Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2013, Article ID 952628, 4 pages
http://dx.doi.org/10.1155/2013/952628
Research Article
Sharp Large Deviation for the Energy of 𝛼-Brownian Bridge
Shoujiang Zhao,1 Qiaojing Liu,1 Fuxiang Liu,1 and Hong Yin2
1
2
School of Science, China Three Gorges University, Yichang 443002, China
School of Information, Renmin University of China, Beijing 100872, China
Correspondence should be addressed to Qiaojing Liu; qjliu2002@163.com
Received 26 April 2013; Accepted 23 October 2013
Academic Editor: Yaozhong Hu
Copyright © 2013 Shoujiang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the sharp large deviation for the energy of 𝛼-Brownian bridge. The full expansion of the tail probability for energy is
obtained by the change of measure.
1. Introduction
In this paper we consider the sharp large deviation principle (SLDP) of energy 𝑆𝑡 , where
We consider the following 𝛼-Brownian bridge:
𝛼
𝑑𝑋𝑡 = −
𝑋 𝑑𝑡 + 𝑑𝑊𝑡 ,
𝑇−𝑡 𝑡
𝑡
𝑆𝑡 = ∫
𝑋0 = 0,
0
(1)
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 OrnsteinUhlenbeck (cf. Bercu and Rouault [7], Bercu et al. [8], and
Bercu et al. [9, 10]).
𝑋𝑠2
(𝑠 − 𝑇)2
𝑑𝑠.
(2)
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 /𝜆 𝑡 , 𝑡 ∈ [0, 𝑇)} satisfies the large deviation principle with speed 𝜆 𝑡 and good rate
function 𝐼(⋅) defined by the following:
{ 1 ((2𝛼0 − 1) 𝑥 − 1)2 , if 𝑥 > 0;
𝐼 (𝑥) = { 8𝑥
if 𝑥 ≤ 0,
{+∞,
(3)
where 𝜆 𝑡 = log(𝑇/(𝑇 − 𝑡)).
Theorem 2. {𝑆𝑡 /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,
𝑃 (𝑆𝑡 ≥ 𝑐𝜆 𝑡 ) =
exp {−𝐼 (𝑐) 𝜆 𝑡 + 𝐻 (𝑎𝑐 )}
√2𝜋𝑎𝑐 𝛽𝑡
𝑝
𝑏𝑐,𝑘
1
× (1 + ∑
+ 𝑂 ( 𝑝+1 )) ,
𝜆
𝜆𝑡
𝑘=1 𝑡
(4)
2
International Journal of Stochastic Analysis
where
𝜎𝑐2 = 4𝑐2 ,
𝑎𝑐 =
Proof. By Itô’s formula and Girsanov’s formula (see Jacob and
Shiryaev [11]), for all 𝑢 ∈ D𝐿 and 𝑡 ∈ [0, 𝑇),
𝛽𝑡 = 𝜎𝑐 √𝜆 𝑡 ,
log
(1 − 2𝛼)2 𝑐2 − 1
,
8𝑐2
(5)
= (𝛼 − 𝛽) ∫
1
1 − (1 − 2𝛼) 𝑐
𝐻 (𝑎𝑐 ) = − log (
).
2
2
𝑙ℎ
ℎ2 ℎ12
𝑙
1
(−
−
+ 4 2 + 3 21
2
𝜎𝑐
2
2 8𝜎𝑐 2𝜎𝑐
𝑡
0
𝛼2 − 𝛽2 𝑡 𝑋𝑠2
𝑋𝑠
𝑑𝑠,
𝑑𝑋𝑠 −
∫
2
𝑠−𝑇
2
0 (𝑠 − 𝑇)
𝑡
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
𝑏𝑐,1 =
𝑑𝑃𝛼
|
𝑑𝑃𝛽 [0,𝑡]
𝑋𝑠
𝑑𝑋𝑠
∫
0 𝑠−𝑇
=
𝑡
𝑋𝑠2
𝑋𝑡2
1
𝑡
+∫
(
𝑑𝑠 − log (1 − )) .
2
2 (𝑡 − 𝑇)
𝑇
0 (𝑠 − 𝑇)
Therefore,
5𝑙2
𝑙
ℎ
1
− 34 + 1 − 3 2 − 2 ) ,
24𝜎𝑐 𝑎𝑐 2𝑎𝑐 𝜎𝑐 𝑎𝑐
(6)
= log E𝛽 exp {
Given 𝛼 > 1/2, we first consider the following logarithmic
moment generating function of 𝑆𝑡 ; that is,
×∫
0
(𝑠 − 𝑇)
𝑑𝑠} ,
2
∀𝜆 ∈ R.
𝑑𝑠}
𝑑𝑃𝛼
| )
𝑑𝑃𝛽 [0,𝑡]
1 2
(𝛽 − 𝛼2 + 𝛼 − 𝛽 + 2𝑢)
2
+
𝑋𝑠2
(𝑠 − 𝑇)
2
𝛼−𝛽 2 𝛼−𝛽
𝑡
𝑋 −
log (1 − )
2 (𝑡 − 𝑇) 𝑡
2
𝑇
2. Large Deviation for Energy
𝑡
𝑋𝑠2
𝑡
𝐿 𝑡 (𝑢) = log E𝛽 (exp {𝑢 ∫
0
with 𝑙𝑘 = 𝐿(𝑘) (𝑎𝑐 ), and ℎ𝑘 = 𝐻(𝑘) (𝑎𝑐 ).
𝐿 𝑡 (𝑢) := log E𝛼 exp {𝑢 ∫
𝑡
0
𝑋𝑠2
(𝑠 − 𝑇)2
𝑑𝑠} .
(13)
(7)
And let
If 4𝑢 ≤ (1 − 2𝛼)2 , we can choose 𝛽 such that (𝛽 − 1/2)2 − (𝛼 −
1/2)2 + 2𝑢 = 0. Then
1 − 2𝛼 − 𝜑 (𝜆)
𝜆𝑡
4
(8)
𝐿 𝑡 (𝑢) = −
be the effective domain of 𝐿 𝑡 . By the same method as in Zhao
and Liu [5], we have the following lemma.
−
1
1
log { (1 + ℎ (𝑢))}
2
2
Lemma 3. Let D𝐿 be the effective domain of the limit 𝐿 of 𝐿 𝑡 ;
then for all 𝑢 ∈ D𝐿 , one has
−
1
1 − ℎ (𝑢)
exp {2𝜑 (𝑢) 𝜆 𝑡 }} ,
log {1 +
2
1 + ℎ (𝑢)
D𝐿 𝑡 := {𝑢 ∈ R, 𝐿𝑡 (𝑢) < +∞}
𝐿 𝑡 (𝑢)
𝐻 (𝑢) 𝑅 (𝑢)
= 𝐿 (𝑢) +
+
,
𝜆𝑡
𝜆𝑡
𝜆𝑡
(9)
1 − 2𝛼 − 𝜑 (𝑢)
,
4
1
1
𝐻 (𝜆) = − log { (1 + ℎ (𝑢))} ,
2
2
where 𝜑(𝑢) = −√(1 − 2𝛼)2 − 8𝑢, and ℎ(𝑢) = (1 − 2𝛼)/𝜑(𝑢).
Therefore,
−
(10)
1
1 − ℎ (𝑢)
exp {2𝜑 (𝑢) 𝜆 𝑡 }} ,
𝑅 (𝑢) = − log {1 +
2
1 + ℎ (𝑢)
1
1
log { (1 + ℎ (𝑢))}
2𝜆 𝑡
2
1 − ℎ (𝑢)
1
exp {2𝜑 (𝑢) 𝜆 𝑡 }}
log {1 +
−
2𝜆 𝑡
1 + ℎ (𝑢)
= 𝐿 (𝑢) +
where 𝜑(𝑢) = −√(1 − 2𝛼)2 − 8𝑢 and ℎ(𝑢) = (1 − 2𝛼)/𝜑(𝑢).
Furthermore, the remainder 𝑅(𝑢) satisfies
𝑅 (𝑢) = 𝑂𝑡 → 𝑇 (exp {2𝜑 (𝑢) 𝜆 𝑡 }) .
(14)
1 − 2𝛼 − 𝜑 (𝑢)
𝐿 𝑡 (𝑢)
= −
𝜆𝑡
4
with
𝐿 (𝑢) = −
(12)
(11)
(15)
𝐻 (𝑢) 𝑅 (𝑢)
+
.
𝜆𝑡
𝜆𝑡
Proof of Theorem 1. From Lemma 3, we have
𝐿 (𝑢) = lim𝑡 → 𝑇
𝐿 𝑡 (𝑢) 1 − 2𝛼 − 𝜑 (𝑢)
=
,
𝜆𝑡
4
(16)
International Journal of Stochastic Analysis
3
and 𝐿(⋅) is steep; by the Gärtner-Ellis theorem (Dembo and
Zeitouni [12]), 𝑆𝑡 /𝜆 𝑡 satisfies the large deviation principle
with speed 𝜆 𝑡 and good rate function 𝐼(⋅) defined by the
following:
{ 1 ((2𝛼 − 1) 𝑥 − 1)2 ,
𝐼 (𝑥) = { 8𝑥
{+∞,
if 𝑥 > 0;
Lemma 7. When 𝑡 approaches 𝑇, Φ𝑡 belongs to 𝐿2 (R) and,
for all 𝑢 ∈ R,
Φ𝑡 (𝑢) = exp {−
𝑖𝑢
× exp {(𝐿 𝑡 (𝑎𝑐 + ) − 𝐿 𝑡 (𝑎𝑐 ))} .
𝛽𝑡
(17)
if + 𝑥 ≤ 0.
𝑖𝑢√𝜆 𝑡 𝑐
}
𝜎𝑐
(23)
Moreover,
Remark 4. Theorem 1 can also be obtained by using
Theorem 1 in Zhao and Liu [5].
𝐵𝑡 = E𝑄 exp {−𝑎𝑐 𝛽𝑡 𝑈𝑡 𝐼{𝑈𝑡 ≥0} } = 𝐶𝑡 + 𝐷𝑡 ,
(24)
with
3. Sharp Large Deviation for Energy
For 𝑐 > 1/(2𝛼 − 1), let
(1 − 2𝛼)2 𝑐2 − 1
𝑎𝑐 =
,
8𝑐2
𝜎𝑐2 = 𝐿󸀠󸀠 (𝑎𝑐 ) = 4𝑐3 ,
1
𝐻 (𝑎𝑐 ) = − log (1 − (1 − 2𝛼) 𝑐) .
2
(18)
1
𝑖𝑢 −1
(1 +
) Φ𝑡 (𝑢) 𝑑𝑢,
∫
2𝜋𝑎𝑐 𝛽𝑡 |𝑢|>𝑠𝑡
𝑎𝑐 𝛽𝑡
(25)
󵄨󵄨 󵄨󵄨
1/3
󵄨󵄨𝐷𝑡 󵄨󵄨 = 𝑂 (exp {−𝐷𝜆 𝑡 }) ,
1/6
𝑇
(26)
)) ,
𝑇−𝑡
for some positive constant 𝑠, and 𝐷 is some positive constant.
exp {𝐿 (𝑎𝑐 ) − 𝑐𝑎𝑐 𝜆 𝑡 +𝑐𝑎𝑐 𝜆 𝑡 − 𝑎𝑐 𝑆𝑡 } 𝑑𝑄𝑡
Proof. For any 𝑢 ∈ R,
= exp {𝐿 (𝑎𝑐 ) − 𝑐𝑎𝑐 𝜆 𝑡 } E𝑄 exp {−𝑎𝑐 𝛽𝑡 𝑈𝑡 𝐼{𝑈𝑡 ≥0} } = 𝐴 𝑡 𝐵𝑡 ,
(19)
Φ𝑡 (𝑢) = E (exp {𝑖𝑢𝑈𝑡 } exp {𝑎𝑐 𝑆𝑡 − 𝐿 𝑡 (𝑎𝑐 )})
= exp {−
where E𝑄 is the expectation after the change of measure
𝑑𝑄𝑡
= exp {𝑎𝑐 𝑆𝑡 − 𝐿 𝑡 (𝑎𝑐 )} ,
𝑑𝑃
𝑈𝑡 =
𝑆𝑡 − 𝑐𝜆 𝑡
,
𝛽𝑡
(20)
𝛽𝑡 = 𝜎𝑐 √𝜆 𝑡 .
(21)
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,
𝑝
𝜓
1
−(𝑝+1)
(1 + ∑ 𝑘𝑘 + 𝑂 (𝜆 𝑡
)) .
𝜆
𝑎𝑐 𝜎𝑐 √2𝜋𝜆 𝑡
𝑘=1 𝑡
(22)
Proof of Theorem 2. 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.
(27)
𝑖𝑢
) − 𝐿 𝑡 (𝑎𝑐 ))} .
𝛽𝑡
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
−(𝜅/2)𝜆 𝑡
𝜏𝑢2
󵄨2
󵄨󵄨
)
󵄨󵄨Φ𝑡 (𝑢)󵄨󵄨󵄨 ≤ (1 +
𝜆𝑡
Lemma 5. For all 𝑐 > 1/(2𝛼−1), when 𝑡 approaches 𝑇 enough,
𝐴 𝑡 = exp {−𝐼 (𝑐) 𝜆 𝑡 + 𝐻 (𝑎𝑐 )} (1 + 𝑂 ((𝑇 − 𝑡)𝑐 )) .
𝑖𝑢√𝜆 𝑡 𝑐
}
𝜎𝑐
× exp {(𝐿 𝑡 (𝑎𝑐 +
By Lemma 3, we have the following expression of 𝐴 𝑡 .
𝐵𝑡 =
𝐷𝑡 =
𝑠𝑡 = 𝑠(log (
𝑃 (𝑆𝑡 ≥ 𝑐𝜆 𝑡 )
𝑆𝑡 ≥𝑐𝜆 𝑡
1
𝑖𝑢 −1
(1 +
) Φ𝑡 (𝑢) 𝑑𝑢,
∫
2𝜋𝑎𝑐 𝛽𝑡 |𝑢|≤𝑠𝑡
𝑎𝑐 𝛽𝑡
where
Then
=∫
𝐶𝑡 =
;
(28)
therefore, Φ𝑡 (⋅) belongs to 𝐿2 (R), and by Parseval’s formula,
for some positive constant 𝑠, let
𝑠𝑡 = 𝑠(log (
1/6
𝑇
)) ;
𝑇−𝑡
(29)
we get
𝐵𝑡 =
1
1
𝑖𝑢 −1
(1 +
) Φ𝑡 (𝑢) 𝑑𝑢 +
∫
2𝜋𝑎𝑐 𝛽𝑡 |𝑢|≤𝑠𝑡
𝑎𝑐 𝛽𝑡
2𝜋𝑎𝑐 𝛽𝑡
𝑖𝑢 −1
×∫
(1 +
) Φ𝑡 (𝑢) 𝑑𝑢
𝑎𝑐 𝛽𝑡
|𝑢|>𝑠𝑡
= : 𝐶𝑡 + 𝐷𝑡 ,
󵄨󵄨 󵄨󵄨
1/3
󵄨󵄨𝐷𝑡 󵄨󵄨 = 𝑂 (exp {−𝐷𝜆 𝑡 }) ,
where 𝐷 is some positive constant.
(30)
(31)
(32)
4
International Journal of Stochastic Analysis
Proof of Lemma 6. By Lemma 3, we have
−2𝑐
𝑘
𝐿(𝑘)
𝐻(𝑘) (𝑎𝑐 ) 𝑂 (𝜆 𝑡 (𝑇 − 𝑡) )
𝑡 (𝑎𝑐 )
= 𝐿(𝑘) (𝑎𝑐 ) +
+
. (33)
𝜆𝑡
𝜆𝑡
𝜆𝑡
Noting that 𝐿󸀠 (𝑎𝑐 ) = 0, 𝐿󸀠󸀠 (𝑎𝑐 ) = 𝜎𝑐2 and
𝐿󸀠󸀠 (𝑎𝑐 ) 𝑖𝑢 2
𝑢2
( ) 𝜆𝑡 = − ,
2
𝛽𝑡
2
(34)
for any 𝑝 > 0, by Taylor expansion, we obtain
log Φ𝑡 (𝑢) = −
2𝑝+3
(𝑘)
𝑖𝑢 𝑘 𝐿 (𝑎𝑐 )
𝑢2
+ 𝜆𝑡 ∑ ( )
2
𝛽𝑡
𝑘!
𝑘=3
2𝑝+1
+ ∑(
𝑘=1
+ 𝑂(
(𝑘)
𝑖𝑢 𝑘 𝐻 (𝑎𝑐 )
)
𝛽𝑡
𝑘!
max (1, |𝑢|2𝑝+4 )
𝑝+1
𝜆𝑡
(35)
);
therefore, there exist integers 𝑞(𝑝), 𝑟(𝑝) and a sequence 𝜑𝑘,𝑙
independent of𝑝; when 𝑡 approaches 𝑇, we get
Φ𝑡 (𝑢) = exp {−
2𝑝 𝑞(𝑝)
𝜑𝑘,𝑙 𝑢𝑙
𝑢2
1
} (1 +
∑ ∑ 𝑘/2
2
√𝜆 𝑡 𝑘=0 𝑙=𝑘+1 𝜆 𝑡
+ 𝑂(
max (1, |𝑢|𝑟(𝑝) )
𝑝+1
𝜆𝑡
(36)
)) ,
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.
Acknowledgment
This research was supported by the National Natural Science
of Tianyuan Foundation under Grant 11226202.
References
[1] M. J. Brennan and E. S. Schwartz, “Arbitrage in stock index
futures,” The Journal of Business, vol. 63, pp. 7–31, 1990.
[2] M. Barczy and G. Pap, “Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes,”
Journal of Statistical Planning and Inference, vol. 140, no. 6, pp.
1576–1593, 2010.
[3] M. Barczy and G. Pap, “Explicit formulas for Laplace transforms
of certain functionals of some time inhomogeneous diffusions,”
Journal of Mathematical Analysis and Applications, vol. 380, no.
2, pp. 405–424, 2011.
[4] H. Jiang and S. Zhao, “Large and moderate deviations in testing
time inhomogeneous diffusions,” Journal of Statistical Planning
and Inference, vol. 141, no. 9, pp. 3160–3169, 2011.
[5] S. Zhao and Q. Liu, “Large deviations for parameter estimators
of 𝛼-Brownian bridge,” Journal of Statistical Planning and Inference, vol. 142, no. 3, pp. 695–707, 2012.
[6] R. R. Bahadur and R. Ranga Rao, “On deviations of the sample
mean,” Annals of Mathematical Statistics, vol. 31, pp. 1015–1027,
1960.
[7] B. Bercu and A. Rouault, “Sharp large deviations for the Ornstein-Uhlenbeck process,” Rossiı̆skaya Akademiya Nauk. Teoriya Veroyatnosteı̆ i ee Primeneniya, vol. 46, no. 1, pp. 74–93, 2001.
[8] B. Bercu, F. Gamboa, and M. Lavielle, “Sharp large deviations for
Gaussian quadratic forms with applications,” European Series in
Applied and Industrial Mathematics. Probability and Statistics,
vol. 4, pp. 1–24, 2000.
[9] B. Bercu, L. Coutin, and N. Savy, “Sharp large deviations for
the fractional Ornstein-Uhlenbeck process,” SIAM Theory of
Probability and Its Applications, vol. 55, pp. 575–610, 2011.
[10] B. Bercu, L. Coutin, and N. Savy, “Sharp large deviations for
the non-stationary Ornstein-Uhlenbeck process,” Stochastic
Processes and their Applications, vol. 122, no. 10, pp. 3393–3424,
2012.
[11] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer Press, Berlin, Germany, 1987.
[12] A. Dembo and O. Zeitouni, Large Deviations Techniques and
Applications, vol. 38, Springe, Berlin, Germany, 2nd edition,
1998.
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