212 Electronic Communications in Probability
Theorem 1.2. 1. P
θ ,γ,x
q
T λ
T
ˆ θ
T
− θ ∈ · , T
≥ 1 satisfies the large deviation principle with
speed λ
T
and rate function I
1
u =
u
2
4 θ
, that is, for any closed set F in R, lim sup
n →∞
1 λ
T
log P
θ ,γ,x
r T
λ
T
ˆ θ
T
− θ ∈ F ≤ − inf
u ∈F
u
2
4 θ
and open set G in R, lim inf
n →∞
1 λ
T
log P
θ ,γ,x
r T
λ
T
ˆ θ
T
− θ ∈ G ≥ − inf
u ∈G
u
2
4 θ
. 2.
P
θ ,γ,x
q
T λ
T
ˆ γ
T
− γ ∈ · , T
≥ 1 satisfies the large deviation principle with speed
λ
T
and rate function I
2
u =
θ u
2
2 θ +2γ
2
, that is, for any closed set F in R, lim sup
n →∞
1 λ
T
log P
θ ,γ,x
r T
λ
T
ˆ γ
T
− γ ∈ F ≤ − inf
u ∈F
θ u
2
2 θ + 2γ
2
and open set G in R, lim inf
n →∞
1 λ
T
log P
θ ,γ,x
r T
λ
T
ˆ γ
T
− γ ∈ G ≥ − inf
u ∈G
θ u
2
2 θ + 2γ
2
. 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 ˆ
θ
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 ˆ θ
T
and ˆ γ
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: X
t
=
x −
γ θ
e
−θ t
+ γ
θ + e
−θ t
Z
t
e
θ s
dW
s
. 2.1
From this expression, it is easily seen that for any t ≥ 0,
µ
t
:=E
θ ,γ,x
X
t
=
x −
γ θ
e
−θ t
+ γ
θ ,
2.2 σ
2 t
:=Var
θ ,γ,x
X
t
= 1
2 θ
1 − e
−2θ t
2.3
Deviation inequalities and MDP for estimators in OU model 213
and for any 0 ≤ s ≤ t,
Cov
θ ,γ,x
X
s
, X
t
= 1
2 θ
1 − e
−2θ s
e
−θ t−s
. 2.4
Therefore E
θ ,γ,x
ˆ µ
T
= 1
T E
θ ,γ,x
Z
T
X
t
d t =
1 θ T
x
− γ
θ
1 − e
−θ T
+ γ
θ ,
2.5 Var
θ ,γ,x
ˆ µ
T
= 1
T
2
E
θ ,γ,x
Z
T
e
−θ t
Z
t
e
θ s
dW
s
d t
2
2.6
= 1
θ
2
T
2
T −
1 2
θ e
−2θ T
− 1 + 2
θ e
−θ T
− 1 and so for all T
≥ 1, Var
θ ,γ,x
ˆ µ
T
≤ 1
2 θ
3
T 2θ + 1
2.7 and
E
θ ,γ,x
ˆ σ
2 T
= 1
2 θ
+ 1
4 θ
2
T 1 − e
−2θ T
−1 + 2θ
x −
γ θ
2
− 1
θ
2
T
2
1 − e
−θ T 2
x
− γ
θ
2
1 − e
−θ T
− 1
θ
2
T
2
T −
1 2
θ e
−2θ T
− 1 + 2
θ e
−θ T
− 1 which implies
E
θ ,γ,x
ˆ σ
2 T
− 1
2 θ
≤ 1
θ
2
T θ
x
− γ
θ
2
+ 2
θ .
2.8
Lemma 2.1. For any 0 ≤ α ≤ θ
2
4, for all T ≥ 1, E
θ ,γ,x
exp α
Z
T
X
2 t
d t ∞,
and there exist finite positive constants L
1
and L
2
such that for all 0 ≤ α ≤ θ
2
4 and T ≥ 1, E
θ ,γ,x
exp α
Z
T
X
2 t
d t ≤ L
1
e
L
2
αT
. Proof.
For any 0 ≤ α ≤ θ
2
4, set κ = p
θ
2
− 2α. Then by Girsanov theorem, we have d P
θ ,γ,x
d P
κ,γ,x
= exp −
Z
T
θ − κX
t
d X
t
− Z
T
αX
2 t
− γθ − κX
t
d t
214 Electronic Communications in Probability
and so E
θ ,γ,x
exp α
Z
T
X
2 t
d t =E
κ,γ,x
d P
θ ,γ,x
d P
κ,γ,x
exp α
Z
T
X
2 t
d t =E
κ,γ,x
exp −θ + κ
Z
T
X
t
d X
t
+ γ Z
T
θ − κX
t
d t =E
κ,γ,x
exp −θ − κ
2 X
2 T
− T + γ Z
T
θ − κX
t
d t ≤ exp
θ − κT 2
E
κ,γ,x
exp γ
Z
T
θ − κX
t
d t where the last inequality is due to
θ ≥ κ. Now we have to estimate E
κ,γ,x
exp{γ R
T
θ − κX
t
d t }.
Since under P
κ,γ,x
, ˆ
µ
T
∼ N 1
κT x −
γ κ
1 − e
−κT
+ γ
κ ,
1 κ
2
T
2
T −
1 2
κ e
−2κT
− 1 + 2
κ e
−κT
− 1 ,
we have E
κ,γ,x
exp γ
Z
T
θ − κX
t
d t = exp
γθ − κ κ
x
− γ
κ
1 − e
−κT
+ γT
· exp ¨ γ
2
θ − κ
2
2 κ
2
T −
1 2
κ e
−2κT
− 1 + 2
κ e
−κT
− 1 «
. Noting
θ p
2 ≤ κ ≤ θ , 0 ≤ θ − κ = 2αθ + κ ≤ 2αθ and θ − κ
2
≤ αθ for all 0 ≤ α ≤ θ
2
4, we complete the proof of the lemma.
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
F X := 1
p T
Z
T
gX
s
ds − E Z
T
gX
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 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 lim
kgk
L2
→0
| f w + g − f w − D
g
f w |
kgk
L
2
= 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
1 b
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
Ent
P
θ ,γ,x
f
2
≤ 2
θ
2
E
θ ,γ,x
Z
T
|∇
t
f |
2
d t ,
f ∈ C
1 b
W L
2
2.9 where the entropy of f
2
is given by Ent
P
θ ,γ,x
f
2
= E
θ ,γ,x
f
2
log f
2
− E
θ ,γ,x
f
2
log E
θ ,γ,x
f
2
.
Lemma 2.2. For any
|α| ≤ θ
2
4, E
θ ,γ,x
exp α
Z
T
X
2 t
d t − E
θ ,γ,x
Z
T
X
2 t
d t ≤ E
θ ,γ,x
exp 4
α
2
θ
2
Z
T
X
2 t
d t and
E
θ ,γ,x
exp
¦ αT
ˆ
µ
2 T
− E
θ ,γ,x
ˆ µ
2 T
© ≤ E
θ ,γ,x
exp 4
α
2
θ
2
Z
T
X
2 t
d t .
Proof. We apply Theorem 2.7 in [12] to prove the conclusions of the lemma. Take
A
1
= {α f ; |α| ≤ θ
2
4} and A
2
= {αh; |α| ≤ θ
2
4}, where f w =
Z
T
w
2 t
d t, hw =
1 T
Z
T
w
t
d t
2
. Define
Γ
1
g
1
= 4
θ
2
g
2 1
f , g
1
∈ A
1
; Γ
2
g
2
= 4
θ
2
g
2 2
h , g
2
∈ A
2
. Then for any
λ ∈ [−1, 1], g
1
∈ A
1
and g
2
∈ A
2
, λg
1
∈ A
1
, λg
2
∈ A
2
, Γ
1
λg
1
= λ
2
Γ
1
g
1
, Γ
2
λg
2
= λ
2
Γ
2
g
2
and by Lemma 2.1 E
θ ,γ,x
exp {λΓ
1
g
1
} ∞, E
θ ,γ,x
exp {λΓ
2
g
2
} ∞.
Choose a sequence of real C
∞
-functions Φ
n
, n ≥ 1 with compact support such that lim
n →∞
sup
|x|≤M
|Φ
n
x− e
x
| = 0 for all M ∈ 0, ∞. For any g
1
= α f ∈ A
1
and g
2
= αh ∈ A
2
, set F
n
w = Φ
n
g
1
w2 ,
H
n
w = Φ
n
g
2
w2 .
216 Electronic Communications in Probability
Then for any g ∈ L
2
[0, T ], R, lim
kgk
L2
→0
|F
n
w + g − F
n
w − αΦ
′ n
g
1
w2 〈w, g〉
L
2
| kgk
L
2
= 0 and
lim
kgk
L2
→0
|H
n
w + g − H
n
w − αΦ
′ n
g
2
w2
1 T
R
T
w
t
d t R
T
g
t
d t |
kgk
L
2
= 0. Therefore, F
n
, H
n
∈ C
1 b
W L
2
, ∇F
n
= αΦ
′ n
g
1
w2 w, and
∇H
n
= α
T Z
T
w
t
d tΦ
′ n
g
2
w2 and so by 2.9, we have
Ent
P
θ ,γ,x
F
2 n
≤
2 θ
2
E
θ ,γ,x
Z
T
|αw
t
|
2
d t
Φ
′ n
g
1
w2
2
and Ent
P
θ ,γ,x
H
2 n
≤
2 θ
2
E
θ ,γ,x
1
T α
Z
T
w
t
d t
2
Φ
′ n
g
2
w2
2
.
Letting n → ∞ and by Lemma 2.1, we get
Ent
P
θ ,γ,x
e
g
1
≤ 1
2 E
θ ,γ,x
Γ
1
g
1
e
g
1
, Ent
P
θ ,γ,x
e
g
2
≤ 1
2 E
θ ,γ,x
Γ
2
g
2
e
g
2
, 2.10
and so the conclusions of the lemma hold by Theorem 2.7 in [12] and T ˆ µ
2 T
≤ R
T
X
2 t
d t.
2.3 Deviation inequalities
