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Statistics and Probability Letters 79 (2009) 2076–2085
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Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro
Least squares estimator for discretely observed Ornstein–Uhlenbeck
processes with small Lévy noisesI
Hongwei Long ∗
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-0991, USA
article
abstract
info
Article history:
Received 15 September 2008
Received in revised form 26 June 2009
Accepted 29 June 2009
Available online 3 July 2009
We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck
processes with small Lévy noises, observed at n regularly spaced time points ti = i/n, i =
1, . . . , n on [0, 1]. Least squares method is used to obtain an estimator of the drift
parameter. The consistency and the rate of convergence of the least squares estimator (LSE)
are established when a small dispersion parameter ε → 0 and n → ∞ simultaneously.
The asymptotic distribution of the LSE in our general setting is shown to be the convolution
of a normal distribution and a stable distribution. The obtained results are different from
the classical cases where asymptotic distributions are normal.
© 2009 Elsevier B.V. All rights reserved.
MSC:
primary 62F12
62M05
secondary 60G52
60J75
1. Introduction
Let (Ω , F , P) be a basic probability space equipped with a right continuous and increasing family of σ -algebras (Ft , t ≥
0). Let (Lt , t ≥ 0) be an (Ft )-adapted Lévy process with special decomposition
Lt = aBt + bZt ,
where a, b are known constants, (Bt , t ≥ 0) is a standard Brownian motion and (Zt , t ≥ 0) is a standard α -stable Lévy
motion independent of (Bt , t ≥ 0), with Z1 ∼ Sα (1, β, 0) and β ∈ [−1, 1] is a skewness parameter. For technical reasons,
we assume that 1 < α < 2. The generalized Ornstein–Uhlenbeck process X = (Xt , t ≥ 0), starting from x0 ∈ R is defined
as the unique strong solution to the following linear stochastic differential equation (SDE)
dXt = −θ Xt dt + ε dLt ,
t ∈ [0, 1];
X 0 = x0 .
(1.1)
Assume that this process is observed at regularly spaced time points {ti = i/n, i = 1, 2, . . . , n}. The only unknown quantity
in SDE (1.1) is the parameter θ . We denote the true value of the parameter by θ0 . The purpose of this paper is to study the
least squares estimator for the true value θ0 based on the sampling data (Xti )ni=1 . Generalized O–U processes have been used
to describe geophysical models in Ditlevsen (1999a,b). In these models, climate change is related to an SDE driven by an
α -stable process (with α ≈ 1.75).
In the case of diffusion processes driven by Brownian motions, a popular method is the maximum likelihood estimator
(MLE), based on the Girsanov density, when the processes can be observed continuously (see Prakasa Rao (1999), Liptser
and Shiryaev (2001), Kutoyants (2004)). When a diffusion process is observed only at discrete times, in most cases the
transition density, and hence the likelihood function of the observations, is not explicitly computable. In order to overcome
I This work is supported by FAU Start-up funding at the C.E. Schmidt College of Science.
∗
Tel.: +1 561 297 0810; fax: +1 561 297 2436.
E-mail address: [email protected].
0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2009.06.018
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this difficulty, some approximate likelihood methods have been proposed by Lo (1988), Pedersen (1995a,b), Poulsen (1999),
and Aït-Sahalia (2002). For a comprehensive review on MLE and other related methods, we refer to Sørensen (2004). Least
squares estimator (LSE) is asymptotically equivalent to the MLE. For the LSE, the convergence in probability is proved
in Dorogovcev (1976) and Le Breton (1976), the strong consistency is studied in Kasonga (1988), and the asymptotic
distribution was studied in Prakasa Rao (1983). For a more recent comprehensive discussion, we refer to Prakasa Rao (1999),
Kutoyants (2004) and the references therein.
The parametric estimation problems for diffusion processes with jumps based on discrete observations have been
studied by Shimizu and Yoshida (2006) and Shimizu (2006) via maximum likelihood method. They established consistency
and asymptotic normality for the proposed estimators. The driving jump processes considered in Shimizu and Yoshida
(2006) and Shimizu (2006) include a large class of Lévy processes such as compound Poisson processes, gamma, inverse
Gaussian, variance gamma, normal inverse Gaussian or some generalized tempered stable processes. Masuda (2005) dealt
with the consistency and asymptotic normality of the TFE (trajectory-fitting estimator) and LSE when the driving process
is zero-mean adapted process (including Lévy process) with finite moments. The parametric estimation for Lévy-driven
Ornstein–Uhlenbeck processes was also studied by Brockwell et al. (2007), Valdivieso et al. (2008), and Spiliopoulos (2008).
However, the aforementioned papers were unable to cover an important class of driving Lévy processes, namely α -stable
Lévy motions with α ∈ (0, 2). Recently, Hu and Long (2007, 2009) have started the study on parameter estimation for
Ornstein–Uhlenbeck processes driven by α -stable Lévy motions.
The asymptotic theory of parametric estimation for diffusion processes with small white noise based on continuoustime observations is well developed (see, e.g., Kutoyants (1984, 1994), Yoshida (1992a, 2003), Uchida and Yoshida (2004a)).
There have been many applications of small noise asymptotics to mathematical finance, see for example Yoshida (1992b),
Takahashi (1999), Kunitomo and Takahashi (2001), Takahashi and Yoshida (2004), Uchida and Yoshida (2004b). From a
practical point of view, in parametric inference, it is more realistic and interesting to consider asymptotic estimation for
diffusion processes with small noise based on discrete observations. Substantial progress has been made in this direction.
Genon-Catalot (1990) and Laredo (1990) studied the efficient estimation of drift parameters of small diffusions from discrete
observations when ε → 0 and n → ∞. Sørensen (2000) used martingale estimating function to establish consistency and
asymptotic normality of the estimators of drift and diffusion coefficient parameters when ε → 0 and n is fixed. Sørensen
and Uchida (2003) and Gloter and Sørensen (2008) used a contrast function to study the efficient estimation for unknown
parameters in both drift and diffusion coefficient functions. Uchida (2004, 2008) used the martingale estimating function
approach to study the estimation of drift parameters for small diffusions under weaker conditions. Thus, in the cases of small
diffusions, the asymptotic distributions of the estimators are normal under suitable conditions on ε and n.
However, there has been no study on parametric inference for stochastic processes with small Lévy noises yet. The
difficulty in such cases arises from the infinite variance property of α -stable processes. Recently, Hu and Long (2009)
discussed the long time behavior of the LSE of discretely observed O–U processes driven by α -stable Lévy noises with
fixed dispersion coefficient (ε = 1) under ergodicity condition (θ0 > 0). In this paper, in contrast to Hu and Long (2009),
we assume that the time interval is fixed and θ0 is not necessarily positive. We are interested in the study of parameter
estimation for generalized O–U processes satisfying the SDE (1.1) based on discrete observations. We shall employ the least
squares method to obtain an asymptotically consistent estimator.
To obtain the LSE, we introduce the following contrast function
ρn,ε (θ) =
n
X
Xt − Xt + θ Xt · 1ti−1 2 ,
i
i−1
i−1
(1.2)
i =1
where 1ti−1 = ti − ti−1 = 1/n. Then the LSE θ̂n,ε is defined as θ̂n,ε = arg minθ ρn,ε (θ ), which can be explicitly represented
as
n
P
θ̂n,ε = −
(Xti − Xti−1 )Xti−1
i=1
n−1
n
P
i=1
.
(1.3)
Xt2i−1
The Eq. (1.1) can be solved explicitly so that we can represent the LSE θ̂n,ε as
θ̂n,ε = n(1 − e
−θ0 /n
ε
)−
n
P
Xti−1 ·
R ti
i=1
n− 1
ti−1
n
P
i=1
e−θ0 (ti −s) dLs
.
(1.4)
Xt2i−1
In this paper, we consider the asymptotics of the LSE θ̂n,ε with high frequency (n → ∞) and small dispersion (ε → 0).
Our goal is to prove that θ̂n,ε → θ0 in probability and to establish the rate of convergence and the asymptotic distributions.
We obtain some new asymptotic distributions for the LSE in our general setting, which are the convolutions of normal
distributions and stable distributions. Of course, when b = 0 (pure diffusion cases), our results coincide with classical ones.
In the paper, we shall use C to denote a generic constant which may vary from place to place.
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H. Long / Statistics and Probability Letters 79 (2009) 2076–2085
The paper is organized as follows. In Section 2 we establish the consistency of the LSE θ̂n,ε . In Section 3, we study the rate of
convergence for the LSE and obtain the asymptotic distribution, which will be different from the classical cases. In Section 4,
we provide some simulation result. The simulation study for hyperbolic processes indicates some possible extensions to
general nonlinear SDEs driven by small Lévy noises. Finally, some concluding remarks are given in Section 5.
2. Consistency of the least squares estimator
Recall that a random variable η is said to have a stable distribution with index of stability α ∈ (0, 2], scale parameter
σ ∈ (0, ∞), skewness parameter β ∈ [−1, 1], and location parameter µ ∈ (−∞, ∞) if it has characteristic function of the
following form:
o
n

απ α
α

1
−
i
β
sgn
(
u
)
tan
+
i
µ
u
, if α 6= 1,
exp
−σ
|
u
|

2
φη (u) = E exp{iuη} =
2

exp −σ |u| 1 + iβ sgn(u) log |u| + iµu , if α = 1.
π
We denote η ∼ Sα (σ , β, µ). When µ = 0, we say η is strictly α -stable. If in addition β = 0, we call η symmetric α -stable. We
refer to Janicki and Weron (1994), Samorodnitsky and Taqqu (1994) and Sato (1999) for more details on stable distributions.
Throughout this paper, it is assumed that Z1 ∼ Sα (1, β, 0) for some β ∈ [−1, 1].
We say that a stochastic process Y = (Yt , t ≥ 0) is an (Ft )-adapted Lévy process if Yt ∈ Ft for all t ≥ 0 and
(1) Y0 = 0 a.s.
(2) Y has independent increments, i.e., Yt − Ys is independent of Fs , 0 ≤ s < t < ∞.
(3) Y has stationary increments, i.e., Yt − Ys has the same distribution as Yt −s for 0 ≤ s < t < ∞.
(4) Y is stochastically continuous (or continuous in probability), i.e, for all ε > 0 and for all s ≥ 0,
lim P(|Yt − Ys | > ε) = 0.
t →s
Note that every Lévy process has a unique modification which is càdlàg (right continuous with left limits) and which is also
a Lévy process. Therefore we assume that our Lévy processes are càdlàg. We refer to Sato (1999) for more details on Lévy
processes. If the distribution of Yt − Ys is Sα ((t − s)1/α , β, 0) for any 0 ≤ s < t < ∞, then we say that Y is a standard
α -stable Lévy motion.
Now we impose the following
on n and ε :
conditions
1
(C1) n → ∞, ε → 0 and ε n 4
∨
1
1
p−2
α−1 ∨ 1 − 1
p
2
n 2α
→
0.
(C2) n → ∞, ε → 0 and ε
→ 0.
(C3) n → ∞ and ε → 0.
Here p ∈ [1, α) is chosen to be as close to α as possible. We shall use notation ‘‘→P ’’ to denote ‘‘convergence in probability’’.
Theorem 2.1. (i) Let a 6= 0 and b 6= 0. Then, under condition (C1), we have θ̂n,ε →P θ0 .
(ii) Let a = 0 and b 6= 0. Then, under condition (C2), we have θ̂n,ε →P θ0 .
(iii) Let a 6= 0 and b = 0. Then, under condition (C3), we have θ̂n,ε →P θ0 .
Before proving Theorem 2.1, we shall establish some preliminary results. We first give the following moment equality
for stable integrals, which will be a crucial tool for proofs of our main results.
Lemma 2.2. Let f (·) : [0, 1] → R+ be a deterministic function satisfying 0 f α (s)ds < ∞. Then, for any 0 < q < α , there
exists a positive constant C (q, α, β) depending only on q, α and β such that for each t ∈ [0, 1]
R1
q 1/q
Z t
Z t
1/α
α
E
f (s)dZs = C (q, α, β)
f (s)ds
.
0
(2.1)
0
Proof. Let τt = 0 f α (s)ds for t ∈ [0, 1]. By the inner clock property of stable (stochastic) integrals (see Theorem 4.1
and Theorem 4.2 of Kallenberg (1992) or Theorem 3.1 in Rosinski and Woyczynski (1986) for the R
symmetric case), there
t
exists a strictly α -stable process Z 0 with the same finite dimensional distributions as Z such that 0 f (s)dZs = Z 0 (τt ) ∼
(τt )1/α Sα (1, β, 0) almost surely. Hence, by using Zolotarev’s theorem (see Zolotarev (1957)), we find that
Rt
q 1/q
Z t
Z t
1/α
1/α
0 q 1/q
α
E
f (s)dZs E|Z1 |
f (s)ds
,
= (τt )
= C (q, α, β)
0
0
where
"
C (q, α, β) =
2q+1 Γ (
q
q
)Γ (−q/α) απ 2α
απ 1 + β 2 tan2
cos
arctan β tan
√
2
α
2
α π Γ (−q/2)
This completes the proof.
q +1
2
#1/q
.
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By using (1.4), we find
θ̂n,ε − θ0 = [n(1 − e−θ0 /n ) − θ0 ] −
ε
n
P
e−θ0 (ti −s) dLs
R ti
Xi−1
ti−1
i=1
n−1
n
P
Xi2−1
i=1
Φ1 (n, ε)
:= Λn −
.
Φ2 (n, ε)
(2.2)
It is obvious that Λn → 0 as n → ∞. We shall study the asymptotic behavior of Φ1 (n, ε) and Φ2 (n, ε), respectively.
Proposition 2.3. (i) Let a 6= 0 and b 6= 0. Then, as n → ∞, ε → 0 and ε n1/4 → 0, we have Φ1 (n, ε) →P 0.
(ii) Let a = 0 and b 6= 0. Then, as n → ∞, ε → 0 and ε n(α−1)/(2α) → 0, we have Φ1 (n, ε) →P 0.
(iii) Let a 6= 0 and b = 0. Then, as n → ∞ and ε → 0, we have Φ1 (n, ε) →P 0.
Proof. (i) Note that
−θ0 ti−1
Xti−1 = e
x0 + ε
ti−1
Z
e−θ0 (ti−1 −s) dLs .
(2.3)
0
Then, we have the following decomposition
Φ1 (n, ε) = ε x0
n
X
e−θ0 ti−1
Z
n
X
e−θ0 ti−1
Z
n Z
X
i=1
+ abε2
ti−1
e−θ0 (ti −s) dBs + bε x0
i =1
ti−1
e−θ0 (ti−1 −s) dBs
e−θ0 (ti−1 −s) dZs
Z
0
ti
ti
e−θ0 (ti −s) dLs
ti−1
0
n
X
e−θ0 ti−1
Z
ti
e−θ0 (ti −s) dZs
ti−1
e−θ0 (ti −s) dBs + abε 2
ti−1
ti−1
Z
e−θ0 (ti−1 −s) dLs
i=1
ti
Z
0
n Z
X
n Z
X
i=1
ti
ti−1
i =1
+ a2 ε 2
e−θ0 (ti −s) dLs + ε 2
ti−1
i =1
= aε x0
ti
n Z
X
i=1
e−θ0 (ti −s) dBs + b2 ε 2
ti−1
ti−1
Z
e−θ0 (ti−1 −s) dZs
Z
ti
e−θ0 (ti −s) dZs
ti−1
0
n Z
X
i=1
e−θ0 (ti−1 −s) dBs
ti−1
0
ti
e−θ0 (ti −s) dZs
ti−1
6
:=
X
Φ1,k (n, ε).
(2.4)
k=1
We shall study the convergence of Φ1,k (n, ε) for k = 1, 2, . . . , 6, respectively. For Φ1,1 (n, ε), by Chebyshev’s inequality and
Ito’s isometry property, we find that for any given δ > 0
2
Z ti
n
X
e−θ0 ti−1
e−θ0 (ti −s) dBs P (|Φ1,1 (n, ε)| > δ) ≤ δ −2 E aε x0
ti−1
i=1
= δ −2 a2 x20 ε2
n
e2θ0 /n − 1 X −2θ0 ti
e
,
2θ0
i =1
(2.5)
which tends to zero as n → ∞ and ε → 0. For Φ1,2 (n, ε), by the Markov inequality and Lemma 2.2, we find that for any
given δ > 0
P (|Φ1,2 (n, ε)| > δ) ≤
≤
≤
=
Z ti
n
X
δ −1 E bε x0
e−θ0 ti−1
e−θ0 (ti −s) dZs ti−1
i=1
Z
n
1X
−1
−θ0 (ti−1 +ti ) θ0 s
δ |bkx0 |ε E e
e 1(ti−1 ,ti ] (s)dZs 0 i=1
!1/α
Z 1 X
n
α
−1
−θ0 (ti−1 +ti ) θ0 s
δ |bkx0 |ε C
e
e 1(ti−1 ,ti ] (s) ds
0 i=1
!1/α
n
eαθ0 /n − 1 X −αθ0 ti
−1
δ |b||x0 |ε C
e
,
αθ0
i=1
(2.6)
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H. Long / Statistics and Probability Letters 79 (2009) 2076–2085
which tends to zero as n → ∞ and ε → 0. For Φ1,3 (n, ε), by Chebyshev’s inequality and Ito’s isometry property, we find
that for any given δ > 0
2
Z ti
n Z t
2 2 X i−1 −θ0 (ti−1 −s)
−θ0 (ti −s)
P (|Φ1,3 (n, ε)| > δ) ≤ δ E a ε
e
dBs e
dBs ·
ti−1
i =1 0
Z
Z
n
ti
ti−1
X
e−2θ0 (ti −s) ds
e−2θ0 (ti−1 −s) ds ·
= δ − 2 a4 ε 4
−2
= δ − 2 a4 ε 4
ti−1
0
i =1
n
1 − e−2θ0 /n X 1 − e−2θ0 ti−1
2θ0
2θ0
i =1
,
(2.7)
which tends to zero as n → ∞ and ε → 0. For Φ1,4 (n, ε), by the Markov inequality, independence of stochastic integrals
with respect to Brownian motion and α -stable Lévy motion, and Lemma 2.2, we find that for any given δ > 0
Z ti
n Z t
2 X i−1 −θ0 (ti−1 −s)
−θ0 (ti −s)
P (|Φ1,4 (n, ε)| > δ) ≤ δ E abε
e
dZs e
dBs ·
ti−1
i=1 0
−1
≤ δ −1 |ab|ε 2
n Z
X
= C δ |ab|ε
Z
!1/α
ti
e−αθ0 (ti −s) ds
·C
ti−1
1 − e−αθ0 /n
2
e−2θ0 (ti−1 −s) ds
1/2
0
i =1
−1
ti−1
1/α X
n αθ0
1 − e−2θ0 ti−1
1/2
2θ0
i=1
= O ε2 n1−1/α ,
(2.8)
α−1
2α
→ 0. Applying similar techniques to Φ1,5 (n, ε) and Φ1,6 (n, ε), we get
P (|Φ1,5 (n, ε)| > δ) ≤ O ε 2 n1/2 ,
which tends to zero as ε n
(2.9)
which tends to zero as ε n1/4 → 0, and
P (|Φ1,6 (n, ε)| > δ) ≤ O ε 2 n1−1/α ,
(2.10)
which tends to zero as ε n(α−1)/(2α) → 0. Note that ε n1/4 → 0 implies ε n(α−1)/(2α) → 0 since (α − 1)/(2α) < 1/4 when
1 < α < 2. Therefore, by combining (2.4)–(2.10), we conclude that (i) holds.
(ii) If a = 0, b 6= 0 (pure jump case), then there are only two terms Φ1,2 (n, ε) and Φ1,6 (n, ε) left in the decomposition (2.4)
of Φ1 (n, ε). From the proof of (i), it immediately follows that Φ1 (n, ε) →P 0 as n → ∞, ε → 0 and ε n(α−1)/(2α) → 0.
(iii) If a 6= 0, b = 0 (pure diffusion case), then there are only two terms Φ1,1 (n, ε) and Φ1,3 (n, ε) left in the
decomposition (2.4) of Φ1 (n, ε). Again, from the proof of (i), we immediately conclude that Φ1 (n, ε) →P 0 as n → ∞
and ε → 0.
Next, we consider the asymptotic behavior of Φ2 (n, ε).
Proposition 2.4. (i) If a 6= 0 and b 6= 0, then we have Φ2 (n, ε) →P x20
2 1 −2θ0 s
ds as n
P x0 0 e
2 1 −2θ0 s
x
e
ds as n
P 0 0
(ii) If a = 0 and b 6= 0, then we have Φ2 (n, ε) →
(iii) If a 6= 0 and b = 0, then we have Φ2 (n, ε) →
R
R
R1
0
1
e−2θ0 s ds as n → ∞, ε → 0 and ε n p
→ ∞, ε → 0 and ε n
→ ∞ and ε → 0.
1
1
p−2
− 12
→ 0.
→ 0.
Proof. (i) By (2.3), we have
Φ2 (n, ε) =
=
n
1X
x0 e
n i=1
n
1X
n i=1
−θ0 ti−1
+ε
ti−1
Z
e
−θ0 (ti−1 −s)
2
dLs
0
x20 e−2θ0 ti−1 + 2aε x0
Z
n
1 X −2θ0 ti−1 ti−1 θ0 s
e
e dBs
n i=1
0
Z ti−1
2
Z
n
n
1 X −2θ0 ti−1 ti−1 θ0 s
1 X 2 2 −2θ0 ti−1
θ0 s
+2bε x0
e
e dZs +
a ε e
e dBs
n i =1
n i =1
0
0
+
:=
n
1X
n i=1
6
X
i=1
2 2 −2θ0 ti−1
b ε e
Φ2,i (n, ε).
ti−1
Z
e
0
θ0 s
2
dZs
+
n
1X
n i=1
2 −2θ0 ti−1
2abε e
ti−1
Z
e
0
θ0 s
ti−1
Z
dBs ·
eθ0 s dZs
0
(2.11)
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2081
We study the limiting behavior of Φ2,i (n, ε), i = 1, . . . , 6, respectively. It is clear that
Φ2,1 (n, ε) →
x20
1
Z
e−2θ0 s ds,
(2.12)
0
as n → ∞. For Φ2,2 (n, ε), by the Markov inequality, it follows that
Z ti−1
n
X
P (|Φ2,2 (n, ε)| > δ) ≤ δ −1 E 2aε x0 n−1
eθ0 s dBs e−2θ0 ti−1
0
i=1
−1 −1
≤ 2|a|ε|x0 |δ n
n
X
e
ti−1
Z
−2θ0 ti−1
e
2 θ0 s
1/2
ds
,
(2.13)
0
i =1
which tends to zero as n → ∞ and ε → 0. For Φ2,3 (n, ε), by the Markov inequality and Lemma 2.2, we have
Z ti−1
n
X
θ0 s
−1
−2θ0 ti−1
P (|Φ2,3 (n, ε)| > δ) ≤ δ E 2aε x0 n
e dZs e
0
i=1
−1
−1 −1
≤ 2|b|ε|x0 |δ n
n
X
e
−2θ0 ti−1
ti−1
Z
C
e
αθ0 s
1/α
ds
,
(2.14)
0
i =1
which tends to zero as n → ∞ and ε → 0. Similar to the proof of (2.13), we get
P (|Φ2,4 (n, ε)| > δ) ≤ a2 ε 2 δ −1 n−1
n
X
e−2θ0 ti−1
ti−1
Z
e2θ0 s ds,
(2.15)
0
i =1
which tends to zero as n → ∞ and ε → 0. For Φ2,5 (n, ε), by the Markov inequality and Lemma 2.2, we find that for
1≤p<α
Z ti−1
2 p/2
n
X
P (|Φ2,5 (n, ε)| > δ) ≤ δ −p/2 E b2 ε 2 n−1
e−2θ0 ti−1
eθ0 s dZs 0
i=1
Z
p
n
X
ti−1 θ s ≤ |b|p εp δ −p/2 n−p/2
e−pθ0 ti−1 E e 0 dZs 0
i=1
≤ |b|p εp δ −p/2 n−p/2
n
X
e−pθ0 ti−1 C
= O(ε n
(2.16)
1
P (|Φ2,6 (n, ε)| > δ) ≤ 2δ
|ab|ε
≤ 2δ |ab|ε
−1
p/α
),
− 12
which tends to zero as n → ∞, ε → 0 and ε n p
independence of the two stochastic integrals, we have
−1
eαθ0 s ds
0
i=1
p 1−p/2
ti−1
Z
21
n
X
n i=1
21
n
X
n i=1
−2θ0 ti−1
e
−2θ0 ti−1
→ 0. Finally, for Φ2,6 (n, ε), by the Markov inequality and the
Z
E ti−1
0
Z
e dBs · E ti−1
Z
e
e
0
ti−1
θ0 s
2θ0 s
1/2
ds
0
e dZs θ0 s
ti−1
Z
·C
e
αθ0 s
1/α
ds
,
(2.17)
0
which tends to zero as ε → 0. Combining (2.11)–(2.17), we complete the proof of (i).
(ii) If a = 0, b 6= 0 (pure jump case), then there are only three terms Φ2,1 (n, ε), Φ2,3 (n, ε) and Φ2,5 (n, ε) left in the
decomposition (2.11) of Φ2 (n, ε). From the proof of (i), it immediately follows that (ii) holds.
(iii) If a 6= 0, b = 0 (pure diffusion case), then there are only three terms Φ2,1 (n, ε), Φ2,2 (n, ε) and Φ2,4 (n, ε) left in the
decomposition (2.11) of Φ2 (n, ε). Again, from the proof of (i), we conclude that (iii) holds. Proof of Theorem 2.1. (i) By using the fact that Λn → 0 as n → ∞ and combining Proposition 2.3-(i) and Proposition 2.4(i), we immediately conclude that
θ̂n,ε − θ0 = Λn −
Φ1 (n, ε)
→P 0
Φ2 (n, ε)
(2.18)
under condition (C1).
Similarly, the statements (ii) and (iii) follow from Proposition 2.3-(ii) and (iii), Proposition 2.4-(ii) and (iii), respectively. This
completes the proof. Author's personal copy
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H. Long / Statistics and Probability Letters 79 (2009) 2076–2085
3. Asymptotics of the least squares estimator
To obtain the rate of convergence we need to impose
the following conditions.
1
(C4) As n → ∞ and ε → 0, nε → ∞ and ε n 2
1
1
p−2
∨
α−1 ∨ 1 − 1
p
2
n α
→
0.
(C5) As n → ∞ and ε → 0, nε → ∞ and ε
→ 0.
(C6) As n → ∞ and ε → 0, nε → ∞.
Note that the choice of p is the same as in the previous section. It is clear that (C4) implies (C5) while (C5) implies (C6).
We use notation ‘‘⇒’’ to denote ‘‘convergence in distribution’’. Throughout, N denotes a random variable with the standard
normal distribution, and U denotes a random variable with the standard α -stable distribution Sα (1, β, 0) independent of N.
Our main result of this section is as follows.
Theorem 3.1. (i) If a 6= 0 and b 6= 0, then, under condition (C4), we have
ε (θ̂n,ε − θ0 ) ⇒ −ax0
−1
−1
1/2
2θ0
−1
N − bx0
1 − e−2θ0
2θ0
1 − e−αθ0
1 − e−2θ0
1/α
αθ0
U.
(3.1)
(ii) If a = 0 and b 6= 0, then, under condition (C5), we have
1/α
2θ0
1 − e−αθ0
−1
−1
ε (θ̂n,ε − θ0 ) ⇒ −bx0
U.
1 − e−2θ0
αθ0
(3.2)
(iii) If a 6= 0 and b = 0, then, under condition (C6), we have
1/2
2θ0
−1
−1
ε (θ̂n,ε − θ0 ) ⇒ −ax0
N.
1 − e−2θ0
(3.3)
Remark 3.2. The statement (iii) of Theorem 3.1 is a classical result in parameter estimation for small diffusions. The
statements (i) and (ii) of Theorem 3.1 are new, which give the rate of convergence and the asymptotic distributions of
LSE when there are small Lévy stable noises in the observable Ornstein–Uhlenbeck processes.
Theorem 3.1 will be proved by establishing several preliminary lemmas and propositions. We first give an explicit
expression for ε −1 (θ̂n,ε − θ0 ). By using (2.2), we have
n
P
ε −1 (θ̂n,ε − θ0 ) = ε −1 [n(1 − e−θ0 /n ) − θ0 ] −
Xti−1
i=1
R ti
ti−1
n− 1
e−θ0 (ti −s) dLs
n
P
i =1
:= ε −1 Λn −
Xt2i−1
ε −1 Φ1 (n, ε)
.
Φ2 (n, ε)
(3.4)
For convenience, we let
Ψ1 (n, ε) = ε −1 Φ1 (n, ε) =
6
X
ε −1 Φ1,i (n, ε) :=
i=1
6
X
Ψ1,i (n, ε).
(3.5)
i=1
We shall study the asymptotic behavior of Ψ1,i (n, ε), i = 1, . . . , 6, respectively.
Lemma 3.3. Under condition (C6), we have ε −1 Λn → 0.
Proof. Basic calculation yields that ε −1 Λn ≤
C θ02
nε
, which tends to zero under condition (C6).
Proposition 3.4. If a 6= 0, then, as n → ∞, we have
Ψ1,1 (n, ε) ⇒ ax0
Proof. Let Wi =
R ti
ti−1
1 − e−2θ0
2θ0
1/2
N.
(3.6)
R
ti
eθ0 s dBs , i = 1, . . . , n. Then, it is easy to see that Wi ∼ N 0,
independent normal random variables. It follows that
ti−1
e2θ0 s ds , and W1 , . . . , Wn are
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H. Long / Statistics and Probability Letters 79 (2009) 2076–2085
Ψ1,1 (n, ε) = ax0
n
X
e
−θ0 (ti−1 +ti )
Wi ∼ N
0,
a2 x20
n
X
i =1
⇒ ax0
e
−2θ0 (ti−1 +ti )
1/2
2θ0
as n → ∞. This completes the proof.
e
2θ0 s
ds
ti−1
i=1
1 − e−2θ0
!
ti
Z
2083
N,
(3.7)
Proposition 3.5. If b 6= 0, then, as n → ∞, we have
Ψ1,2 (n, ε) ⇒ bx0
Proof. Let Ui =
R ti
ti−1
1 − e−αθ0
1/α
αθ0
U.
(3.8)
eθ0 s dZs , i = 1, . . . , n. Then, by the inner clock property for the α -stable stochastic integral (see Rosinski
and Woyczynski, 1986; Kallenberg, 1992), we find that Ui ∼
R
ti
ti−1
eαθ0 s ds
1/α
Sα (1, β, 0), and U1 , . . . , Un are independent
α -stable random variables. By basic property of α -stable random variables (see Janicki and Weron, 1994), we find that
!1/α
Z ti
n
n
X
X
Ψ1,2 (n, ε) = bx0
e−θ0 (ti−1 +ti ) Ui ∼ bx0
e−αθ0 (ti−1 +ti )
eαθ0 s ds
Sα (1, β, 0)
i =1
⇒ bx0
ti−1
i=1
1 − e−αθ0
1/α
αθ0
U,
(3.9)
as n → ∞. The independence of U and N follows from the fact that Ψ1,2 (n, ε) and Ψ1,1 (n, ε) are independent. This completes
the proof. Proposition 3.6. If a 6= 0 and b 6= 0, then the following convergence results hold:
(i) Ψ1,3 (n, ε) →P 0 as n → ∞ and ε → 0;
(ii) Ψ1,4 (n, ε) →P 0 as n → ∞, ε → 0 and ε n1−1/α → 0;
(iii) Ψ1,5 (n, ε) →P 0 as n → ∞, ε → 0 and ε n1/2 → 0;
(iv) Ψ1,6 (n, ε) →P 0 as n → ∞, ε → 0 and ε n1−1/α → 0.
Proof. We can easily get the probability estimates for Ψ1,i (n, ε), i = 3, 4, 5, 6 from the corresponding probability estimates
in (2.7)–(2.10) for Φ1,i (n, ε), i = 3, 4, 5, 6 by simply replacing ε 2 with ε , which shall immediately follow the claims
(i)–(iv). Remark 3.7. (i) Note that ε n1/2 → 0 implies ε n1−1/α → 0. So, if a 6= 0 and b 6= 0, then Proposition 3.6 gives that
6
X
Ψ1,i (n, ε) →P 0
i =3
as n → ∞, ε → 0 and ε n1/2 → 0.
(ii) If a = 0 and b 6= 0 (pure jump case), then Ψ1,i (n, ε) = 0, i = 3, 4, 5, and Proposition 3.6 gives Ψ1,6 (n, ε) →P 0 as
n → ∞, ε → 0 and ε n1−1/α → 0.
(iii) If a 6= 0 and b = 0 (pure diffusion case), then Ψ1,i (n, ε) = 0, i = 4, 5, 6, and Proposition 3.6 gives Ψ1,3 (n, ε) →P 0 as
n → ∞ and ε → 0.
Finally, we are in a position to prove Theorem 3.1.
Proof of Theorem 3.1. (i) By combining Lemma 3.3, Proposition 3.4, Proposition 3.5, Proposition 3.6, Remark 3.7-(i) and
Proposition 2.4-(i), and using Slutsky’s theorem, it follows that under condition (C4)
ε −1 (θ̂n,ε − θ0 ) ⇒ −
ax0
1−e−2θ0
2θ0
1/2
x20
−1
= −ax0
2θ0
1 − e−2θ0
N + bx0
1−e−2θ0
2θ0
1/2
1−e−αθ0
αθ0
1/α
U
−1
N − bx0
2θ0
1 − e−2θ0
1 − e−αθ0
αθ0
1/α
U.
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H. Long / Statistics and Probability Letters 79 (2009) 2076–2085
Table 1
Sample means and mean absolute deviations (MAD) of the estimators θ̂n,ε for hyperbolic processes based on 30 independent samples.
ε
n
β = −0.5
−2
10
10−3
10−4
10−2
10−3
10−4
100
1000
β=0
β = 0.5
Mean
MAD
Mean
MAD
Mean
MAD
1.96452
1.99652
1.99965
2.02567
2.00257
2.00026
0.01899
0.00186
0.00019
0.01671
0.00167
0.00017
1.97032
1.99708
1.99971
2.01525
2.00153
2.00015
0.02029
0.00200
0.00020
0.01964
0.00197
0.00020
1.97690
1.99772
1.99977
2.00367
2.00038
2.00004
0.02210
0.00220
0.00022
0.02324
0.00233
0.00023
(ii) By using Lemma 3.3, Proposition 3.5, Remark 3.7-(ii), Proposition 2.4-(ii) and Slutsky’s theorem, we conclude that under
condition (C5)
ε (θ̂n,ε − θ0 ) ⇒ −
−1
bx0
1−e−αθ0
1/α
αθ0
x20
1−e−2θ0
2θ0
U
−1
= −bx0
2θ0
1 − e−2θ0
1 − e−αθ0
αθ0
1/α
U.
(iii) By using Lemma 3.3, Proposition 3.4, Remark 3.7-(iii), Proposition 2.4-(iii) and Slutsky’s theorem, we conclude that
under condition (C6)
ε −1 (θ̂n,ε − θ0 ) ⇒ −
ax0
x20
1−e−2θ0
2θ0
1/2
1−e−2θ0
2θ0
N
1
= −ax−
0
2θ0
1 − e−2θ0
1/2
N. 4. Simulation
In this section, we would like to use simulation to indicate some possible extensions to more general SDEs driven by
small Lévy noises. We consider the hyperbolic process defined by the following SDE:
dXt = θ0 p
Xt
1 + Xt2
dt + ε dLt ,
t ∈ [0, 1];
X 0 = x0 ,
(4.1)
where x0 and ε are known constants, θ0 is an unknown parameter, and Lt = aBt + bZt is the special Lévy process defined
as before. We are interested in estimating θ0 by using least squares method based on discrete observations (Xti )ni=1 . In the
simulation, we let θ0 = 2, x0 = 1.2, a = 1, b = 2, α = 1.5 and use different choices for the values of β , n and ε . The sample
means and mean absolute deviations of the estimators for this hyperbolic process based on 30 independent samples are
shown in Table 1.
We see from Table 1 that when n is large enough and ε is small enough, the obtained estimators are very close to the true
parameter value θ0 = 2. Hence, we expect that the theoretical results established in this paper could be extended to more
general nonlinear SDEs driven by small Lévy noises.
5. Concluding remarks
In this paper, we consider LSE for Ornstein–Uhlenbeck processes driven by a special class of Lévy processes (linear
combination of Brownian motions and α -stable motions). Some new asymptotic results are established in Theorems 2.1
and 3.1. Of course, the class of one-dimensional Ornstein–Uhlenbeck processes studied in the paper is limited in application.
The simulation study for the hyperbolic processes done in Section 4 shows that we should be able to extend the theoretical
results of this paper to more general models described by nonlinear SDEs driven by small Lévy processes with possibly
a high-dimensional unknown parameter vector. It is also possible to use other methods such as approximate likelihood
method, martingale estimating function method, and M-estimation method, to study parameter estimation for stochastic
differential equations driven by small Lévy noises. We expect to obtain some new and interesting results in this direction,
although it would be much more challenging to do so. All these extensions will be in the scope of our ongoing and future
projects.
Acknowledgments
The author is very grateful to the referees and the editor for their insightful and valuable comments which have greatly
improved the presentation of the paper. The author also thanks Mr. Sandun Perera (Florida Atlantic University) for his
assistance with the simulation study.
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2085
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