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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 byPθ,γ,xthe distribution of the solution of (1.1).

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

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

[15])

ˆ

θ_T=−T

RT

0 Xtd Xt+ (XT−x)

RT

0 Xtd t

TR₀TX2

td t−

RT

0 Xtd t

2 (1.2)

=θ+WTµˆT−

RT

0 XtdWt

Tσˆ2

T

,

ˆ

γ_T=−

RT

0 Xtd t

RT

0 Xtd Xt+ (XT−x)

RT

0 X 2

td t TR₀TX2

td t−

RT

0 Xtd t

2 (1.3)

=γ+WT

T +

ˆ

µ_T(W_Tµˆ_T−RT

0 XtdWt)

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 thatθˆT andγˆT 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 Xtd Xt,

RT

0 X 2

td t)by Girsanov’s formula and obtained large deviations forθˆT 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→ ∞}, _pλT

T →0. (1.5)

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

T≥1,

Pθ,γ,x

€

|θˆT−θ| ≥r

Š

≤C0exp

¦

−C1r T Eθ,γ,x( ˆσ2T)min

1,C2r

©

+C₀exp¦₋C₃T E_θ,γ,x( ˆσ2_T)©

and

Pθ,γ,x |γˆT−γ| ≥r

≤C0exp

¦

−C1r T Eθ,γ,x( ˆσ2T)min

1,C2r

©

+C₀exp¦₋C₃T Eθ,γ,x( ˆσ2T)

©

.

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).

Pθ,γ,x

_q

T

λT( ˆθT−θ)∈ ·

,T_≥1

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

2

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

lim sup n→∞

1

λ_T logPθ,γ,x

r

T

λ_T( ˆθT−θ)∈F

!

≤ −inf u∈F

u2 4θ and open set G inR_,

lim inf n→∞

1

λ_TlogPθ,γ,x

r

T

λ_T( ˆθT−θ)∈G

!

≥ −inf u∈G

u2

4θ. (2).

P_θ,γ,x

_q

T

λT

(ˆγ_T−γ)_{∈ ·}

,T_≥1

satisfies the large deviation principle with speedλ_T and rate function I2(u) = θu

2

2(θ+2γ2₎, that is, for any closed set F inR, lim sup

n→∞

1

λ_TlogPθ,γ,x

r

T

λ_T(ˆγT−γ)∈F

!

≤ −inf u∈F

θu2 2(θ+2γ2₎

and open set G inR_,

lim inf n→∞

1

λ_TlogPθ,γ,x

r

T

λ_T(ˆγT−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: Xt=



x− γ

θ

‹

e−θt+ γ

θ +e

−θt

Z t

0

eθsdWs. (2.1)

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



x− γ

θ

‹

e−θt+ γ

θ, (2.2)

σ2

t :=Varθ,γ,x(Xt) = 1 2θ(1−e

and for any 0_≤s_≤t,

Cov_θ,γ,x(Xs,Xt) = 1 2θ(1−e

−2θs_)e−θ(t−s)_{. (2.4)}

Therefore

Eθ,γ,x( ˆµT) = 1 TEθ,γ,x

ZT

0

Xtd t

!

= 1

θT



x₋ γ θ

‹

(1₋e−θT) +γ

θ, (2.5)

Varθ,γ,x µˆT

= 1

T2Eθ,γ,x



 

ZT

0

e−θt

Zt

0

eθsdW_sd t

!2



 (2.6)

= 1

θ2_T2

T₋ 1 2θ(e

−2θT

−1) +2

θ(e

−θT

−1)

and so for allT≥1,

Varθ,γ,x µˆT

≤ _2θ13_T (2θ+1) (2.7)

and

E_θ,γ,x( ˆσ2_T) = 1

2θ + 1

4θ2_T(1−e− 2θT₎

−1+2θ



x₋γ θ

‹2

− 1

θ2_T2(1−e−

θT₎2_x−γ

θ

‹2

(1−e−θT)

− 1

θ2_T2

T₋ 1 2θ(e

−2θT_{−1) +}2 θ(e

−θT₋₁₎

which implies

Eθ,γ,x

( ˆσ2_T)₋ 1

2θ

≤

1 θ2_T

θ



x−γ

θ

‹2 +2

θ

. (2.8)

Lemma 2.1. For any0_≤α_≤θ2_{/4, for all T ≥1,}

Eθ,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,

Eθ,γ,x exp α

ZT

0

X2_td t

!!

≤L1eL2αT.

Proof. For any 0≤α_≤θ2_{/4, setκ=}p_θ2_{−2α. Then by Girsanov theorem, we have}

d Pθ,γ,x d Pκ,γ,x

=exp

(

−

Z T

0

(θ₋κ)Xtd Xt−

ZT

0

(αX2_{t −}γ(θ₋κ)Xt)d t

and so

Eθ,γ,x exp α

ZT

0

X_t2d t

!!

=Eκ,γ,x

d Pθ,γ,x d Pκ,γ,x

exp

(

α

ZT

0

X2_td t

)!

=Eκ,γ,x exp

(

(₋θ+κ)

ZT

0

Xtd Xt+γ

ZT

0

(θ₋κ)Xtd t

)!

=Eκ,γ,x exp

(

−(θ₋κ)

2 (X

2

T−T) +γ

Z T

0

(θ₋κ)Xtd t

)!

≤exp

(θ₋κ)T 2

Eκ,γ,x exp

(

γ

Z T

0

(θ₋κ)Xtd t

)!

where the last inequality is due toθ_≥κ. Now we have to estimateEκ,γ,x(exp{γ

RT

0(θ−κ)Xtd t}).

Since underPκ,γ,x,

ˆ

µ_T∼N

1 κT(x−

γ κ)(1−e

−κT_{) +}γ κ,

1 κ2_T2

T₋ 1 2κ(e

−2κT

−1) +2

κ(e

−κT

−1)

, we have

Eκ,γ,x exp

(

γ

Z T

0

(θ₋κ)Xtd 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 /p2_≤κ_≤θ, 0_≤θ₋κ=2α/(θ+κ)_≤2α/θand(θ₋κ)2_{≤αθ for all 0≤α≤θ}2_/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):=_p1

T

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)

differentiable with respect to theL2_{-norm, if it can be extend toL}2_{([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}

lim

kgkL2→0

|f(w+g)₋f(w)₋D_gf(w)_|

kg_kL2 =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

θ2Eθ,γ,x

Z T

0

|∇tf|2d t

!

, f _∈C_b1(W/L2) (2.9) where the entropy of f2_{is given by}

EntPθ,γ,x(f

2_{) =E}

θ,γ,x(f2logf2)−Eθ,γ,x(f2)logEθ,γ,x(f2). Lemma 2.2. For any_|α_{| ≤}θ2/4,

Eθ,γ,x exp

(

α

ZT

0

X2_td t−Eθ,γ,x

Z T

0

X2_td t

!!)!

≤Eθ,γ,x exp

(

4α2

θ2

Z T

0

X2_td t

)!

and

Eθ,γ,x

€

exp¦αT€µˆ2_T−Eθ,γ,x( ˆµ2T)

Š©Š

≤Eθ,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}andA

2={αh; |α| ≤θ2/4}, where

f(w) =

Z T

0

w2_td t, h(w) = 1

T

ZT

0

wtd t

!2

. Define

Γ1(g1) =

4 θ2

g2 1

f , g1∈ A1; Γ2(g2) = 4 θ2

g2 2

h , g2∈ A2.

Then for anyλ _∈[₋1, 1], g_{1 ∈ A1} and g_{2 ∈ A2}, λg_{1 ∈ A1}, λg_{2 ∈ A2}, Γ₁(λg₁) = λ2_Γ 1(g1), Γ2(λg2) =λ2Γ2(g2)and by Lemma 2.1

Eθ,γ,x exp{λΓ1(g1)}

<_∞, Eθ,γ,x exp{λΓ2(g2)}

<_∞.

Choose a sequence of realC∞-functionsΦ_n,n≥1 with compact support such that limn→∞sup|x|≤M|Φn(x)− ex|=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),

lim

kgkL2→0

|F_n(w+g)₋F_n(w)₋αΦ′_n g1(w)/2

〈w,g_〉L2|

kgkL2

=0 and

lim

kgkL2→0

|H_n(w+g)₋H_n(w)₋αΦ′_n g₂(w)/2₁

T

RT

0 wtd t

RT

0 gtd t|

kg_kL2

=0. Therefore,Fn,Hn∈C1b(W/L

2_),∇F

n=αΦ′n g1(w)/2

w, and

∇Hn= α T

ZT

0

wtd tΦ′n g2(w)/2

and so by (2.9), we have

EntPθ,γ,x

€

F_n2Š_≤ 2 θ2Eθ,γ,x

ZT

0

|αwt|2d t

€

Φ′_n g1(w)/2

Š2

!

and

Ent_Pθ,γ,x€H_n2Š_≤ 2 θ2Eθ,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 EntPθ,γ,x(e

g1₎

≤ 1₂Eθ,γ,x Γ1(g1)eg1

, EntPθ,γ,x(e g2₎

≤1₂Eθ,γ,x Γ2(g2)eg2

, (2.10)

and so the conclusions of the lemma hold by Theorem 2.7 in[12]andTµˆ2

T≤

RT

0 X 2

td t.

ƒ

2.3

Deviation inequalities

SinceXT∼N

€

µT,σ2T

Š

, and underPθ,γ,x

ˆ

µ_T∼N

1 θT(x−

γ θ)(1−e

−θT_{) +}γ θ,

1 θ2_T2

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)

≥r

Š

≤2 exp¦₋θr2©, (2.11)

Pθ,γ,x

€ µˆ

T−Eθ,γ,x( ˆµT)

≥r

Š

≤2 exp

¨

−θ

3_Tr2

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, Pθ,γ,x

Z T

0

X2_td t₋Eθ,γ,x

ZT

0

X2_td t

!

≥r T

!

≤C0exp

−C1r Tmin

1,C2r

and

Pθ,γ,x

€ µˆ2

T−Eθ,γ,x( ˆµ2T)

≥r

Š

≤C0exp

−C1r Tmin

1,C2r

.

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

€

|σˆ2_T−Eθ,γ,x( ˆσ2T)| ≥r

Š

≤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

!!)!

≤L₁eL2α2_T .

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

Pθ,γ,x

Z T

0

X2_td t₋Eθ,γ,x(

Z T

0

X2_td t)_{≤ −}r T

!

≤L₁e−(αr−L2α2)T_. Now, by

sup

|α|≤θ2_/4{

αr₋L₂α2_{} ≥}θ

2_r

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, Pθ,γ,x

‚

WT



ˆ

µT− 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( ˆµT−Eθ,γ,x( ˆµT))

≥r T/2 ©

∪

¨

W_T



Eθ,γ,x( ˆµT)− γ θ

‹

≥

r T/2

«

⊂¦|WT| ≥

p

r T/2©_∪¦( ˆµ_T−E_θ,γ,x( ˆµ_T))

≥pr

©

∪

( W_T

≥

θr T 2|€x−_θγŠ|

)

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



WT( ˆµT−

γ θ)

≥r T

‹

≤2 exp

−Tr

8

+2 exp

¨

− θ

3_Tr

2θ+1

«

+2 exp







− θ

2_r2_T

8€x_−θγŠ2







.

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, Pθ,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

Xt−β

dWt− α2

2

ZT

0

Xt−β

2

d t

)

, T≥0

is_FT-martingale, where_FT:=σ(W_t,t_≤T). Therefore, by Hölder inequality, we can get that for anyε_∈(0, 1],

Eθ,γ,x exp

(

α

ZT

0

Xt−β

dWt

)!

≤ Eθ,γ,x exp

(

(1+ε)2_α2

2ε

Z T

0

Xt−β

2

d t

)!! ε

1+ε

Eθ,γ,x

M_T((1+ε)α)

₁₊1_ε

= 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,

Eθ,γ,x exp

(

α

Z T

0

Xt−β

dWt

)!

≤ Eθ,γ,x exp

(

2α2

ZT

0

(Xt−β)2d t

)!!1

2

≤ Eθ,γ,x exp

(

4α2

ZT

0

X2_td t

)!!1

4

Eθ,γ,x exp

(

4α2

ZT

0

(₋2βX_t+β2_{)d t}

)!!1

4

≤L1eL2α

2_T .

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

Proof of Theorem 1.1

We only show the first inequality. The second one is similar. By

ˆ

θT−θ=

W_T€µˆ_T−γ

θ

Š

−R₀T€X_t− γ

θ

Š

dW_t Tσˆ2

T for anyr>0 andT≥1,

P_θ,γ,x€_|θˆ_T−θ_{| ≥}rŠ

≤Pθ,γ,x

€ σˆ2

T−Eθ,γ,x( ˆσ2T)

≥E_θ,γ,x( ˆσ2

T)/2

Š

+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



Xt− γ θ ‹ dWt (3.1)

≤|WT

€

ˆ

µ_T−γ

θ

Š

|

Tσˆ2

T

+|

2θσˆ2_T−1_|

RT 0 €

Xt−

γ θ Š dWt

Tσˆ2

T and for

(ˆγ_T−γ)₋WT

T + 2γ T Z T 0 

Xt− γ θ ‹ dWt (3.2)

≤|µˆT||WT

€

ˆ

µ_T−γ

θ

Š

|

Tσˆ2

T

+|

2γσˆ2

T−µˆT|

RT 0 €

X_t−γ

θ Š dW_t

Tσˆ2

T

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

lim sup T→∞

1

λ_TlogPθ,γ,x

 µˆT−

γ θ W T ≥ p

Tλ_Tr

‹

=_−∞,

lim sup T→∞

1

λ_TlogPθ,γ,x

µˆT−

γ θ ZT 0 

Xt− γ θ ‹ dWt

≥pTλ_Tr

! =_−∞ and lim sup T→∞ 1 λT

logP_θ,γ,x

ˆ

σ2_T− 1 2θ Z T 0 

X_t− γ θ ‹ dW_t

≥pTλ_Tr

!

(2). For anyδ >0, lim sup

T→∞

1

λ_T logPθ,γ,x

( ˆθ_T−θ)₋2θ

T

Z T

0



Xt− γ θ ‹ dWt ≥δ r λ_T T ! =_−∞ and lim sup T→∞ 1

λ_TlogPθ,γ,x

(ˆγT−γ)− WT T − 2γ T ZT 0 

Xt− γ θ ‹ dWt ≥δ r λ_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

≥pTλ_Tr

) ⊂ ¨ ˆ

σ2_T− 1 2θ ≥ r L « ∪    1 p

Tλ_T

Z T 0 

Xt− γ θ ‹ dWt ≥L    .

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

T→∞

1

λ_TlogPθ,γ,x

‚ ˆ σ2 T− 1 2θ ≥ r L Œ =_−∞ and lim sup T→∞ 1

λ_T logPθ,γ,x

  1 p Tλ_T Z T 0 

Xt− γ θ ‹ dWt ≥L 

≤ −L2C₁C₂.

Hence,

lim sup T→∞

1

λ_TlogPθ,γ,x

ˆ σ2 T− 1 2θ Z T 0 

Xt− γ θ ‹ dWt

≥pTλ_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 r λT T ! ⊂   

|W_T



ˆ

µ_T−γ θ

‹

| ≥δσˆ2_T

p

Tλ_T 2    ∪   

|2θσˆ2_T−1_|

ZT 0 

X_t− γ θ ‹ dW_t

≥δσˆ2_T

p

Tλ_T 2    ⊂   

|W_T



ˆ

µ_T−γ θ

‹

| ≥δEθ,γ,x( ˆσ2T)

p

Tλ_T 4







∪¦|σˆ2_T−Eθ,γ,x( ˆσ2T)| ≥Eθ,γ,x( ˆσ2T)/2

©

∪







|2θσˆ2_T−1_|

Z T 0 

Xt− γ θ ‹ dWt

≥δEθ,γ,x( ˆσ2T)

p

Tλ_T 4





and

(ˆγT−γ)− 2γ

T

ZT

0



Xt− γ θ ‹ dWt ≥δ r λ_T T ! ⊂   

|µˆT||WT



ˆ

µT− γ θ

‹

| ≥δσˆ2_T

p

Tλ_T 2    ∪   

|2γσˆ2_T−µˆT|

Z T 0 

Xt− γ θ ‹ dWt

≥δσˆ2_T

p

Tλ_T 2    ⊂ § µˆT−

γ θ ≥ γ 2θ ª

∪¦|σˆ2_T−E_θ,γ,x( ˆσ2_T)_{| ≥}E_θ,γ,x( ˆσ2_T)/2©

∪







3γ 2θ|WT



ˆ

µT− γ θ

‹

| ≥δEθ,γ,x( ˆσ2T)

p Tλ_T 4    ∪     2γσˆ

2 T− γ θ + µˆT−

γ θ ‹ Z T 0 

Xt− γ θ ‹ dWt

≥δE_θ,γ,x( ˆσ2

T)

p

Tλ_T 4







.

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

ƒ

Lemma 3.2. For eachβ,κ _∈ R _fixed,

§

Pθ,γ,x



κ

p

TλT

RT

0 Xt−β

dWt∈ ·

‹

,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,

lim T→∞

1

T logPθ,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,_{Pθ,γ,x(

q

T

λT( ˆθT−θ)∈ ·),T ≥1}and{Pθ,γ,x(

q

T

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

(

Pθ,γ,x

r T λ_T 2θ T ZT 0 

Xt− γ θ

‹

dWt∈ ·

!

,T ≥1

)

and

(

Pθ,γ,x

r T λ_T WT T + 2γ T Z T 0 

X_t−γ θ ‹ dW_t ! ∈ · !

,T_≥1

)

, respectively. Noting forγ₆= 0, WT

T + 2γ T RT 0 €

Xt−

γ θ

Š

dWt = 2

γ

T

RT

0

Xt−

γ θ+

1 2γ

dWt, Theorem 1.2 follows from Lemma 3.2.

ƒ

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

References

[1] B. Bercu, A. Rouault. Sharp large deviations for the Ornstein-Uhlenbeck process.Theory of Prob. and its Appl., 46(2002), 1-19. MR1968706

[2] S. G. Bobkov, F. Götze. Exponential Integrability and Transportation Cost Related to Loga-rithmic Sobolev Inequalities.Journal of Functional Analysis, 163(1999), 1-28. MR1682772 [3] P. Cattiaux, A. Guillin. Deviation bounds for additive functionals of Markov process.ESAIM:

Probability and Statistics. 12(2008), 12-29. MR2367991

[4] A. Dembo. Moderate Deviations for Martingales with Bounded Jumps.Electronic Commu-nications in Probability, 1 (1996),11-17. MR1386290

[5] A. Dembo, D. Zeitouni.Large Deviations Techniques and Applications, Springer-Verlag, 1998. MR1619036

[6] J. D. Deuschel, D. W. Stroock.Large Deviations, New York, 1989. MR0997938

[7] H. Djellout, A. Guillin, L. M. Wu. Transportation cost-information inequalities and appli-cations to random dynamical system and diffusions.Ann. Probab., 32(2004), 2702-2732. MR2078555

[8] H. Djellout, A. Guillin, L. M. Wu. Moderate deviations for non-linear functionals and em-pirical spectral density of moving average processes.Ann. Inst. H. Poincar´e Probab. Statist., 42(2006), 393-416. MR2242954

[9] D. Florens-Landais, H. Pham, Large deviations in estimate of an Ornstein-Uhlenbeck model. Journal of Applied Probability, 36(1999), 60-77. MR1699608

[10] M. Gourcy, L. M. Wu, Logarithmic Sobolev inequalities of diffusions for the L2 metric. Potential Analysis, 25(2006), 77-102. MR2238937

[11] A. Guillin, R. Liptser, Examples of moderate deviation principles for diffusion processes. Discrete and continuous dynamical systems-series B, 6(2006), 803-828. MR2223909 [12] M. Ledoux, Concentration of Measure and Logarithmic Sobolev Inequalities. S´eminaire de

probabilit´es XXXIII, Lecture Notes in Mathematics, 1709(1999), 120-216. MR1767995 [13] M. Ledoux, The Concentration of Measure Phenomenon.Mathematical Surveys and

Mono-graphs 89, American Mathematical Society, 2001. MR1849347

[14] P. Lezaud, Chernoff and Berry-Essen’s inequalities for Markov processes.ESAIMS: Probabil-ity and Statistics, 5(2001), 183-201. MR1875670

[15] Yury A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics, London, 2004, MR2144185

[16] B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes. Oxford University Presss, New York,1999.

[17] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag, 1991. MR1083357

[18] L. M. Wu, A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri. Poincaré, Probabilités et Statistiques. 36(2000), 435-445. MR1785390

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



WT( ˆµT−

γ θ)

≥r T

‹

≤2 exp

−Tr 8

+2 exp ¨

− θ

3_Tr

2θ+1 «

+2 exp 



 − θ

2_r2_T

8€x_−θγŠ2 



 .

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,

Pθ,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

Xt−β

dWt−

α2

2 ZT

0

Xt−β 2

d t )

, T≥0

is_FT-martingale, where_FT:=σ(W_t,t_≤T). Therefore, by Hölder inequality, we can get that for anyε_∈(0, 1],

Eθ,γ,x exp (

α

ZT

0

Xt−β

dWt )!

≤ Eθ,γ,x exp (

(1+ε)2_α2

2ε

Z T

0

Xt−β 2

d t )!! ε

1+ε

Eθ,γ,x

M_T((1+ε)α) ₁₊1_ε

= 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,

Eθ,γ,x exp (

α

Z T

0

Xt−β

dWt )!

≤ Eθ,γ,x exp (

2α2

ZT

0

(Xt−β)2d t )!!1

2

≤ Eθ,γ,x exp (

4α2

ZT

0

X2_td t )!!1

4

Eθ,γ,x exp (

4α2

ZT

0

(₋2βXt+β2)d t )!!1

4

≤L1eL2α 2_T

.

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

Proof of Theorem 1.1

We only show the first inequality. The second one is similar. By

ˆ θT−θ=

W_T€µˆ_T−γ

θ Š

−R₀T€X_t− γ θ Š

dW_t Tσˆ2

T for anyr>0 andT≥1,

P_θ,γ,x€_|θˆ_T−θ_{| ≥}rŠ ≤Pθ,γ,x

€ σˆ2

T−Eθ,γ,x( ˆσ2T)

≥E_θ,γ,x( ˆσ2 T)/2

Š

+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

 Xt−

γ θ

‹ dWt

(3.1)

≤|WT €

ˆ µ_T−γ

θ Š

| Tσˆ2

T

+|

2θσˆ2_T−1_|

RT

0

€ Xt−

γ θ Š

dWt Tσˆ2

T and for

(ˆγ_T−γ)₋WT

T + 2γ

T Z T

0

 Xt−

γ θ

‹ dWt

(3.2)

≤|µˆT||WT €

ˆ µ_T−γ

θ Š

| Tσˆ2

T

+|

2γσˆ2

T−µˆT|

RT

0

€ X_t−γ

θ Š

dW_t Tσˆ2

T

.

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

T→∞

1

λ_TlogPθ,γ,x

 µˆT−

γ θ

W

T ≥

p Tλ_Tr

‹

=_−∞,

lim sup T→∞

1

λ_TlogPθ,γ,x

µˆT−

γ θ

ZT

0

 Xt−

γ θ

‹ dWt

≥pTλ_Tr !

=_−∞

and

lim sup T→∞

1

λT

logP_θ,γ,x

ˆ σ2_T− 1

2θ

Z T

0

 X_t− γ

θ

‹ dW_t

≥pTλ_Tr !

(2). For anyδ >0, lim sup

T→∞

1

λ_T logPθ,γ,x

( ˆθ_T−θ)₋2θ

T Z T

0

 Xt−

γ θ ‹ dWt ≥δ r λ_T T ! =_−∞ and lim sup T→∞ 1

λ_TlogPθ,γ,x

(ˆγT−γ)− WT T − 2γ T ZT 0  Xt−

γ θ ‹ dWt ≥δ r λ_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

≥pTλ_Tr ) ⊂ ¨ ˆ σ2_T− 1

2θ ≥ r L « ∪    1 p

Tλ_T

Z T 0  Xt−

γ θ ‹ dWt ≥L    .

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

T→∞

1

λ_TlogPθ,γ,x

‚ ˆ σ2 T− 1 2θ ≥ r L Œ =_−∞ and lim sup T→∞ 1

λ_T logPθ,γ,x



 1 p

Tλ_T

Z T 0  Xt−

γ θ ‹ dWt ≥L 

≤ −L2C₁C₂. Hence,

lim sup T→∞

1

λ_TlogPθ,γ,x

ˆ σ2 T− 1 2θ Z T 0  Xt−

γ θ ‹ dWt

≥pTλ_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 r λT T ! ⊂    |W_T



ˆ µ_T−γ

θ

‹ | ≥δσˆ2_T

p Tλ_T

2    ∪   

|2θσˆ2_T−1_| ZT 0  X_t− γ

θ ‹ dW_t

≥δσˆ2_T

p Tλ_T

2    ⊂    |W_T



ˆ µ_T−γ

θ

‹

| ≥δEθ,γ,x( ˆσ2T) p

Tλ_T

4 





∪¦|σˆ2_T−Eθ,γ,x( ˆσ2T)| ≥Eθ,γ,x( ˆσ2T)/2 ©

∪ 





|2θσˆ2_T−1_| Z T 0  Xt−

γ θ ‹ dWt

≥δEθ,γ,x( ˆσ2T) p

Tλ_T

4 



and

(ˆγT−γ)− 2γ

T ZT

0

 Xt−

γ θ ‹ dWt ≥δ r λ_T T ! ⊂    |µˆ_T||WT



ˆ µT−

γ θ

‹ | ≥δσˆ2_T

p Tλ_T

2    ∪   

|2γσˆ2_T−µˆ_T|

Z T 0  Xt−

γ θ ‹ dWt

≥δσˆ2_T

p Tλ_T

2    ⊂ § µˆT−

γ θ ≥ γ 2θ ª

∪¦|σˆ2_T−E_θ,γ,x( ˆσ2_T)_{| ≥}E_θ,γ,x( ˆσ2_T)/2©

∪ 



 3γ

2θ|WT



ˆ µT−

γ θ

‹

| ≥δEθ,γ,x( ˆσ2T) p

Tλ_T

4    ∪     2γσˆ

2 T− γ θ + µˆT−

γ θ ‹ Z T 0  Xt−

γ θ ‹ dWt

≥δEθ,γ,x( ˆσ2T) p

Tλ_T

4 



 .

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

ƒ

Lemma 3.2. For eachβ,κ _∈ R _fixed, §

Pθ,γ,x 

κ p

TλT

RT

0 Xt−β

dWt∈ ·

‹ ,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, lim

T→∞

1

T logPθ,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,_{Pθ,γ,x( q

T

λT( ˆθT−θ)∈ ·),T ≥1}and{Pθ,γ,x(

q T

λT(ˆγT−γ)∈ ·),T ≥1}are

expo-nential equivalent to (

Pθ,γ,x r T λ_T 2θ T ZT 0  Xt−

γ θ

‹ dWt∈ ·

! ,T ≥1

)

and

( Pθ,γ,x

r T λ_T WT T + 2γ T Z T 0  X_t−γ

θ ‹ dW_t ! ∈ · ! ,T_≥1

) , respectively. Noting forγ₆= 0, WT

T + 2γ T RT 0 € Xt−

γ θ Š

dWt = 2 γ T

RT

0

Xt−

γ θ+

1 2γ

dWt, Theorem 1.2 follows from Lemma 3.2.

ƒ

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

References

[1] B. Bercu, A. Rouault. Sharp large deviations for the Ornstein-Uhlenbeck process.Theory of Prob. and its Appl., 46(2002), 1-19. MR1968706

[2] S. G. Bobkov, F. Götze. Exponential Integrability and Transportation Cost Related to Loga-rithmic Sobolev Inequalities.Journal of Functional Analysis, 163(1999), 1-28. MR1682772

[3] P. Cattiaux, A. Guillin. Deviation bounds for additive functionals of Markov process.ESAIM: Probability and Statistics. 12(2008), 12-29. MR2367991

[4] A. Dembo. Moderate Deviations for Martingales with Bounded Jumps.Electronic Commu-nications in Probability, 1 (1996),11-17. MR1386290

[5] A. Dembo, D. Zeitouni.Large Deviations Techniques and Applications, Springer-Verlag, 1998. MR1619036

[6] J. D. Deuschel, D. W. Stroock.Large Deviations, New York, 1989. MR0997938

[7] H. Djellout, A. Guillin, L. M. Wu. Transportation cost-information inequalities and appli-cations to random dynamical system and diffusions.Ann. Probab., 32(2004), 2702-2732. MR2078555

[8] H. Djellout, A. Guillin, L. M. Wu. Moderate deviations for non-linear functionals and em-pirical spectral density of moving average processes.Ann. Inst. H. Poincar´e Probab. Statist., 42(2006), 393-416. MR2242954

[9] D. Florens-Landais, H. Pham, Large deviations in estimate of an Ornstein-Uhlenbeck model. Journal of Applied Probability, 36(1999), 60-77. MR1699608

[10] M. Gourcy, L. M. Wu, Logarithmic Sobolev inequalities of diffusions for the L2 metric. Potential Analysis, 25(2006), 77-102. MR2238937

[11] A. Guillin, R. Liptser, Examples of moderate deviation principles for diffusion processes. Discrete and continuous dynamical systems-series B, 6(2006), 803-828. MR2223909

[12] M. Ledoux, Concentration of Measure and Logarithmic Sobolev Inequalities. S´eminaire de probabilit´es XXXIII, Lecture Notes in Mathematics, 1709(1999), 120-216. MR1767995

[13] M. Ledoux, The Concentration of Measure Phenomenon.Mathematical Surveys and Mono-graphs 89, American Mathematical Society, 2001. MR1849347

[14] P. Lezaud, Chernoff and Berry-Essen’s inequalities for Markov processes.ESAIMS: Probabil-ity and Statistics, 5(2001), 183-201. MR1875670

[15] Yury A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes. Springer Series in Statistics, London, 2004, MR2144185

[16] B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes. Oxford University Presss, New York,1999.

[17] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag, 1991. MR1083357

[18] L. M. Wu, A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri. Poincaré, Probabilités et Statistiques. 36(2000), 435-445. MR1785390