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