1. Introduction
We consider the following α-Brownian bridge:
(1)dXt=-αT-tXtdt+dWt, X0=0,
where W is a standard Brownian motion, t∈[0,T), T∈(0,∞), and the constant α>1/2. Let Pα denote the probability distribution of the solution {Xt,t∈[0,T)} of (1). The α-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].

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 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 St, where
(2)St=∫0tXs2(s-T)2ds.

Our main results are the following.

Theorem 1.
Let {Xt,t∈[0,T)} be the process given by the stochastic differential equation (1). Then {St/λt,t∈[0,T)} satisfies the large deviation principle with speed λt and good rate function I(·) defined by the following:
(3)I(x)={18x((2α0-1)x-1)2,if x>0;+∞,if x≤0,
where λt=log(T/(T-t)).

Theorem 2.
{
S
t
/
λ
t
,
t
∈
[
0
,
T
)
}
satisfies SLDP; that is, for any c>1/(2α-1), there exists a sequence bc,k such that, for any p>0, when t approaches T enough,
(4)P(St≥cλt)=exp{-I(c)λt+H(ac)}2πacβt×(1+∑k=1pbc,kλt+O(1λtp+1)),
where
(5)σc2=4c2, βt=σcλt,ac=(1-2α)2c2-18c2,H(ac)=-12log(1-(1-2α)c2).
The coefficients bc,k may be explicitly computed as functions of the derivatives of L and H (defined in Lemma 3) at point ac. For example, bc,1 is given by
(6)bc,1=1σc2(-h22-h122+l48σc2+l3h12σc2 -5l3224σc4+h1ac-l32acσc2-1ac2),
with lk=L(k)(ac), and hk=H(k)(ac).

2. Large Deviation for Energy
Given α>1/2, we first consider the following logarithmic moment generating function of St; that is,
(7)Lt(u):=log𝔼αexp{u∫0tXs2(s-T)2ds}, ∀λ∈ℝ.
And let
(8)𝒟Lt:={u∈ℝ, Lt(u)<+∞}
be the effective domain of Lt. By the same method as in Zhao and Liu [5], we have the following lemma.

Lemma 3.
Let 𝒟L be the effective domain of the limit L of Lt; then for all u∈𝒟L, one has
(9)Lt(u)λt=L(u)+H(u)λt+R(u)λt,
with
(10) L(u)=-1-2α-φ(u)4,H(λ)=-12log{12(1+h(u))},R(u)=-12log{1+1-h(u)1+h(u)exp{2φ(u)λt}},
where φ(u)=-(1-2α)2-8u and h(u)=(1-2α)/φ(u). Furthermore, the remainder R(u) satisfies
(11)R(u)=Ot→T (exp{2φ(u)λt}).

Proof.
By Itô’s formula and Girsanov’s formula (see Jacob and Shiryaev [11]), for all u∈𝒟L and t∈[0,T),
(12)logdPαdPβ|[0,t] =(α-β)∫0tXss-TdXs-α2-β22∫0tXs2(s-T)2ds,∫0tXss-TdXs =12(Xt2(t-T)+∫0tXs2(s-T)2ds-log(1-tT)).
Therefore,
(13)Lt(u)=log𝔼β(exp{u∫0tXs2(s-T)2ds}dPαdPβ|[0,t])=log𝔼βexp{Xs2(s-T)2α-β2(t-T)Xt2-α-β2log(1-tT) +12(β2-α2+α-β+2u) ×∫0tXs2(s-T)2ds}.
If 4u≤(1-2α)2, we can choose β such that (β-1/2)2-(α-1/2)2+2u=0. Then
(14)Lt(u)=-1-2α-φ(λ)4λt-12log{12(1+h(u))}-12log{1+1-h(u)1+h(u)exp{2φ(u)λt}},
where φ(u)=-(1-2α)2-8u, and h(u)=(1-2α)/φ(u). Therefore,
(15)Lt(u)λt=-1-2α-φ(u)4-12λtlog{12(1+h(u))} -12λtlog{1+1-h(u)1+h(u)exp{2φ(u)λt}}=L(u)+H(u)λt+R(u)λt.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.
From Lemma 3, we have
(16)L(u)=limt→TLt(u)λt=1-2α-φ(u)4,
and L(·) is steep; by the Gärtner-Ellis theorem (Dembo and Zeitouni [12]), St/λt satisfies the large deviation principle with speed λt and good rate function I(·) defined by the following:
(17)I(x)={18x((2α-1)x-1)2,if x>0;+∞,if +x≤0.

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

3. Sharp Large Deviation for Energy
For c>1/(2α-1), let
(18)ac=(1-2α)2c2-18c2, σc2=L′′(ac)=4c3,H(ac)=-12log(1-(1-2α)c).
Then
(19)P(St≥cλt) =∫St≥cλtexp{L(ac)-cacλt+cacλt-acSt}dQt =exp{L(ac)-cacλt}𝔼Qexp{-acβtUtI{Ut≥0}}=AtBt,
where 𝔼Q is the expectation after the change of measure
(20)dQtdP=exp{acSt-Lt(ac)},Ut=St-cλtβt, βt=σcλt.

By Lemma 3, we have the following expression of At.

Lemma 5.
For all c>1/(2α-1), when t approaches T enough,
(21)At=exp{-I(c)λt+H(ac)}(1+O((T-t)c)).
For Bt, one gets the following.

Lemma 6.
For all c>1/(2α-1), the distribution of Ut under Qt converges to N(0,1) distribution. Furthermore, there exists a sequence ψk such that, for any p>0 when t approaches T enough,
(22)Bt=1acσc2πλt(1+∑k=1pψkλtk+O(λt-(p+1))).

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>.
The theorem follows from Lemma 5 and Lemma 6.

It only remains to prove Lemma 6. Let Φt(·) be the characteristic function of Ut under Qt; then we have the following.

Lemma 7.
When t approaches T, Φt belongs to L2(ℝ) and, for all u∈ℝ,
(23)Φt(u)=exp{-iuλtcσc}×exp{(Lt(ac+iuβt) -Lt(ac))}.
Moreover,
(24)Bt=𝔼Qexp{-acβtUtI{Ut≥0}}=Ct+Dt,
with
(25)Ct=12πacβt∫|u|≤st(1+iuacβt)-1Φt(u) du,Dt=12πacβt∫|u|>st(1+iuacβt)-1Φt(u) du,|Dt|=O(exp{-Dλt1/3}),
where
(26)st=s(log(TT-t))1/6,
for some positive constant s, and D is some positive constant.

Proof.
For any u∈ℝ,
(27)Φt(u)=𝔼(exp{iuUt}exp{acSt-Lt(ac)})=exp{-iuλtcσc}×exp{(Lt(ac+iuβt) -Lt(ac))}.
By the same method as in the proof of Lemma 2.2 in [7] by Bercu and Rouault, there exist two positive constants τ and κ such that
(28)|Φt(u)|2≤(1+τu2λt)-(κ/2)λt;
therefore, Φt(·) belongs to L2(ℝ), and by Parseval’s formula, for some positive constant s, let
(29)st=s(log(TT-t))1/6;
we get
(30)Bt=12πacβt∫|u|≤st(1+iuacβt)-1Φt(u)du+12πacβt×∫|u|>st(1+iuacβt)-1Φt(u)du(31)=:Ct+Dt,(32)|Dt|=O(exp{-Dλt1/3}),
where D is some positive constant.

Proof of Lemma <xref ref-type="statement" rid="lem3.2">6</xref>.
By Lemma 3, we have
(33)Lt(k)(ac)λt=L(k)(ac)+H(k)(ac)λt+O(λtk(T-t)-2c)λt.
Noting that L′(ac)=0, L′′(ac)=σc2 and
(34)L′′(ac)2(iuβt)2λt=-u22,
for any p>0, by Taylor expansion, we obtain
(35)logΦt(u)=-u22+λt∑k=32p+3(iuβt)kL(k)(ac)k!+∑k=12p+1(iuβt)kH(k)(ac)k!+O(max(1,|u|2p+4)λtp+1);
therefore, there exist integers q(p), r(p) and a sequence φk,l independent of p; when t approaches T, we get
(36)Φt(u)=exp{-u22}(1+1λt∑k=0 2p ∑l=k+1q(p)φk,lulλtk/2 1+1λt∑k=0 2p ∑l=k+1q(p)φk,lulλtk/2+O(max(1,|u|r(p))λtp+1)),
where O is uniform as soon as |u|≤st.

Finally, we get the proof of Lemma 6 by Lemma 7 together with standard calculations on the N(0,1) distribution.