distribution. Furthermore, there exists a sequence such that, for any > 0 when approaches enough, Proof of Theorem 2. The theorem follows from Lemma 5 and Lemma 6.
It only remains to prove Lemma 6. Let Φ (⋅) be the characteristic function of under ; then we have the following.

Lemma 7.
When approaches , Φ belongs to 2 (R) and, for all ∈ R, Moreover, for some positive constant , and is some positive constant.
Proof. For any ∈ R, By the same method as in the proof of Lemma 2.2 in [7] by Bercu and Rouault, there exist two positive constants and such that therefore, Φ (⋅) belongs to 2 (R), and by Parseval's formula, for some positive constant , let we get = : + , where is some positive constant.
for any > 0, by Taylor expansion, we obtain therefore, there exist integers ( ), ( ) and a sequence , independent of ; when approaches , we get where is uniform as soon as | | ≤ . Finally, we get the proof of Lemma 6 by Lemma 7 together with standard calculations on the (0, 1) distribution.