DEVIATION INEQUALITIES AND MODERATE DEVIATIONS FOR ESTIMATORS OF PARAMETERS IN AN ORNSTEIN-UHLENBECK PROCESS WITH LINEAR DRIFT

ELECTRONIC COMMUNICATIONS in PROBABILITY

DEVIATION INEQUALITIES AND MODERATE DEVIATIONS FOR ESTIMATORS OF PARAMETERS IN AN ORNSTEIN-UHLENBECK PROCESS WITH LINEAR DRIFT

FUQING GAO¹

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R.China email: fqgao@whu.edu.cn

HUI JIANG

School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China email: huijiang@nuaa.edu.cn

SubmittedDecember 29, 2008, accepted in final formApril 21, 2009 AMS 2000 Subject classification: 60F12, 62F12, 62N02

Keywords: Deviation inequality, logarithmic Sobolev inequality, moderate deviations, Ornstein- Uhlenbeck process

Abstract

Some deviation inequalities and moderate deviation principles for the maximum likelihood esti- mators of parameters in an Ornstein-Uhlenbeck process with linear drift are established by the logarithmic Sobolev inequality and the exponential martingale method.

1 Introduction and main results

1.1 Introduction

We consider the following Ornstein-Uhlenbeck process

d X_t= (−θX_t+γ)d t+dW_t, X₀=x (1.1) whereW is a standard Brownian motion andθ,γare unknown parameters withθ∈(0,+∞). We denote byP_θ,γ,xthe distribution of the solution of (1.1).

It is known that the maximum likelihood estimators (MLE) of the parameters θ andγare (cf.

1RESEARCH SUPPORTED BY THE NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA (10871153)

210

[15])

θˆ_T=−TRT

0 X_td X_t+ (X_T−x)RT 0 X_td t TR_T

0 X²_td t−R_T

0 X_td t

₂ (1.2)

=θ+W_Tµˆ_T−RT 0 X_tdW_t Tσˆ²_T ,

ˆ

γ_T=−RT

0 X_td tRT

0 X_td X_t+ (X_T−x)RT 0 X²_td t TR_T

0 X²_td t−R_T

0 X_td t

₂ (1.3)

=γ+W_T

T +µˆ_T(W_Tµˆ_T−RT

0 X_tdW_t) Tσˆ²_T , where

ˆ µ_T= 1

T Z T

0

X_td t, σˆ²_T= 1 T

ZT

0

X²_td t−µˆ²_T. (1.4) It is known thatθˆ_T andγˆ_T are consistent estimators ofθ andγand have asymptotic normality (cf.[15]).

Forγ≡0 case, Florens-Landais and Pham([9]) calculated the Laplace functional of(RT 0 X_td X_t, RT

0 X²_td t)by Girsanov’s formula and obtained large deviations forθˆ_T by Gärtner-Ellis theorem.

Bercu and Rouault ([1]) presented a sharp large deviation for θˆ_T. Lezaud ([14]) obtained the deviation inequality of quadratic functional of the classical OU processes. We refer to [8] and [11]for the moderate deviations of some non-linear functionals of moving average processes and diffusion processes. In this paper we use the logarithmic Sobolev inequality (LSI) to study the deviation inequalities and the moderate deviations ofθˆ_T andγˆ_Tforγ6=0 case.

1.2 Main results

Throughout this paper, letλ_T,T≥1 be a positive sequence satisfying λ_T→ ∞, λ_T

pT →0. (1.5)

Theorem 1.1. There exist finite positive constants C₀,C₁,C₂and C₃such that for all r >0and all T≥1,

P_θ,γ,x€

|θˆ_T−θ| ≥rŠ

≤C₀exp¦

−C₁r T E_θ,γ,x( ˆσ²_T)min

1,C₂r © +C₀exp¦

−C₃T E_θ,γ,x( ˆσ²_T)© and

P_θ,γ,x |γˆ_T−γ| ≥r

≤C₀exp¦

−C₁r T E_θ,γ,x( ˆσ²_T)min

1,C₂r © +C₀exp¦

−C₃T E_θ,γ,x( ˆσ²_T)© .

Remark 1.1. In this theorem and the remainder of the paper, all the constants involved depend on θ,γand the initial point x.

Theorem 1.2. (1).

P_θ,γ,x

q

T

λT( ˆθ_T−θ)∈ ·

,T≥1

satisfies the large deviation principle with speedλ_Tand rate function I₁(u) = ^u2

4θ, that is, for any closed set F inR, lim sup

n→∞

1

λ_T logP_θ,γ,x r T

λ_T( ˆθ_T−θ)∈F

!

≤ −inf

u∈F

u² 4θ and open set G inR,

lim inf

n→∞

1

λ_TlogP_θ,γ,x rT

λ_T( ˆθ_T−θ)∈G

!

≥ −inf

u∈G

u² 4θ. (2).

P_θ,γ,x

q

T

λ_T(ˆγ_T−γ)∈ ·

,T≥1

satisfies the large deviation principle with speedλ_T and rate function I₂(u) =_2(θ+2γ^θu2 2), that is, for any closed set F inR,

lim sup

n→∞

1

λ_TlogP_θ,γ,x r T

λ_T(ˆγ_T−γ)∈F

!

≤ −inf

u∈F

θu² 2(θ+2γ²) and open set G inR,

lim inf

n→∞

1

λ_TlogP_θ,γ,x r T

λ_T(ˆγ_T−γ)∈G

!

≥ −inf

u∈G

θu² 2(θ+2γ²).

Inγ=0 case, the deviation inequalities of quadratic functionals of the classical OU process are obtained in [14]. For the large deviations and the moderate deviations ofθˆ_T, we refer to[1], [9]and[11]. The proofs of Theorem 1.1 and Theorem 1.2 are based on the LSI with respect to L²-norm in the Wiener space and Herbst’s argument (cf.[10],[12]).

2 Deviation inequalities

In this section, we give some deviation inequalities for the estimatorsθˆ_Tandγˆ_Tby the logarithmic Sobolev inequality and the exponential martingale method. For deviation bounds for additive functionals of Markov processes, we refer to[3]and[18].

2.1 Moments

It is known that the solution of equation (1.1) has the following expression:

X_t=

 x− γ

θ

‹

e^−θt+ γ θ +e^−θt

Z t

0

e^θsdW_s. (2.1)

From this expression, it is easily seen that for anyt≥0, µ_t:=E_θ,γ,x(X_t) =

 x− γ

θ

‹

e^−θt+ γ

θ, (2.2)

σ²_t :=Var_θ,γ,x(X_t) = 1

2θ(1−e^−2θt) (2.3)

and for any 0≤s≤t,

Cov_θ,γ,x(X_s,X_t) = 1

2θ(1−e^−2θs)e^−θ(t−s). (2.4) Therefore

E_θ,γ,x( ˆµ_T) =1 TE_θ,γ,x

Z_T

0

X_td t

!

= 1 θT

 x− γ

θ

‹

(1−e^−θT) +γ

θ, (2.5)

Var_θ,γ,x µˆ_T

= 1 T²E_θ,γ,x





 ZT

0

e^−θt Zt

0

e^θsdW_sd t

!₂



 (2.6)

= 1

θ²T²

T− 1

2θ(e^−2θT−1) +2

θ(e^−θT−1)

and so for allT≥1,

Var_θ,γ,x µˆ_T

≤ 1

2θ³T (2θ+1) (2.7)

and

E_θ,γ,x( ˆσ²_T) = 1 2θ + 1

4θ²T(1−e^−2θT)

−1+2θ

 x−γ

θ

‹2

− 1

θ²T²(1−e^−θT)²

 x−γ

θ

‹2

(1−e^−θT)

− 1 θ²T²

T− 1

2θ(e^−2θT−1) +2

θ(e^−θT−1)

which implies

E_θ,γ,x( ˆσ²_T)− 1 2θ

≤ 1

θ²T

θ

 x−γ

θ

‹₂ +2

θ

. (2.8)

Lemma 2.1. For any0≤α≤θ²/4, for all T ≥1, E_θ,γ,x exp α

Z _T

0

X_t²d t

!!

<∞,

and there exist finite positive constants L₁and L₂such that for all0≤α≤θ²/4and T≥1, E_θ,γ,x exp α

ZT

0

X²_td t

!!

≤L₁e^L2αT.

Proof. For any 0≤α≤θ²/4, setκ=p

θ²−2α. Then by Girsanov theorem, we have d P_θ,γ,x

d P_κ,γ,x =exp (

− Z _T

0

(θ−κ)X_td X_t− Z_T

0

(αX²_t −γ(θ−κ)X_t)d t )

and so

E_θ,γ,x exp α ZT

0

X_t²d t

!!

=E_κ,γ,x d P_θ,γ,x d P_κ,γ,x exp

( α

ZT

0

X²_td t )!

=E_κ,γ,x exp (

(−θ+κ) Z_T

0

X_td X_t+γ Z_T

0

(θ−κ)X_td t )!

=E_κ,γ,x exp

(−(θ−κ)

2 (X_T²−T) +γ Z T

0

(θ−κ)X_td t )!

≤exp

(θ−κ)T 2

E_κ,γ,x exp (

γ Z T

0

(θ−κ)X_td t )!

where the last inequality is due toθ≥κ. Now we have to estimateE_κ,γ,x(exp{γRT

0(θ−κ)X_td t}).

Since underP_κ,γ,x, ˆ

µ_T∼N 1

κT(x−γ

κ)(1−e^−κT) +γ κ, 1

κ²T²

T− 1

2κ(e^−2κT−1) +2

κ(e^−κT−1)

, we have

E_κ,γ,x exp (

γ Z T

0

(θ−κ)X_td t )!

=exp

γ(θ−κ) κ



x−γ κ

‹

(1−e^−κT) +γT

‹

·exp

¨γ²(θ−κ)² 2κ²

T− 1

2κ(e^−2κT−1) +2

κ(e^−κT−1)

« . Notingθ /p

2≤κ≤θ, 0≤θ−κ=2α/(θ+κ)≤2α/θand(θ−κ)²≤αθ for all 0≤α≤θ²/4, we complete the proof of the lemma.

ƒ

2.2 Logarithmic Sobolev inequality

Since the LSI with respect to the Cameron-Martin metric does not produce the concentration inequality of correct order in large timeT for the functionals

F(X):= 1 pT

Z T

0

g(X_s)ds−E ZT

0

g(X_s)ds

!!

,

in order to get the concentration inequality of correct order for the functionals F(X), as pointed out by Djellout, Guillin and Wu ([7]) we should establish the LSI with respect to theL²-metric.

Let us introduce the logarithmic Sobolev inequality onWwith respect to the gradient inL²([0,T],R) ([10]). Letµbe the Wiener measure onW =C([0,T],R). A function f :W →Ris said to be

differentiable with respect to theL²-norm, if it can be extend toL²([0,T],R)and for anyw∈W, there exists a bounded linear operatorD f(w):g→D_gf(w)onL²([0,T],R)such that

kglimkL2→0

|f(w+g)−f(w)−D_gf(w)| kgkL2

=0.

If f :W →R is differentiable with respect to the L²-norm, then there exists a unique element

∇f(w) = (∇tf(w),t∈[0,T])inL²([0,T],R)such that

D_gf(w) =〈∇f(w),g〉L², f or al l g∈L²([0,T],R).

Denote by C_b¹(W/L²)the space of all bounded function f on W, differentiable with respect to the L²-norm, such that ∇f is also continuous and bounded fromW equipped with L²-norm to L²([0,T],R). Applying Theorem 2.3 in[10]to the Ornstein-Uhlenbeck process with linear drift, we have

Ent_Pθ,γ,x(f²)≤ 2 θ²E_θ,γ,x

Z T

0

|∇tf|²d t

!

, f ∈C_b¹(W/L²) (2.9) where the entropy of f²is given by

Ent_Pθ,γ,x(f²) =E_θ,γ,x(f²logf²)−E_θ,γ,x(f²)logE_θ,γ,x(f²).

Lemma 2.2. For any|α| ≤θ²/4,

E_θ,γ,x exp (

α ZT

0

X²_td t−E_θ,γ,x Z T

0

X²_td t

!!)!

≤E_θ,γ,x exp (4α²

θ² Z T

0

X²_td t )!

and

E_θ,γ,x€ exp¦

αT€ ˆ

µ²_T−E_θ,γ,x( ˆµ²_T)Š©Š

≤E_θ,γ,x exp (4α²

θ² ZT

0

X_t²d t )!

.

Proof. We apply Theorem 2.7 in [12] to prove the conclusions of the lemma. Take A1 = {αf; |α| ≤θ²/4}andA2={αh; |α| ≤θ²/4}, where

f(w) = Z _T

0

w²_td t, h(w) = 1 T

Z_T

0

w_td t

!2

. Define

Γ₁(g₁) = 4 θ²

g₁²

f , g₁∈ A1; Γ₂(g₂) = 4 θ²

g²₂

h , g₂∈ A2.

Then for anyλ ∈[−1, 1], g₁ ∈ A1 and g₂ ∈ A2, λg₁ ∈ A1, λg₂ ∈ A2, Γ₁(λg₁) = λ²Γ₁(g₁), Γ₂(λg₂) =λ²Γ₂(g₂)and by Lemma 2.1

E_θ,γ,x exp{λΓ₁(g₁)}

<∞, E_θ,γ,x exp{λΓ₂(g₂)}

<∞.

Choose a sequence of realC^∞-functionsΦ_n,n≥1 with compact support such that lim_n→∞sup_|x|≤M|Φ_n(x)− e^x|=0 for allM∈(0,∞). For anyg₁=αf ∈ A1andg₂=αh∈ A2, set

F_n(w) = Φ_n g₁(w)/2

, H_n(w) = Φ_n g₂(w)/2 .

Then for any g∈L²([0,T],R),

kglimkL2→0

|F_n(w+g)−F_n(w)−αΦ^′_n g₁(w)/2

〈w,g〉L²| kgkL²

=0 and

kglimkL2→0

|H_n(w+g)−H_n(w)−αΦ^′_n g₂(w)/2₁

T

RT

0 w_td tRT 0 g_td t| kgkL²

=0.

Therefore,F_n,H_n∈C¹_b(W/L²),∇F_n=αΦ^′_n g₁(w)/2 w, and

∇H_n= α T

ZT

0

w_td tΦ^′_n g₂(w)/2

and so by (2.9), we have Ent_Pθ,γ,x€

F_n²Š

≤ 2 θ²E_θ,γ,x

Z_T

0

|αw_t|²d t€

Φ^′_n g₁(w)/2Š2

!

and

Ent_Pθ,γ,x€ H_n²Š

≤ 2 θ²E_θ,γ,x





 1

T α

Z T

0

w_td t

!₂

€Φ^′_n g₂(w)/2Š2





. Lettingn→ ∞and by Lemma 2.1, we get

Ent_Pθ,γ,x(e^g1)≤ 1

2E_θ,γ,x Γ₁(g₁)e^g1

, Ent_Pθ,γ,x(e^g2)≤1

2E_θ,γ,x Γ₂(g₂)e^g2

, (2.10)

and so the conclusions of the lemma hold by Theorem 2.7 in[12]andTµˆ²_T≤RT 0 X²_td t.

ƒ

2.3 Deviation inequalities

SinceX_T∼N€

µ_T,σ²_TŠ

, and underP_θ,γ,x ˆ

µ_T∼N 1

θT(x− γ

θ)(1−e^−θT) +γ θ, 1

θ²T²

T− 1

2θ(e^−2θT−1) +2

θ(e^−θT−1)

, it is easily to get from Chebyshev inequality, for anyr>0,

P_θ,γ,x€

X_T−E_θ,γ,x(X_T) ≥rŠ

≤2 exp¦

−θr²©

, (2.11)

P_θ,γ,x€

µˆ_T−E_θ,γ,x( ˆµ_T) ≥rŠ

≤2 exp

¨

−θ³Tr² 2θ+1

«

(2.12) where we used (2.7).

Lemma 2.3. There exist finite positive constants C₀,C₁,C₂such that for all r>0and all T≥1, P_θ,γ,x

Z _T

0

X²_td t−E_θ,γ,x Z_T

0

X²_td t

!

≥r T

!

≤C₀exp

−C₁r Tmin

1,C₂r and

P_θ,γ,x€

µˆ²_T−E_θ,γ,x( ˆµ²_T) ≥rŠ

≤C₀exp

−C₁r Tmin

1,C₂r .

In particular, there exist finite positive constants C₀,C₁,C₂such that for all r>0and all T≥1, P_θ,γ,x€

|σˆ²_T−E_θ,γ,x( ˆσ²_T)| ≥rŠ

≤C₀exp

−C₁r Tmin

1,C₂r .

Proof. We only prove the first inequality. By Lemma 2.2 and Lemma 2.1, there exist finite positive constantsL₁andL₂such that for allT ≥1, for any|α| ≤θ²/4,

E_θ,γ,x exp (

α ZT

0

X²_td t−E_θ,γ,x Z T

0

X_t²d t

!!)!

≤L₁e^L2α2T. Therefore, by Chebyshev inequality, for anyr>0,T≥1 and|α| ≤θ²/4,

P_θ,γ,x Z T

0

X_t²d t−E_θ,γ,x( ZT

0

X_t²d t)≥r T

!

≤L₁e^{−(αr−L2α2)T}

and

P_θ,γ,x Z T

0

X²_td t−E_θ,γ,x( Z T

0

X²_td t)≤ −r T

!

≤L₁e^{−(αr−L2α2)T}. Now, by

sup

|α|≤θ²/4{αr−L₂α²} ≥θ²r 8 min

1, 2r

L₂θ²

, we obtain the first inequality of the lemma from the above estimates.

ƒ

Lemma 2.4. There exist finite positive constants C₀,C₁and C₂such that for all r>0and all T≥1, P_θ,γ,x

‚

W_T

 ˆ µ_T− γ

θ

‹ ≥r T

Œ

≤C₀exp

−C₁r Tmin

1,C₂r . Proof. Since for anyr>0 andT≥1,

¨

W_T

 ˆ µ_T− γ

θ

‹ ≥r T

«

⊂¦

W_T( ˆµ_T−E_θ,γ,x( ˆµ_T))

≥r T/2©

∪

¨

W_T



E_θ,γ,x( ˆµ_T)− γ θ

‹

≥r T/2

«

⊂¦

|W_T| ≥p r T/2©

∪¦

( ˆµ_T−E_θ,γ,x( ˆµ_T)) ≥p

r©

∪ (

W_T

≥ θr T 2|€

x−_θ^γŠ

| )

,

by (2.12) andW_T∼N(0,T), we get P_θ,γ,x



W_T( ˆµ_T− γ θ)

≥r T

‹

≤2 exp

−Tr 8

+2 exp

¨

− θ³Tr 2θ+1

« +2 exp







− θ²r²T 8€

x−_θ^γŠ2





 .

ƒ

Lemma 2.5. For eachβ ∈R fixed, there exist finite positive constants C₀,C₁,C₂ such that for all r>0and all T≥1,

P_θ,γ,x

Z T

0

X_t−β dW_t

≥r T

!

≤C₀exp

−C₁r Tmin

1,C₂r . Proof. It is known that forα∈R,

M_T^(α)=exp (

α Z T

0

X_t−β

dW_t−α² 2

ZT

0

X_t−β2

d t )

, T≥0

isFT-martingale, whereFT:=σ(W_t,t≤T). Therefore, by Hölder inequality, we can get that for anyε∈(0, 1],

E_θ,γ,x exp (

α Z_T

0

X_t−β dW_t

)!

≤ E_θ,γ,x exp

((1+ε)²α² 2ε

Z T

0

X_t−β2

d t

)!!_1+ε^ε

E_θ,γ,x

M_T^((1+ε)α) _1+ε¹

= E_θ,γ,x exp (

(1+ε)²α² 2ε

Z T

0

X_t−β₂ d t

)!! ^ε

1+ε

.

In particular, takeε=1, then by Lemma 2.1, there exists finite positive constantsL₁=L₁(θ,β,γ,x) andL₂=L₂(θ,β,γ,x)such that for allT ≥1, for anyα²≤θ²/16, by Cauchy-Schwartz inequality,

E_θ,γ,x exp (

α Z T

0

X_t−β dW_t

)!

≤ E_θ,γ,x exp (

2α² ZT

0

(X_t−β)²d t )!!¹

2

≤ E_θ,γ,x exp (

4α² ZT

0

X²_td t )!!¹

4

E_θ,γ,x exp (

4α² ZT

0

(−2βX_t+β²)d t )!!¹

4

≤L₁e^L2α2T.

Therefore, by Chebyshev inequality, the conclusion of the lemma holds.

ƒ

Proof of Theorem 1.1

We only show the first inequality. The second one is similar. By θˆ_T−θ= W_T€

ˆ µ_T−_θ^γŠ

−RT 0

€X_t−_θ^㊠dW_t Tσˆ²_T

for anyr>0 andT≥1, P_θ,γ,x€

|θˆ_T−θ| ≥rŠ

≤P_θ,γ,x€

σˆ²_T−E_θ,γ,x( ˆσ²_T)

≥E_θ,γ,x( ˆσ²_T)/2Š +P_θ,γ,x

W_T

 ˆ µ_T− γ

θ

‹

− ZT

0

 X_t− γ

θ

‹ dW_t

≥E_θ,γ,x( ˆσ²_T)r T/2

!

Therefore, by Lemmas 2.3, 2.4 and 2.5, we obtain the first inequality of the theorem.

ƒ

3 Moderate deviations

In this section, we show Theorem 1.2. By (1.2) and (1.3), we have the following estimates

( ˆθ_T−θ) +2θ T

ZT

0

 X_t− γ

θ

‹ dW_t

(3.1)

≤|W_T€ ˆ µ_T−_θ^γŠ

|

Tσˆ²_T +|2θσˆ²_T−1|

RT 0

€X_t−^γ_θŠ dW_t

Tσˆ²_T

and for

(ˆγ_T−γ)−W_T T +2γ

T Z T

0

 X_t−γ

θ

‹ dW_t

(3.2)

≤|µˆ_T||W_T€ ˆ µ_T−_θ^γŠ

|

Tσˆ²_T +|2γσˆ²_T−µˆ_T|

RT 0

€X_t−_θ^㊠dW_t

Tσˆ²_T .

Lemma 3.1. (1). For any r>0, lim sup

T→∞

1

λ_TlogP_θ,γ,x

 µˆ_T− γ

θ W_T

≥p Tλ_Tr

‹

=−∞,

lim sup

T→∞

1

λ_TlogP_θ,γ,x µˆ_T−γ

θ

ZT

0

 X_t− γ

θ

‹ dW_t

≥p Tλ_Tr

!

=−∞

and

lim sup

T→∞

1

λ_TlogP_θ,γ,x ˆ σ²_T− 1

2θ

Z T

0

 X_t− γ

θ

‹ dW_t

≥p Tλ_Tr

!

=−∞.

(2). For anyδ >0, lim sup

T→∞

1

λ_T logP_θ,γ,x

( ˆθ_T−θ)−2θ T

Z T

0

 X_t−γ

θ

‹ dW_t

≥δ rλ_T

T

!

=−∞

and

lim sup

T→∞

1

λ_TlogP_θ,γ,x

(ˆγ_T−γ)−W_T T −2γ

T Z_T

0

 X_t− γ

θ

‹ dW_t

≥δ rλ_T

T

!

=−∞. Proof. (1). We only give the proof of the third assertion in (1). The rest is similar. For anyL>0,

( ˆ σ²_T− 1

2θ

Z T

0

 X_t− γ

θ

‹ dW_t

≥p Tλ_Tr

)

⊂

¨ ˆ σ²_T− 1

2θ ≥ r

L

«

∪





 1 pTλ_T

Z _T

0

 X_t−γ

θ

‹ dW_t

≥L





 .

By Lemma 2.3, and Lemma 2.5, we have lim sup

T→∞

1

λ_TlogP_θ,γ,x

‚ ˆ σ²_T− 1

2θ ≥ r

L

Œ

=−∞

and

lim sup

T→∞

1

λ_T logP_θ,γ,x



 1 pTλ_T

Z _T

0

 X_t− γ

θ

‹ dW_t

≥L



≤ −L²C₁C₂. Hence,

lim sup

T→∞

1

λ_TlogP_θ,γ,x ˆ σ²_T− 1

2θ

Z T

0

 X_t−γ

θ

‹ dW_t

≥p Tλ_Tr

!

≤ −L²C₁C₂. LettingL→ ∞, we obtain the third conclusion.

(2). It follows from (3.1) and (3.2) that

( ˆθ_T−θ)−2θ T

Z T

0

 X_t−γ

θ

‹ dW_t

≥δ rλ_T

T

!

⊂







|W_T

 ˆ µ_T−γ

θ

‹

| ≥δσˆ²_T pTλ_T

2







∪







|2θσˆ²_T−1|

ZT

0

 X_t− γ

θ

‹ dW_t

≥δσˆ²_T pTλ_T

2







⊂







|W_T

 ˆ µ_T−γ

θ

‹

| ≥δE_θ,γ,x( ˆσ²_T) pTλ_T

4







∪¦

|σˆ²_T−E_θ,γ,x( ˆσ²_T)| ≥E_θ,γ,x( ˆσ²_T)/2©

∪







|2θσˆ²_T−1|

Z _T

0

 X_t− γ

θ

‹ dW_t

≥δE_θ,γ,x( ˆσ²_T) pTλ_T

4







and

(ˆγ_T−γ)−2γ T

Z_T

0

 X_t− γ

θ

‹ dW_t

≥δ rλ_T

T

!

⊂







|µˆ_T||W_T

 µˆ_T− γ

θ

‹

| ≥δσˆ²_T pTλ_T

2







∪







|2γσˆ²_T−µˆ_T|

Z _T

0

 X_t−γ

θ

‹ dW_t

≥δσˆ²_T pTλ_T

2







⊂

§ µˆ_T− γ

θ ≥ γ

2θ ª

∪¦

|σˆ²_T−E_θ,γ,x( ˆσ²_T)| ≥E_θ,γ,x( ˆσ²_T)/2©

∪





 3γ 2θ|W_T

 ˆ µ_T− γ

θ

‹

| ≥δE_θ,γ,x( ˆσ²_T) pTλ_T

4







∪









2γσˆ²_T−γ θ +

µˆ_T−γ

θ

‹

Z _T

0

 X_t−γ

θ

‹ dW_t

≥δE_θ,γ,x( ˆσ²_T) pTλ_T

4





 .

Therefore, by Lemmas 2.3 and (1), we get the conclusions.

ƒ

Lemma 3.2. For eachβ,κ ∈ R fixed,

§ P_θ,γ,x

 pκ

TλT

R_T

0 X_t−β dW_t∈ ·

‹ ,T ≥1

ª

satisfies the LDP with speedλ_Tand rate function J(u) = ^θ2u2

κ²(θ+2(γ−θ β)²). Proof. By (2.12) and Lemma 2.3, we can get for anyδ >0,

T→∞lim 1

T logP_θ,γ,x 1 T

Z T

0

X_t−β2

d t− 1

2θ + 1

θ²(γ−θ β)²

≥δ

!

<0. (3.3) Therefore, Proposition 1 in[4]yields the conclusion of the lemma.

ƒ

Proof of Theorem 1.2

By Lemma 3.1,{P_θ,γ,x( qT

λT( ˆθ_T−θ)∈ ·),T ≥1}and{P_θ,γ,x( q T

λT(ˆγ_T−γ)∈ ·),T ≥1}are expo- nential equivalent to

( P_θ,γ,x

r T λ_T

2θ T

ZT

0

 X_t− γ

θ

‹ dW_t∈ ·

! ,T ≥1

)

and (

P_θ,γ,x r T

λ_T W_T

T +2γ T

Z T

0

 X_t−γ

θ

‹ dW_t

!

∈ ·

! ,T≥1

) , respectively. Noting forγ6= 0, ^WT

T +^2γ

T

RT 0

€X_t−_θ^γŠ

dW_t = ^2γ

T

RT 0

X_t−_θ^γ+_2γ¹

dW_t, Theorem 1.2 follows from Lemma 3.2.

ƒ

AcknowledgmentsThe authors are grateful to referees for their comments and suggestions.

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