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: [email protected]
HUI JIANG
School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China email: [email protected]
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 Xt= (−θXt+γ)d t+dWt, 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.
1RESEARCH SUPPORTED BY THE NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA (10871153)
[15])
ˆ
θT=−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
γT=−
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
µT(WTµˆT−RT
0 XtdWt)
Tσˆ2
T
,
where
ˆ
µT= 1
T
Z T
0
Xtd t, σˆ2T= 1
T
ZT
0
X2td t−µˆ2T. (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 θˆT. 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 ofθˆT andγˆTforγ6=0 case.
1.2
Main results
Throughout this paper, letλT,T≥1 be a positive sequence satisfying λT→ ∞, 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
≤C0exp
¦
−C1r T Eθ,γ,x( ˆσ2T)min
1,C2r
©
+C0exp¦−C3T Eθ,γ,x( ˆσ2T)©
and
Pθ,γ,x |γˆT−γ| ≥r
≤C0exp
¦
−C1r T Eθ,γ,x( ˆσ2T)min
1,C2r
©
+C0exp¦−C3T Eθ,γ,x( ˆσ2T)
©
.
Theorem 1.2. (1).
Pθ,γ,x
q
T
λT( ˆθT−θ)∈ ·
,T≥1
satisfies the large deviation principle with speedλTand rate function I1(u) = u
2
4θ, that is, for any closed set F inR,
lim sup n→∞
1
λT logPθ,γ,x
r
T
λT( ˆθT−θ)∈F
!
≤ −inf u∈F
u2 4θ
and open set G inR,
lim inf n→∞
1
λTlogPθ,γ,x
r
T
λT( ˆθT−θ)∈G
!
≥ −inf u∈G
u2
4θ.
(2).
Pθ,γ,x
q
T
λT
(ˆγT−γ)∈ ·
,T≥1
satisfies the large deviation principle with speedλT and rate function I2(u) = θu
2
2(θ+2γ2), that is, for any closed set F inR, lim sup
n→∞
1
λTlogPθ,γ,x
r
T
λT(ˆγT−γ)∈F
!
≤ −inf u∈F
θu2 2(θ+2γ2)
and open set G inR,
lim inf n→∞
1
λTlogPθ,γ,x
r
T
λT(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 ofθˆT, 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 estimatorsθˆTandγˆTby 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, µt:=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,
Covθ,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
eθsdWsd t
!2
(2.6)
= 1
θ2T2
T− 1 2θ(e
−2θT
−1) +2
θ(e
−θT
−1)
and so for allT≥1,
Varθ,γ,x µˆT
≤ 2θ13T (2θ+1) (2.7)
and
Eθ,γ,x( ˆσ2T) = 1
2θ + 1
4θ2T(1−e− 2θT)
−1+2θ
x−γ θ
2
− 1
θ2T2(1−e−
θT)2x−γ
θ
2
(1−e−θT)
− 1
θ2T2
T− 1 2θ(e
−2θT−1) +2 θ(e
−θT−1)
which implies
Eθ,γ,x
( ˆσ2T)− 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,
Eθ,γ,x exp α
Z T
0
Xt2d t
!!
<∞,
and there exist finite positive constants L1and L2such that for all0≤α≤θ2/4and T≥1,
Eθ,γ,x exp α
ZT
0
X2td t
!!
≤L1eL2αT.
Proof. For any 0≤α≤θ2/4, setκ=pθ2−2α. Then by Girsanov theorem, we have
d Pθ,γ,x d Pκ,γ,x
=exp
(
−
Z T
0
(θ−κ)Xtd Xt−
ZT
0
(αX2t −γ(θ−κ)Xt)d t
and so
Eθ,γ,x exp α
ZT
0
Xt2d t
!!
=Eκ,γ,x
d Pθ,γ,x d Pκ,γ,x
exp
(
α
ZT
0
X2td 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
µT∼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
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≤θ2/4,
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 timeT for the functionals
F(X):=p1
T
Z T
0
g(Xs)ds−E
ZT
0
g(Xs)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 toL2([0,T],R)and for anyw∈W,
there exists a bounded linear operatorD f(w):g→Dgf(w)onL2([0,T],R)such that
lim
kgkL2→0
|f(w+g)−f(w)−Dgf(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
Dgf(w) =〈∇f(w),g〉L2, f or al l g∈L2([0,T],R).
Denote by Cb1(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
EntPθ,γ,x(f2)≤ 2
θ2Eθ,γ,x
Z T
0
|∇tf|2d t
!
, f ∈Cb1(W/L2) (2.9) where the entropy of f2is 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
X2td t−Eθ,γ,x
Z T
0
X2td t
!!)!
≤Eθ,γ,x exp
(
4α2
θ2
Z T
0
X2td t
)!
and
Eθ,γ,x
exp¦αTµˆ2T−Eθ,γ,x( ˆµ2T)
©
≤Eθ,γ,x exp
(
4α2
θ2
ZT
0
Xt2d 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
w2td 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], g1 ∈ A1 and g2 ∈ A2, λg1 ∈ A1, λg2 ∈ A2, Γ1(λg1) = λ2Γ 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∞-functionsΦn,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
Fn(w) = Φn g1(w)/2
, Hn(w) = Φn g2(w)/2
Then for any g∈L2([0,T],R),
lim
kgkL2→0
|Fn(w+g)−Fn(w)−αΦ′n g1(w)/2
〈w,g〉L2|
kgkL2
=0
and
lim
kgkL2→0
|Hn(w+g)−Hn(w)−αΦ′n g2(w)/21
T
RT
0 wtd t
RT
0 gtd t|
kgkL2
=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
Fn2≤ 2 θ2Eθ,γ,x
ZT
0
|αwt|2d t
Φ′n g1(w)/2
2
!
and
EntPθ,γ,xHn2≤ 2 θ2Eθ,γ,x
1
T α
Z T
0
wtd t
!2
Φ′n g2(w)/22
.
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.
2.3
Deviation inequalities
SinceXT∼N
µT,σ2T
, and underPθ,γ,x
ˆ
µT∼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, Pθ,γ,xXT−Eθ,γ,x(XT)
≥r
≤2 exp¦−θr2©, (2.11)
Pθ,γ,x
µˆ
T−Eθ,γ,x( ˆµT)
≥r
≤2 exp
¨
−θ
3Tr2
2θ+1
«
(2.12)
Lemma 2.3. There exist finite positive constants C0,C1,C2such that for all r>0and all T≥1,
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,
we obtain the first inequality of the lemma from the above estimates.
by (2.12) andWT∼N(0,T), we get Pθ,γ,x
WT( ˆµT−
γ θ)
≥r T
≤2 exp
−Tr
8
+2 exp
¨
− θ
3Tr
2θ+1
«
+2 exp
− θ
2r2T
8x−θγ2
.
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
Xt−β
dWt
≥r T
!
≤C0exp
−C1r Tmin
1,C2r
.
Proof. It is known that forα∈R,
MT(α)=exp
(
α
Z T
0
Xt−β
dWt− α2
2
ZT
0
Xt−β
2
d t
)
, T≥0
isFT-martingale, whereFT:=σ(Wt,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)2α2
2ε
Z T
0
Xt−β
2
d t
)!! ε
1+ε
Eθ,γ,x
MT((1+ε)α)
1+1ε
= Eθ,γ,x exp
(
(1+ε)2α2 2ε
Z T
0
Xt−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
X2td t
)!!1
4
Eθ,γ,x exp
(
4α2
ZT
0
(−2βXt+β2)d t
)!!1
4
≤L1eL2α
2T .
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
Therefore, by Lemmas 2.3, 2.4 and 2.5, we obtain the first inequality of the theorem.
3
Moderate deviations
In this section, we show Theorem 1.2. By (1.2) and (1.3), we have the following estimates
and for
lim sup
T→∞
lim sup T→∞
(2). For anyδ >0,
lim sup T→∞
lim sup T→∞
Proof. (1). We only give the proof of the third assertion in (1). The rest is similar. For anyL>0,
(
By Lemma 2.3, and Lemma 2.5, we have
lim sup T→∞
lim sup T→∞
lim sup T→∞
LettingL→ ∞, we obtain the third conclusion. (2). It follows from (3.1) and (3.2) that
and
Therefore, by Lemmas 2.3 and (1), we get the conclusions.
satisfies the LDP with speedλTand rate function J(u) = θ2u2
Therefore, Proposition 1 in[4]yields the conclusion of the lemma.
Proof of Theorem 1.2
By Lemma 3.1,{Pθ,γ,x( nential equivalent to
(
AcknowledgmentsThe authors are grateful to referees for their comments and suggestions.
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