Elect. Comm. in Probab. 14 (2009), 210–220
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
Submitted December 29, 2008, accepted in final form April 21, 2009
AMS 2000 Subject classification: 60F12, 62F12, 62N02
Keywords: Deviation inequality, logarithmic Sobolev inequality, moderate deviations, OrnsteinUhlenbeck process
Abstract
Some deviation inequalities and moderate deviation principles for the maximum likelihood estimators 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 = (−θ X t + γ)d t + dWt ,
X0 = x
(1.1)
where W is a standard Brownian motion and θ , γ are unknown parameters with θ ∈ (0, +∞). We
denote by Pθ ,γ,x the distribution of the solution of (1.1).
It is known that the maximum likelihood estimators (MLE) of the parameters θ and γ are (cf.
1
RESEARCH SUPPORTED BY THE NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA (10871153)
210
Deviation inequalities and MDP for estimators in OU model
211
[15])
RT
X t d X t + (X T − x) 0 X t d t
θ̂ T =
R T
2
RT
T 0 X t2 d t − 0 X t d t
RT
WT µ̂ T − 0 X t dWt
,
=θ +
T σ̂2T
−T
RT
0
RT
RT
X t d t 0 X t d X t + (X T − x) 0 X t2 d t
γ̂ T =
2
R T
RT
T 0 X t2 d t − 0 X t d t
RT
µ̂ T (WT µ̂ T − 0 X t dWt )
WT
=γ +
+
,
T
T σ̂2T
−
RT
0
where
µ̂ T =
1
T
(1.2)
T
Z
X t d t,
0
σ̂2T
=
1
T
(1.3)
T
Z
0
X t2 d t − µ̂2T .
(1.4)
It is known that θ̂ T and γ̂ T are consistent estimators of θ and γ and have asymptotic normality
(cf. [15]).
RT
For γ ≡ 0 case, Florens-Landais and Pham([9]) calculated the Laplace functional of ( 0 X t d X t ,
RT
X t2 d t) by Girsanov’s formula and obtained large deviations for θ̂ T by Gärtner-Ellis theorem.
0
Bercu and Rouault ([1]) presented a sharp large deviation for θ̂ 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 of θ̂ T and γ̂ T for γ 6= 0 case.
1.2 Main results
Throughout this paper, let λ T , T ≥ 1 be a positive sequence satisfying
λ T → ∞,
λT
p → 0.
T
(1.5)
Theorem 1.1. There exist finite positive constants C0 , C1 , C2 and C3 such that for all r > 0 and all
T ≥ 1,
€
Š
¦
©
Pθ ,γ,x |θ̂ T − θ | ≥ r ≤C0 exp −C1 r T Eθ ,γ,x (σ̂2T ) min 1, C2 r
¦
©
+ C0 exp −C3 T Eθ ,γ,x (σ̂2T )
and
¦
©
Pθ ,γ,x |γ̂ T − γ| ≥ r ≤C0 exp −C1 r T Eθ ,γ,x (σ̂2T ) min 1, C2 r
¦
©
+ C0 exp −C3 T Eθ ,γ,x (σ̂2T ) .
Remark 1.1. In this theorem and the remainder of the paper, all the constants involved depend on
θ , γ and the initial point x.
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Electronic Communications in Probability
q
Theorem 1.2. (1).
Pθ ,γ,x
T
λT
speed λ T and rate function I1 (u) =
(θ̂ T − θ ) ∈ · , T ≥ 1 satisfies the large deviation principle with
u2
,
4θ
that is, for any closed set F in R,
!
r
1
u2
T
lim sup
log Pθ ,γ,x
(θ̂ T − θ ) ∈ F ≤ − inf
u∈F 4θ
λT
n→∞ λ T
and open set G in R,
lim inf
n→∞
q
(2).
Pθ ,γ,x
r
1
λT
log Pθ ,γ,x
T
λT
λT
rate function I2 (u) =
θu
,
2(θ +2γ2 )
lim sup
n→∞
1
λT
!
(θ̂ T − θ ) ∈ G
≥ − inf
u∈G
u2
4θ
.
(γ̂ T − γ) ∈ · , T ≥ 1
2
T
satisfies the large deviation principle with speed λ T and
that is, for any closed set F in R,
r
log Pθ ,γ,x
!
T
λT
(γ̂ T − γ) ∈ F
≤ − inf
u∈F
θ u2
2(θ + 2γ2 )
and open set G in R,
lim inf
n→∞
1
λT
r
log Pθ ,γ,x
T
λT
!
(γ̂ T − γ) ∈ G
≥ − inf
u∈G
θ u2
2(θ + 2γ2 )
.
In γ = 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 of θ̂ 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
L 2 -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 estimators θ̂ T and γ̂ T by 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:
Z t

γ ‹ −θ t γ
e
+ + e−θ t
eθ s dWs .
Xt = x −
θ
θ
0
From this expression, it is easily seen that for any t ≥ 0,

γ ‹ −θ t γ
e
+ ,
µ t :=Eθ ,γ,x (X t ) = x −
θ
θ
1
−2θ t
2
(1 − e
)
σ t :=Varθ ,γ,x (X t ) =
2θ
(2.1)
(2.2)
(2.3)
Deviation inequalities and MDP for estimators in OU model
213
and for any 0 ≤ s ≤ t,
1
Covθ ,γ,x (X s , X t ) =
(1 − e−2θ s )e−θ (t−s) .
2θ
(2.4)
Therefore
Eθ ,γ,x (µ̂ T ) =
1
T
!
T
Z
Eθ ,γ,x
Varθ ,γ,x µ̂ T
1 
θT
x−
γ‹
(1 − e−θ T ) +
θ
!2 
Z t

e−θ t
eθ s dWs d t 
0

=
Xtdt
γ
θ
,
Z T

= 2 Eθ ,γ,x 
T
0
0
1
2 −θ T
1 −2θ T
= 2 2 T−
(e
− 1) + (e
− 1)
2θ
θ
θ T
1
and so for all T ≥ 1,
Varθ ,γ,x µ̂ T ≤
1
2θ 3 T
(2.5)
(2.6)
(2θ + 1)
(2.7)
and
Eθ ,γ,x (σ̂2T ) =

γ ‹2
−2θ T
(1
−
e
)
−1
+
2θ
x
−
2θ
θ
4θ 2 T
‹2

γ
1
(1 − e−θ T )
− 2 2 (1 − e−θ T )2 x −
θ
θ T 1 −2θ T
1
2 −θ T
− 2 2 T−
(e
− 1) + (e
− 1)
2θ
θ
θ T
1
1
+
which implies
Eθ ,γ,x (σ̂2T ) −

γ ‹2 2
.
+
≤ 2
θ x−
2θ
θ
θ
θ T
1
1
(2.8)
Lemma 2.1. For any 0 ≤ α ≤ θ 2 /4, for all T ≥ 1,
exp
Eθ ,γ,x
!!
T
Z
X t2 d t
α
< ∞,
0
and there exist finite positive constants L1 and L2 such that for all 0 ≤ α ≤ θ 2 /4 and T ≥ 1,
exp
Eθ ,γ,x
!!
T
Z
≤ L1 e L2 αT .
X t2 d t
α
0
Proof. For any 0 ≤ α ≤ θ 2 /4, set κ =
d Pθ ,γ,x
d Pκ,γ,x
( Z
= exp −
p
θ 2 − 2α. Then by Girsanov theorem, we have
T
0
(θ − κ)X t d X t −
)
T
Z
(αX t2
0
− γ(θ − κ)X t )d t
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Electronic Communications in Probability
and so
X t2 d t
α
exp
Eθ ,γ,x
!!
T
Z
0
d Pθ ,γ,x
=Eκ,γ,x
d Pκ,γ,x
(
( Z
)!
T
X t2 d t
exp α
0
T
Z
0
0
(
=Eκ,γ,x
exp
≤ exp
−(θ − κ)
2
(θ − κ)T
− T) + γ
( Z
Eκ,γ,x
)!
T
Z
(X T2
2
(θ − κ)X t d t
XtdXt + γ
exp (−θ + κ)
=Eκ,γ,x
)!
T
Z
0
(θ − κ)X t d t
)!
T
exp γ
0
(θ − κ)X t d t
where the last inequality is due to θ ≥ κ. Now we have to estimate Eκ,γ,x (exp{γ
Since under Pκ,γ,x ,
µ̂ T ∼ N
1
κT
(x −
γ
κ
)(1 − e
−κT
)+
γ
1
T−
,
κ κ2 T 2
1
2κ
(e
−2κT
− 1) +
2
κ
RT
0
(θ − κ)X t d t}).
(e
−κT
− 1)
,
we have
( Z
exp γ
Eκ,γ,x
0
= exp
)!
T
(θ − κ)X t d t
γ(θ − κ) 
κ
¨
· exp
x−
γ2 (θ − κ)2
γ‹
κ
(1 − e−κT ) + γT
T−
2κ2
1
2κ
(e
−2κT
‹
− 1) +
2
κ
«
(e
−κT
− 1)
.
p
Noting θ / 2 ≤ κ ≤ θ , 0 ≤ θ − κ = 2α/(θ + κ) ≤ 2α/θ and (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 time T for the functionals
1
F (X ) := p
T
T
Z
0
g(X s )ds − E
!!
T
Z
g(X s )ds
,
0
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 the L 2 -metric.
Let us introduce the logarithmic Sobolev inequality on W with respect to the gradient in L 2 ([0, T ], R)
([10]). Let µ be the Wiener measure on W = C([0, T ], R). A function f : W → R is said to be
Deviation inequalities and MDP for estimators in OU model
215
differentiable with respect to the L 2 -norm, if it can be extend to L 2 ([0, T ], R) and for any w ∈ W ,
there exists a bounded linear operator D f (w) : g → D g f (w) on L 2 ([0, T ], R) such that
| f (w + g) − f (w) − D g f (w)|
lim
kgk L2
kgk L2 →0
= 0.
If f : W → R is differentiable with respect to the L 2 -norm, then there exists a unique element
∇ f (w) = (∇ t f (w), t ∈ [0, T ]) in L 2 ([0, T ], R) such that
D g f (w) = ⟨∇ f (w), g⟩ L 2 , f or al l g ∈ L 2 ([0, T ], R).
Denote by C b1 (W /L 2 ) the space of all bounded function f on W , differentiable with respect to
the L 2 -norm, such that ∇ f is also continuous and bounded from W equipped with L 2 -norm to
L 2 ([0, T ], R). Applying Theorem 2.3 in [10] to the Ornstein-Uhlenbeck process with linear drift,
we have
!
Z T
2
2
2
|∇ t f | d t , f ∈ C b1 (W /L 2 )
(2.9)
Ent Pθ ,γ,x ( f ) ≤ 2 Eθ ,γ,x
θ
0
where the entropy of f 2 is given by
Ent Pθ ,γ,x ( f 2 ) = Eθ ,γ,x ( f 2 log f 2 ) − Eθ ,γ,x ( f 2 ) log Eθ ,γ,x ( f 2 ).
Lemma 2.2. For any |α| ≤ θ 2 /4,
(
Z T
Eθ ,γ,x
X t2 d t
exp α
0
− Eθ ,γ,x
(
!!)!
T
Z
X t2 d t
0
≤ Eθ ,γ,x
exp
(
Z
and
€
¦
Eθ ,γ,x exp αT
€
µ̂2T
Eθ ,γ,x (µ̂2T )
−
Š©Š
≤ Eθ ,γ,x
exp
4α2
4α2
θ2
X t2 d t
0
)!
T
X t2 d t
θ2
)!
T
Z
.
0
Proof. We apply Theorem 2.7 in [12] to prove the conclusions of the lemma. Take A1 =
{α f ; |α| ≤ θ 2 /4} and A2 = {αh; |α| ≤ θ 2 /4}, where
T
Z
w 2t d t,
f (w) =
0
Define
Γ1 (g1 ) =
4 g12
θ2 f
, g 1 ∈ A1 ;
h(w) =
1
T
!2
T
Z
wt d t
.
0
Γ2 (g2 ) =
4 g22
θ2 h
, g 2 ∈ A2 .
Then for any λ ∈ [−1, 1], g1 ∈ A1 and g2 ∈ A2 , λg1 ∈ A1 , λg2 ∈ A2 , Γ1 (λg1 ) = λ2 Γ1 (g1 ),
Γ2 (λg2 ) = λ2 Γ2 (g2 ) and by Lemma 2.1
Eθ ,γ,x exp{λΓ1 (g1 )} < ∞, Eθ ,γ,x exp{λΓ2 (g2 )} < ∞.
Choose a sequence of real C ∞ -functions Φn , n ≥ 1 with compact support such that limn→∞ sup|x|≤M |Φn (x)−
e x | = 0 for all M ∈ (0, ∞). For any g1 = α f ∈ A1 and g2 = αh ∈ A2 , set
Fn (w) = Φn g1 (w)/2 , H n (w) = Φn g2 (w)/2 .
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Electronic Communications in Probability
Then for any g ∈ L 2 ([0, T ], R),
|Fn (w + g) − Fn (w) − αΦ′n g1 (w)/2 ⟨w, g⟩ L 2 |
lim
=0
kgk L 2
kgk L2 →0
and
|H n (w + g) − H n (w) − αΦ′n g2 (w)/2
lim
1
T
RT
0
wt d t
RT
g t d t|
0
kgk L 2
kgk L2 →0
= 0.
Therefore, Fn , H n ∈ C b1 (W /L 2 ), ∇Fn = αΦ′n g1 (w)/2 w, and
∇H n =
α
T
Z
T
w t d tΦ′n g2 (w)/2
0
and so by (2.9), we have
Ent Pθ ,γ,x
€
Fn2
Š
≤
θ2
!
T
Z
2
2
|αw t | d t
Eθ ,γ,x
0
€
Φ′n
g1 (w)/2
Š2
and

€
Ent Pθ ,γ,x H n2
Š
2
1
≤ 2 Eθ ,γ,x 
T
θ
!2
T
Z
α
wt d t
€
Φ′n

Š 2 
g2 (w)/2
.
0
Letting n → ∞ and by Lemma 2.1, we get
Ent Pθ ,γ,x (e g1 ) ≤
1
2
Eθ ,γ,x Γ1 (g1 )e g1 ,
Ent Pθ ,γ,x (e g2 ) ≤
1
2
Eθ ,γ,x Γ2 (g2 )e g2 ,
and so the conclusions of the lemma hold by Theorem 2.7 in [12] and T µ̂2T ≤
RT
0
(2.10)
X t2 d t.
ƒ
2.3 Deviation inequalities
Š
€
Since X T ∼ N µ T , σ2T , and under Pθ ,γ,x
µ̂ T ∼ N
1
θT
(x −
γ
θ
)(1 − e
−θ T
)+
γ
1
,
θ θ2T2
T−
1
2θ
(e
−2θ T
− 1) +
it is easily to get from Chebyshev inequality, for any r > 0,
Š
¦
©
€
Pθ ,γ,x X T − Eθ ,γ,x (X T ) ≥ r ≤ 2 exp −θ r 2 ,
Pθ ,γ,x
where we used (2.7).
€
¨
«
Š
θ 3 Tr 2
µ̂ T − Eθ ,γ,x (µ̂ T ) ≥ r ≤ 2 exp −
2θ + 1
2
θ
(e
−θ T
− 1)
,
(2.11)
(2.12)
Deviation inequalities and MDP for estimators in OU model
217
Lemma 2.3. There exist finite positive constants C0 , C1 , C2 such that for all r > 0 and all T ≥ 1,
!
!
Z T
Z T
2
2
Xt dt
X t d t − Eθ ,γ,x
≥ r T ≤ C0 exp −C1 r T min 1, C2 r
Pθ ,γ,x
0
0
and
Pθ ,γ,x
€
Š
µ̂2T − Eθ ,γ,x (µ̂2T ) ≥ r ≤ C0 exp −C1 r T min 1, C2 r
.
In particular, there exist finite positive constants C0 , C1 , C2 such that for all r > 0 and all T ≥ 1,
€
Š
Pθ ,γ,x |σ̂2T − Eθ ,γ,x (σ̂2T )| ≥ r ≤ C0 exp −C1 r T min 1, C2 r .
Proof. We only prove the first inequality. By Lemma 2.2 and Lemma 2.1, there exist finite positive
constants L1 and L2 such that for all T ≥ 1, for any |α| ≤ θ 2 /4,
!!)!
(
Z T
Z T
Eθ ,γ,x
exp α
0
2
≤L1 e L2 α T .
X t2 d t
X t2 d t − Eθ ,γ,x
0
Therefore, by Chebyshev inequality, for any r > 0, T ≥ 1 and |α| ≤ θ 2 /4,
!
Z T
Z T
Pθ ,γ,x
0
and
X t2 d t − Eθ ,γ,x (
T
Z
Pθ ,γ,x
0
0
0
≤ L1 e−(αr−L2 α
2
)T
!
T
Z
X t2 d t − Eθ ,γ,x (
X t2 d t) ≥ r T
X t2 d t) ≤ −r T
≤ L1 e−(αr−L2 α
2
)T
.
Now, by
sup {αr − L2 α2 } ≥
|α|≤θ 2 /4
θ2r
2r
,
min 1,
8
L2 θ 2
we obtain the first inequality of the lemma from the above estimates.
ƒ
Lemma 2.4. There exist finite positive constants C0 , C1 and C2 such that for all r > 0 and all T ≥ 1,
‚
Œ

γ‹
Pθ ,γ,x WT µ̂ T −
≥ r T ≤ C0 exp −C1 r T min 1, C2 r .
θ
Proof. Since for any r > 0 and T ≥ 1,
¨
«

γ‹
WT µ̂ T −
≥ rT
θ
¨
⊂
¦
©
WT (µ̂ T − Eθ ,γ,x (µ̂ T )) ≥ r T /2 ∪
¦
⊂ |WT | ≥
p
©
r T /2 ∪
¦

WT
Eθ ,γ,x (µ̂ T ) −
p ©
(µ̂ T − Eθ ,γ,x (µ̂ T )) ≥ r ∪
γ‹
θ
«
≥ r T /2
(
WT ≥
θ rT
Š
€
γ
2| x − θ |
)
,
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Electronic Communications in Probability
by (2.12) and WT ∼ N (0, T ), we get
‹

γ
Pθ ,γ,x WT (µ̂ T − ) ≥ r T
θ

≤2 exp −
Tr
¨
+ 2 exp −
8
«
θ 3 Tr
+ 2 exp
2θ + 1

θ 2 r2 T 
− €
.
Š
γ 2

8 x−θ

ƒ
Lemma 2.5. For each β ∈ R fixed, there exist finite positive constants C0 , C1 , C2 such that for all
r > 0 and all T ≥ 1,
!
Z T
Pθ ,γ,x
X t − β dWt ≥ r T ≤ C0 exp −C1 r T min 1, C2 r .
0
Proof. It is known that for α ∈ R,
( Z T
(α)
MT
= exp α
0
X t − β dWt −
α2
2
)
T
Z
0
Xt − β
2
dt
,
T ≥0
is F T -martingale, where F T := σ(Wt , t ≤ T ). Therefore, by Hölder inequality, we can get that for
any ε ∈ (0, 1],
)!
( Z T
X t − β dWt
Eθ ,γ,x exp α
0
(
≤
Eθ ,γ,x
exp
Eθ ,γ,x
exp
0
(1 + ε)2 α2
2ε
)!!
T
Z
2ε
(
=
(1 + ε)2 α2
Xt − β
2
Xt − β
2
)!!
0
dt
T
Z
ε
1+ε
1
1+ε
((1+ε)α)
Eθ ,γ,x M T
ε
1+ε
.
dt
In particular, take ε = 1, then by Lemma 2.1, there exists finite positive constants L1 = L1 (θ , β, γ, x)
and L2 = L2 (θ , β, γ, x) such that for all T ≥ 1, for any α2 ≤ θ 2 /16, by Cauchy-Schwartz inequality,
)!
( Z T
X t − β dWt
Eθ ,γ,x exp α
0
(
≤
exp 2α2
Eθ ,γ,x
≤
≤L1 e
(
4
X t2 d t
0
L 2 α2 T
)!! 1
T
Z
exp 4α2
Eθ ,γ,x
2
(X t − β)2 d t
0
(
)!! 1
T
Z
Eθ ,γ,x
)!! 1
T
Z
exp 4α2
4
(−2β X t + β 2 )d t
0
.
Therefore, by Chebyshev inequality, the conclusion of the lemma holds.
ƒ
Deviation inequalities and MDP for estimators in OU model
219
Proof of Theorem 1.1
We only show the first inequality. The second one is similar. By
θ̂ T − θ =
Š RT €
Š
€
γ
γ
WT µ̂ T − θ − 0 X t − θ dWt
T σ̂2T
for any r > 0 and T ≥ 1,
€
Š
Pθ ,γ,x |θ̂ T − θ | ≥ r
€
Š
≤Pθ ,γ,x σ̂2T − Eθ ,γ,x (σ̂2T ) ≥ Eθ ,γ,x (σ̂2T )/2
!
Z T

γ‹
γ‹
Xt −
−
dWt ≥ Eθ ,γ,x (σ̂2T )r T /2
+ Pθ ,γ,x WT µ̂ T −
θ
θ
0
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
(θ̂ T − θ ) +
≤
T
Z
2θ
T

Xt −
0
€
Š
γ
|WT µ̂ T − θ |
θ
(3.1)
dWt
RT €
|2θ σ̂2T − 1|
+
T σ̂2T
γ‹
0
Xt −
γ
θ
Š
dWt
T σ̂2T
and for
(γ̂ T − γ) −
≤
WT
T
2γ
+
T
€
Š
γ
|µ̂ T ||WT µ̂ T − θ |
T σ̂2T
T
Z

Xt −
0
+
γ‹
θ
|2γσ̂2T − µ̂ T |
(3.2)
dWt
RT €
0
γ
θ
Xt −
Š
dWt
.
T σ̂2T
Lemma 3.1. (1). For any r > 0,
lim sup
T →∞
lim sup
T →∞
1
λT
1
λT

log Pθ ,γ,x
log Pθ ,γ,x
µ̂ T −
γ
θ
θ
WT ≥
T

γ‹
Z
lim sup
T →∞
λT
log Pθ ,γ,x
σ̂2T
−
1
2θ
Xt −
0
and
1
γ
µ̂ T −
T
Z
0

Xt −
θ
p
= −∞,
!
dWt ≥
γ‹
θ
‹
T λT r
p
T λT r
= −∞
!
dWt ≥
p
T λT r
= −∞.
220
Electronic Communications in Probability
(2). For any δ > 0,
lim sup
T →∞
1
λT
(θ̂ T − θ ) −
log Pθ ,γ,x
T
Z
2θ
T
γ‹

Xt −
0
θ
r
dWt ≥ δ
λT
!
= −∞
T
and
lim sup
T →∞
1
λT
(γ̂ T − γ) −
log Pθ ,γ,x
WT
−
T
2γ
T
Z
T

Xt −
0
γ‹
θ
r
dWt ≥ δ
λT
T
!
= −∞.
Proof. (1). We only give the proof of the third assertion in (1). The rest is similar. For any L > 0,
(
)
Z T
‹
p
1
γ
σ̂2T −
dWt ≥ T λ T r
Xt −
2θ
θ
0


¨
«
Z T
‹

 1
γ
1
r
Xt −
∪ p
dWt ≥ L .
⊂ σ̂2T −
≥

 Tλ
2θ
L
θ
0
T
By Lemma 2.3, and Lemma 2.5, we have
lim sup
T →∞
and
lim sup
T →∞
1
λT
‚
1
λT
σ̂2T
log Pθ ,γ,x

log Pθ ,γ,x  p
T
Z
1
T λT
−
1
2θ

0
Xt −
≥
γ‹
θ
r
Œ
= −∞
L

dWt ≥ L  ≤ −L 2 C1 C2 .
Hence,
lim sup
T →∞
1
λT
log Pθ ,γ,x
σ̂2T
−
1
2θ
T
Z

Xt −
0
γ‹
θ
!
dWt ≥
Letting L → ∞, we obtain the third conclusion.
(2). It follows from (3.1) and (3.2) that
r !
Z T
2θ
γ‹
λT
(θ̂ T − θ ) −
dWt ≥ δ
Xt −
T 0
θ
T

 
p
Z


T λT  
γ‹
2
2
| ≥ δσ̂ T
∪ |2θ σ̂ T − 1|
⊂ |WT µ̂ T −

θ
2  


≤ −L 2 C1 C2 .

T
0

|WT
µ̂ T −
γ‹
θ
p
| ≥ δEθ ,γ,x (σ̂2T )

Xt −
γ‹
θ
p
dWt ≥ δσ̂2T
©
T λT  ¦ 2
∪ |σ̂ T − Eθ ,γ,x (σ̂2T )| ≥ Eθ ,γ,x (σ̂2T )/2
4 


T
Z
∪
T λT r


⊂
p


|2θ σ̂2T − 1|
0

Xt −
γ‹
θ
p
dWt ≥ δEθ ,γ,x (σ̂2T )
T λT 
4 
T λT 
2 
Deviation inequalities and MDP for estimators in OU model
221
and
(γ̂ T − γ) −
2γ
T
Z
T
Xt −
0
r
γ‹

dWt ≥ δ
θ

⊂



|µ̂ T ||WT
µ̂ T −
γ‹
θ
| ≥ δσ̂2T
!
T


p
λT

T
Z
T λT  
∪ |2γσ̂2T − µ̂ T |
2  

Xt −
0
γ‹
θ
p
dWt ≥ δσ̂2T
T λT 
2 
©
γ ª ¦ 2
γ
∪ |σ̂ T − Eθ ,γ,x (σ̂2T )| ≥ Eθ ,γ,x (σ̂2T )/2
≥
µ̂ T −
θ
2θ


p

 3γ
T λT 
γ‹
∪
| ≥ δEθ ,γ,x (σ̂2T )
|WT µ̂ T −
 2θ
θ
4 
§
⊂


∪


2γσ̂2T −
γ
θ
+ µ̂ T −
T
Z
γ ‹
θ
γ‹

Xt −
0
θ
p
dWt ≥ δEθ ,γ,x (σ̂2T )
T λT 
.
4 
Therefore, by Lemmas 2.3 and (1), we get the conclusions.
ƒ

§
Lemma 3.2. For each β, κ ∈ R fixed,
Pθ ,γ,x
RT
pκ
T λT 0
θ u
.
κ2 (θ +2(γ−θ β)2 )
2 2
LDP with speed λ T and rate function J(u) =
‹
ª
X t − β dWt ∈ · , T ≥ 1
satisfies the
Proof. By (2.12) and Lemma 2.3, we can get for any δ > 0,
!
Z T
1
1
1
1
2
2
lim
≥ δ < 0.
log Pθ ,γ,x
+ 2 (γ − θ β)
Xt − β dt −
T →∞ T
T 0
2θ
θ
(3.3)
Therefore, Proposition 1 in [4] yields the conclusion of the lemma.
ƒ
Proof of Theorem 1.2
q
By Lemma 3.1, {Pθ ,γ,x (
T
λT
q
(θ̂ T − θ ) ∈ ·), T ≥ 1} and {Pθ ,γ,x (
T
λT
T
!
(γ̂ T − γ) ∈ ·), T ≥ 1} are expo-
nential equivalent to
(
r
T 2θ
Pθ ,γ,x
and
(
r
Pθ ,γ,x
λT T
T
WT
λT
T
respectively. Noting for γ 6= 0,
1.2 follows from Lemma 3.2.
WT
T
Z

Xt −
0
2γ
Z
T

γ‹
θ
dWt ∈ ·
γ‹
!
)
,T ≥1
!
)
dWt ∈ · , T ≥ 1 ,
Xt −
T 0
θ
R €
Š
R 2γ T
γ
2γ T
γ
1
dWt , Theorem
+ T 0 X t − θ dWt = T 0 X t − θ + 2γ
+
222
Electronic Communications in Probability
ƒ
Acknowledgments The authors are grateful to referees for their comments and suggestions.
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