1.Introduction ShoujiangZhao, QiaojingLiu, FuxiangLiu, andHongYin SharpLargeDeviationfortheEnergyof

(1)

Volume 2013, Article ID 952628,4pages 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 Yin

²

1School of Science, China Three Gorges University, Yichang 443002, China

2School of Information, Renmin University of China, Beijing 100872, China

Correspondence should be addressed to Qiaojing Liu; [email protected] 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 of𝛼-Brownian bridge. The full expansion of the tail probability for energy is obtained by the change of measure.

1. Introduction

We consider the following𝛼-Brownian bridge:

𝑑𝑋_𝑡= − 𝛼

𝑇 − 𝑡𝑋_𝑡𝑑𝑡 + 𝑑𝑊_𝑡, 𝑋₀= 0, (1) where𝑊is a standard Brownian motion, 𝑡 ∈ [0, 𝑇), 𝑇 ∈ (0, ∞), and the constant𝛼 > 1/2. Let𝑃_𝛼denote 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 associated with a given future contract in the absence of trans- action 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 of𝛼-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 energy𝑆_𝑡, where

𝑆_𝑡= ∫^𝑡

0

𝑋²_𝑠

(𝑠 − 𝑇)²𝑑𝑠. (2)

Our main results are the following.

Theorem 1. Let{𝑋_𝑡, 𝑡 ∈ [0, 𝑇)}be the process given by the stochastic differential equation(1). Then{𝑆_𝑡/𝜆_𝑡, 𝑡 ∈ [0, 𝑇)}sat- isfies the large deviation principle with speed𝜆_𝑡and good rate function𝐼(⋅)defined by the following:

𝐼 (𝑥) ={ {{

1

8𝑥((2𝛼₀− 1) 𝑥 − 1)², if𝑥 > 0;

+∞, if𝑥 ≤ 0, (3)

where𝜆_𝑡=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 sequence𝑏_𝑐,𝑘such that, for any 𝑝 > 0, when𝑡approaches𝑇enough,

𝑃 (𝑆_𝑡≥ 𝑐𝜆_𝑡) = exp{−𝐼 (𝑐) 𝜆_𝑡+ 𝐻 (𝑎_𝑐)}

√2𝜋𝑎_𝑐𝛽_𝑡

× (1 +∑^𝑝

𝑘=1

𝑏_𝑐,𝑘

𝜆_𝑡 + 𝑂 ( 1 𝜆^𝑝+1_𝑡 )) ,

(4)

(2)

2 International Journal of Stochastic Analysis where

𝜎_𝑐²= 4𝑐², 𝛽_𝑡= 𝜎_𝑐√𝜆_𝑡, 𝑎_𝑐= (1 − 2𝛼)²𝑐²− 1

8𝑐² , 𝐻 (𝑎_𝑐) = −1

2log(1 − (1 − 2𝛼) 𝑐

2 ) .

(5)

The coefficients𝑏_𝑐,𝑘may be explicitly computed as functions of the derivatives of𝐿and𝐻(defined inLemma 3) at point𝑎_𝑐. For example,𝑏_𝑐,1is given by

𝑏_𝑐,1= 1 𝜎_𝑐²(−ℎ₂

2 −ℎ²₁ 2 + 𝑙₄

8𝜎_𝑐² +𝑙₃ℎ₁ 2𝜎_𝑐²

− 5𝑙₃² 24𝜎⁴_𝑐 +ℎ₁

𝑎_𝑐 − 𝑙₃ 2𝑎_𝑐𝜎²_𝑐 − 1

𝑎_𝑐²) ,

(6)

with𝑙_𝑘= 𝐿^(𝑘)(𝑎_𝑐), andℎ_𝑘= 𝐻^(𝑘)(𝑎_𝑐).

2. Large Deviation for Energy

Given𝛼 > 1/2, we first consider the following logarithmic moment generating function of𝑆_𝑡; that is,

𝐿_𝑡(𝑢) :=logE_𝛼exp{𝑢 ∫^𝑡

0

𝑋²_𝑠

(𝑠 − 𝑇)²𝑑𝑠} , ∀𝜆 ∈R. (7) And let

D_𝐿_𝑡:= {𝑢 ∈R, 𝐿_𝑡(𝑢) < +∞} (8) be the effective domain of𝐿_𝑡. By the same method as in Zhao and Liu [5], we have the following lemma.

Lemma 3. LetD_𝐿be the effective domain of the limit𝐿of𝐿_𝑡; then for all𝑢 ∈D_𝐿, one has

𝐿_𝑡(𝑢)

𝜆_𝑡 = 𝐿 (𝑢) +𝐻 (𝑢) 𝜆_𝑡 +𝑅 (𝑢)

𝜆_𝑡 , (9)

with

𝐿 (𝑢) = −1 − 2𝛼 − 𝜑 (𝑢)

4 ,

𝐻 (𝜆) = −1 2log{1

2(1 + ℎ (𝑢))} , 𝑅 (𝑢) = −1

2log{1 + 1 − ℎ (𝑢)

1 + ℎ (𝑢)exp{2𝜑 (𝑢) 𝜆_𝑡}} , (10)

where𝜑(𝑢) = −√(1 − 2𝛼)²− 8𝑢andℎ(𝑢) = (1 − 2𝛼)/𝜑(𝑢).

Furthermore, the remainder𝑅(𝑢)satisfies

𝑅 (𝑢) = 𝑂_{𝑡 → 𝑇} (exp{2𝜑 (𝑢) 𝜆_𝑡}) . (11)

Proof. By Itˆo’s formula and Girsanov’s formula (see Jacob and Shiryaev [11]), for all𝑢 ∈D_𝐿and𝑡 ∈ [0, 𝑇),

log𝑑𝑃_𝛼 𝑑𝑃_𝛽|_[0,𝑡]

= (𝛼 − 𝛽) ∫^𝑡

0

𝑋_𝑠

𝑠 − 𝑇𝑑𝑋_𝑠−𝛼²− 𝛽² 2 ∫^𝑡

0

𝑋²_𝑠 (𝑠 − 𝑇)²𝑑𝑠,

∫^𝑡

0

𝑋_𝑠 𝑠 − 𝑇𝑑𝑋_𝑠

= 1 2 ( 𝑋²_𝑡

(𝑡 − 𝑇) + ∫^𝑡

0

𝑋²_𝑠

(𝑠 − 𝑇)²𝑑𝑠 −log(1 − 𝑡 𝑇)) .

(12)

Therefore,

𝐿_𝑡(𝑢) = logE_𝛽(exp{𝑢 ∫^𝑡

0

𝑋²_𝑠

(𝑠 − 𝑇)²𝑑𝑠}𝑑𝑃_𝛼 𝑑𝑃_𝛽|_[0,𝑡])

= logE_𝛽exp{ 𝛼 − 𝛽

2 (𝑡 − 𝑇)𝑋²_𝑡 −𝛼 − 𝛽

2 log(1 − 𝑡 𝑇) +1

2(𝛽²− 𝛼²+ 𝛼 − 𝛽 + 2𝑢)

× ∫^𝑡

0

𝑋²_𝑠 (𝑠 − 𝑇)²𝑑𝑠} .

(13) If4𝑢 ≤ (1 − 2𝛼)², we can choose𝛽such that(𝛽 − 1/2)²− (𝛼 − 1/2)²+ 2𝑢 = 0. Then

𝐿_𝑡(𝑢) = − 1 − 2𝛼 − 𝜑 (𝜆)

4 𝜆_𝑡

−1 2log{1

2(1 + ℎ (𝑢))}

−1

2log{1 +1 − ℎ (𝑢)

1 + ℎ (𝑢)exp{2𝜑 (𝑢) 𝜆_𝑡}} , (14)

where𝜑(𝑢) = −√(1 − 2𝛼)²− 8𝑢, andℎ(𝑢) = (1 − 2𝛼)/𝜑(𝑢).

Therefore, 𝐿_𝑡(𝑢)

𝜆_𝑡 = −1 − 2𝛼 − 𝜑 (𝑢) 4

− 1 2𝜆_𝑡log{1

2(1 + ℎ (𝑢))}

− 1

2𝜆_𝑡log{1 + 1 − ℎ (𝑢)

1 + ℎ (𝑢)exp{2𝜑 (𝑢) 𝜆_𝑡}}

= 𝐿 (𝑢) +𝐻 (𝑢) 𝜆_𝑡 +𝑅 (𝑢)

𝜆_𝑡 .

(15)

Proof ofTheorem 1. FromLemma 3, we have 𝐿 (𝑢) =lim_{𝑡 → 𝑇}𝐿_𝑡(𝑢)

𝜆_𝑡 =1 − 2𝛼 − 𝜑 (𝑢)

4 , (16)

(3)

and𝐿(⋅)is steep; by the G¨artner-Ellis theorem (Dembo and Zeitouni [12]), 𝑆_𝑡/𝜆_𝑡 satisfies the large deviation principle with speed 𝜆_𝑡 and good rate function 𝐼(⋅) defined by the following:

𝐼 (𝑥) ={ {{

1

8𝑥((2𝛼 − 1) 𝑥 − 1)², if 𝑥 > 0;

+∞, if + 𝑥 ≤ 0. (17)

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

3. Sharp Large Deviation for Energy

For𝑐 > 1/(2𝛼 − 1), let 𝑎_𝑐= (1 − 2𝛼)²𝑐²− 1

8𝑐² , 𝜎_𝑐²= 𝐿^󸀠󸀠(𝑎_𝑐) = 4𝑐³, 𝐻 (𝑎_𝑐) = −1

2log(1 − (1 − 2𝛼) 𝑐) .

(18)

Then

𝑃 (𝑆_𝑡≥ 𝑐𝜆_𝑡)

= ∫𝑆𝑡≥𝑐𝜆𝑡

exp{𝐿 (𝑎_𝑐) − 𝑐𝑎_𝑐𝜆_𝑡 +𝑐𝑎_𝑐𝜆_𝑡− 𝑎_𝑐𝑆_𝑡} 𝑑𝑄_𝑡

=exp{𝐿 (𝑎_𝑐) − 𝑐𝑎_𝑐𝜆_𝑡}E_𝑄exp{−𝑎_𝑐𝛽_𝑡𝑈_𝑡𝐼_{𝑈_𝑡_≥0}} = 𝐴_𝑡𝐵_𝑡, (19) whereE_𝑄is the expectation after the change of measure

𝑑𝑄_𝑡

𝑑𝑃 =exp{𝑎_𝑐𝑆_𝑡− 𝐿_𝑡(𝑎_𝑐)} , 𝑈_𝑡=𝑆_𝑡− 𝑐𝜆_𝑡

𝛽_𝑡 , 𝛽_𝑡= 𝜎_𝑐√𝜆_𝑡.

(20)

ByLemma 3, we have the following expression of𝐴_𝑡. Lemma 5. For all𝑐 > 1/(2𝛼−1), when𝑡approaches𝑇enough,

𝐴_𝑡=exp{−𝐼 (𝑐) 𝜆_𝑡+ 𝐻 (𝑎_𝑐)} (1 + 𝑂 ((𝑇 − 𝑡)^𝑐)) . (21) For𝐵_𝑡, one gets the following.

Lemma 6. For all𝑐 > 1/(2𝛼 − 1), the distribution of𝑈_𝑡under 𝑄_𝑡converges to𝑁(0, 1)distribution. Furthermore, there exists a sequence𝜓_𝑘 such that, for any𝑝 > 0when𝑡approaches𝑇 enough,

𝐵_𝑡= 1

𝑎_𝑐𝜎_𝑐√2𝜋𝜆_𝑡(1 +∑^𝑝

𝑘=1

𝜓_𝑘

𝜆^𝑘_𝑡 + 𝑂 (𝜆^−(𝑝+1)_𝑡 )) . (22) Proof ofTheorem 2. The theorem follows fromLemma 5and Lemma 6.

It only remains to prove Lemma 6. Let Φ_𝑡(⋅) be the characteristic function of 𝑈_𝑡 under 𝑄_𝑡; then we have the following.

Lemma 7. When𝑡 approaches𝑇,Φ_𝑡belongs to𝐿²(R)and, for all𝑢 ∈R,

Φ_𝑡(𝑢) = exp{−𝑖𝑢√𝜆_𝑡𝑐 𝜎_𝑐 }

×exp{(𝐿_𝑡(𝑎_𝑐+𝑖𝑢

𝛽_𝑡) − 𝐿_𝑡(𝑎_𝑐))} .

(23)

Moreover,

𝐵_𝑡=E_𝑄exp{−𝑎_𝑐𝛽_𝑡𝑈_𝑡𝐼_{𝑈_𝑡_≥0}} = 𝐶_𝑡+ 𝐷_𝑡, (24) with

𝐶_𝑡= 1 2𝜋𝑎_𝑐𝛽_𝑡∫

|𝑢|≤𝑠_𝑡(1 + 𝑖𝑢

𝑎_𝑐𝛽_𝑡)⁻¹Φ_𝑡(𝑢) 𝑑𝑢, 𝐷_𝑡= 1

2𝜋𝑎_𝑐𝛽_𝑡∫

|𝑢|>𝑠𝑡

(1 + 𝑖𝑢

𝑎_𝑐𝛽_𝑡)⁻¹Φ_𝑡(𝑢) 𝑑𝑢,

󵄨󵄨󵄨󵄨𝐷^𝑡󵄨󵄨󵄨󵄨 = 𝑂(exp{−𝐷𝜆^1/3_𝑡 }) ,

(25)

where

𝑠_𝑡= 𝑠(log( 𝑇

𝑇 − 𝑡))^1/6, (26) for some positive constant𝑠, and 𝐷is some positive constant.

Proof. For any𝑢 ∈R,

Φ_𝑡(𝑢) =E(exp{𝑖𝑢𝑈_𝑡}exp{𝑎_𝑐𝑆_𝑡− 𝐿_𝑡(𝑎_𝑐)})

= exp{−𝑖𝑢√𝜆_𝑡𝑐 𝜎_𝑐 }

×exp{(𝐿_𝑡(𝑎_𝑐+𝑖𝑢

𝛽_𝑡) − 𝐿_𝑡(𝑎_𝑐))} .

(27)

By the same method as in the proof of Lemma 2.2 in [7]

by Bercu and Rouault, there exist two positive constants𝜏 and𝜅such that

󵄨󵄨󵄨󵄨Φ^𝑡(𝑢)󵄨󵄨󵄨󵄨²≤ (1 +𝜏𝑢² 𝜆_𝑡 )

−(𝜅/2)𝜆𝑡

; (28)

therefore,Φ_𝑡(⋅)belongs to𝐿²(R), and by Parseval’s formula, for some positive constant𝑠, let

𝑠_𝑡= 𝑠(log( 𝑇

𝑇 − 𝑡))^1/6; (29) we get

𝐵_𝑡= 1 2𝜋𝑎_𝑐𝛽_𝑡∫

|𝑢|≤𝑠𝑡

(1 + 𝑖𝑢

𝑎_𝑐𝛽_𝑡)⁻¹Φ_𝑡(𝑢) 𝑑𝑢 + 1 2𝜋𝑎_𝑐𝛽_𝑡

× ∫|𝑢|>𝑠𝑡

(1 + 𝑖𝑢

𝑎_𝑐𝛽_𝑡)⁻¹Φ_𝑡(𝑢) 𝑑𝑢

(30)

= : 𝐶_𝑡+ 𝐷_𝑡, (31)

󵄨󵄨󵄨󵄨𝐷^𝑡󵄨󵄨󵄨󵄨 = 𝑂(exp{−𝐷𝜆^1/3_𝑡 }) , (32) where𝐷 is some positive constant.

(4)

4 International Journal of Stochastic Analysis Proof ofLemma 6. ByLemma 3, we have

𝐿^(𝑘)_𝑡 (𝑎_𝑐)

𝜆_𝑡 = 𝐿^(𝑘)(𝑎_𝑐) +𝐻^(𝑘)(𝑎_𝑐)

𝜆_𝑡 +𝑂 (𝜆^𝑘_𝑡(𝑇 − 𝑡)^−2𝑐) 𝜆_𝑡 . (33) Noting that𝐿^󸀠(𝑎_𝑐) = 0,𝐿^󸀠󸀠(𝑎_𝑐) = 𝜎_𝑐²and

𝐿^󸀠󸀠(𝑎_𝑐) 2 (𝑖𝑢

𝛽_𝑡)²𝜆_𝑡= −𝑢²

2, (34)

for any𝑝 > 0,by Taylor expansion, we obtain logΦ_𝑡(𝑢) = − 𝑢²

2 + 𝜆_𝑡^2𝑝+3∑

𝑘=3

(𝑖𝑢

𝛽_𝑡)^𝑘𝐿^(𝑘)(𝑎_𝑐) 𝑘!

+^2𝑝+1∑

𝑘=1

(𝑖𝑢

𝛽_𝑡)^𝑘𝐻^(𝑘)(𝑎_𝑐) 𝑘!

+ 𝑂 (max(1, |𝑢|^2𝑝+4) 𝜆^𝑝+1_𝑡 ) ;

(35)

therefore, there exist integers𝑞(𝑝),𝑟(𝑝)and a sequence𝜑_𝑘,𝑙 independent of𝑝; when𝑡approaches𝑇, we get

Φ_𝑡(𝑢) = exp{−𝑢²

2} (1 + 1

√𝜆_𝑡

∑2𝑝 𝑘=0

𝑞(𝑝)∑

𝑙=𝑘+1

𝜑_𝑘,𝑙𝑢^𝑙 𝜆^𝑘/2_𝑡

+ 𝑂 (max(1, |𝑢|^𝑟(𝑝)) 𝜆^𝑝+1_𝑡 )) ,

(36)

where𝑂is uniform as soon as|𝑢| ≤ 𝑠_𝑡.

Finally, we get the proof of Lemma 6byLemma 7 together with standard calculations on the𝑁(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 like- lihood 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 of𝛼-Brownian bridge,”Journal of Statistical Planning and Infer- ence, 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 Orn- stein-Uhlenbeck process,”Rossi˘ıskaya Akademiya Nauk. Teor- iya Veroyatnoste˘ı 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 Proc- esses, Springer Press, Berlin, Germany, 1987.

[12] A. Dembo and O. Zeitouni,Large Deviations Techniques and Applications, vol. 38, Springe, Berlin, Germany, 2nd edition, 1998.

(5)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}