#### Abstract

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 Itô’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 Gärtner-Ellis theorem (Dembo and Zeitouni [12]), satisfies the large deviation principle with speed and good rate function defined by the following:

*Remark 4. *Theorem 1 can also be obtained by using Theorem 1 in Zhao and Liu [5].

#### 3. Sharp Large Deviation for Energy

For , let Then where is the expectation after the change of measure

By Lemma 3, we have the following expression of .

Lemma 5. *For all , when approaches enough,
**
For , one gets the following.*

Lemma 6. *For all , the distribution of under converges to distribution. Furthermore, there exists a sequence such that, for any when approaches enough,
*

*Proof of Theorem 2. *The theorem follows from Lemma 5 and Lemma 6.

It only remains to prove Lemma 6. Let be the characteristic function of under ; then we have the following.

Lemma 7. *Whenapproaches,belongs to and, for all,
**
Moreover,
**
with
**
where
**
for some positive constant, andis some positive constant.*

*Proof. *For any,
By the same method as in the proof of Lemma 2.2 in [7] by Bercu and Rouault, there exist two positive constantsandsuch that
therefore, belongs to, and by Parseval’s formula, for some positive constant, let
we get
whereis some positive constant.

*Proof of Lemma 6. *By Lemma 3, we have
Noting that, and
for any by Taylor expansion, we obtain
therefore, there exist integers, and a sequenceindependent of ; whenapproaches, we get
whereis uniform as soon as.

Finally, we get the proof of Lemma 6 by Lemma 7 together with standard calculations on thedistribution.

#### Acknowledgment

This research was supported by the National Natural Science of Tianyuan Foundation under Grant 11226202.