NON-GAUSSIAN QUASI-LIKELIHOOD ESTIMATION OF SDE DRIVEN BY
LOCALLY STABLE LÉVY PROCESS

arXiv:1608.06758v3 [] 8 Jun 2017

HIROKI MASUDA

Abstract. We address estimation of parametric coefficients of a pure-jump Lévy driven univariate
stochastic differential equation (SDE) model, which is observed at high frequency over a fixed time
period. It is known from the previous study [36] that adopting the conventional Gaussian quasi-maximum
likelihood estimator then leads to an inconsistent estimator. In this paper, under the assumption that
the driving Lévy process is locally stable, we extend the Gaussian framework into a non-Gaussian
counterpart, by introducing a novel quasi-likelihood function formally based on the small-time stable
approximation of the unknown transition density. The resulting estimator turns out to be asymptotically
mixed normally distributed without ergodicity and finite moments for a wide range of the driving purejump Lévy processes, showing much better theoretical performance compared with the Gaussian quasimaximum likelihood estimator. Extensive simulations are carried out to show good estimation accuracy.
The case of large-time asymptotics under ergodicity is briefly mentioned as well, where we can deduce
an analogous asymptotic normality result.

1. Introduction
Stochastic differential equation (SDE) driven by a Lévy process is one of basic models to describe
time-varying physical and natural phenomena. There do exist many situations where non-Gaussianity
of distributions of data increments, or of a residual sequence whenever available, is significant in smalltime, making diffusion type models observed at high frequency somewhat inappropriate to reflect reality;
see [3] as well as the enormous references therein, and also [16]. This non-Gaussianity may not be well
modeled even by a diffusion with compound-Poisson jumps as well since jump-time points are then rather
sparse compared with sampling frequency, so that most increments are approximately Gaussian except
for intervals containing jumps. SDE driven by a pure-jump Lévy process may then serve as a good natural
candidate model. For those models, however, a tailor-made estimation procedure seems to be far from
being well developed, which motivated our present study.
In this paper, we consider a solution to the univariate Markovian SDE
(1.1)

dXt = a(Xt , α)dt + c(Xt− , γ)dJt

defined on an underlying complete filtered probability space (Ω, F, (Ft )t∈R+ , P) with
Ft = σ(X0 ) ∨ σ(Js ; s ≤ t),

(1.2)
where:

• The initial random variable X0 is F0 -measurable;
• The driving noise J is a symmetric pure-jump (càdlàg) Lévy process independent of X0 ;
• The trend coefficient a : R × Θα → R and scale coefficient c : R × Θγ → R are assumed to be
known except for the p-dimensional parameter
θ := (α, γ) ∈ Θα × Θγ = Θ ⊂ Rp ,
with Θα ∈ Rpα and Θγ ∈ Rpγ being bounded convex domains.
Our objective here is estimation of θ, when the true value θ0 = (α0 , γ0 ) ∈ Θ does exist and the process X
is observed only at discrete but high-frequency time instants tnj = jhn , j = 0, 1, . . . , n, with nonrandom
sampling step size hn → 0 as n → ∞. We will mostly work under the bounded-domain asymptotics 1:
T
(1.3)
Tn ≡ T , i.e. hn =
n
Date: June 9, 2017.
Key words and phrases. Asymptotic mixed normality, high-frequency sampling, locally stable Lévy process, stable quasilikelihood function, stochastic differential equations.
1The equidistance assumption on the sampling times could be removed as soon as the ratios of min
j≤n (tj − tj−1 ) and
maxj≤n (tj − tj−1 ) are bounded in an appropriate order. This may be shown by the same line as in [36, pp.1604–1605].
1

2

HIROKI MASUDA

for a fixed terminal sampling time T ∈ (0, ∞), that is, we observe not a complete path (Xt )t≤T but the
time-discretized step process
(1.4)

(n)

Xt

:= Xbt/hn chn

over the period [0, T ]; see Section 3.3 for the large-time asymptotics where Tn → ∞ under the ergodicity.
Due to the lack of a closed-form formula for the transition distribution, a feasible approach based
on the genuine likelihood function is rarely available. In this paper, we will introduce a non-Gaussian
quasi-likelihood function 2, which extends the prototype mentioned in [34] and [37], under the locally
(symmetric) β-stable property of J in the sense that
(1.5)

L(h−1/β Jh ) ⇒ Sβ ,

h → 0,

where Sβ stands for the standard symmetric β-stable distribution corresponding to the characteristic
function
ϕ0 (u) := exp(−|u|β );
among others, we refer to [24], [47], [48] and [53]) for comprehensive accounts of general stable distributions. It is known from [6, Proposition 1] that, as long as the linear scaling h Jh for some h → 0
is concerned, the strictly stable distribution is the only possible asymptotic distribution. Many locally
stable Lévy processes with finite variance can exhibit large-time Gaussianity (i.e. central limit effect)
in addition to the small-time non-Gaussianity. In the main results, we will assume the locally stable
property (1.5) with a stronger mode (see Lemmas 2.2 and 2.5) and that the stability index β is known
with
β ∈ [1, 2).
It should be noted that the value β is also known as the Blumenthal-Getoor activity index defined by


Z
β := inf b ≥ 0 :
|z|b ν(dz) < ∞ ,
|z|≤1

which measures degree of J’s jump activity.
The proposed maximum quasi-likelihood estimator θ̂n = (α̂n , γ̂n ) has the property that
√

√
(1.6)
nh1−1/β
(α̂
−
α
),
n(γ̂
−
γ
)
n
0
n
0
n
is asymptotically mixed normally distributed under some conditions, which extends the previous works
[33] and [36] that adopting the Gaussian quasi-maximum likelihood estimator; we refer to [39, Section
2] for some formal comparisons. In particular, the convergence (1.6) clarifies that the activity index β
determines the rate of convergence of estimating the trend parameter α; note that
√ 1−1/β
nh
= T 1−1/β n(2−β)/(2β) → ∞
as n → ∞. It should be emphasized that this estimator can be much more efficient compared with
the Gaussian maximum quasi-likelihood estimator studied in [36]. Most notably, unlike the case of
diffusions, we can estimate not only the scale parameter γ but also the trend parameter α, with the
explicit asymptotic distribution in hand; see [15] for the related local asymptotic normality result. To
prove the asymptotic mixed normality, we will take a doubly approximate procedure based on the EulerMaruyama scheme combined with the stable approximation of L(h−1/β Jh ) for h → 0. Our result provides
us with the first systematic methodology for estimating the possibly non-linear pure-jump Lévy driven
SDE (1.1) based on a non-Gaussian quasi-likelihood.
Here are a couple of further remarks on our model.
(1) The model is semiparametric in the sense that we do not completely specify the Lévy measure
of L(J), while supposing the parametric coefficients; of course, the Lévy measure is an infinitedimensional parameter, so that β alone never determines the distribution L(J) in general. In
estimation of L(X), it would be desirable (whenever possible) to estimate the parameter θ with
leaving the remaining parameters contained in Lévy measure as much as unknown. The proposed
quasi-likelihood, termed as (non-Gaussian) stable quasi-likelihood, will provide us with a widely
applicable tool for this purpose.
2Non-Gaussian quasi-likelihoods have not received much attention compared with the popular Gaussian one. Among
others, we refer to the recent paper [12] for a certain non-Gaussian quasi-likelihood estimation of a possibly heavy-tailed
GARCH model, and also to [52] for self-weighted Laplace quasi-likelihood in a time series context.

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

3

(2) It is assumed from the very beginning that β < 2 so that J contains no Gaussian component.
Normally, the simultaneous presence of a non-degenerate diffusion part and a non-null jump part
makes the parametric-estimation problem much more complicated. The recent papers [25] and
[28] discussed usefulness of pure-jump models. Although they are especially concerned with financial context, pure-jump processes should be useful for model building in many other application
fields where non-Gaussianity of time-varying data is of primary importance. For example, econometrics, signal processing, population dynamics, hydrology, radiophysics, turbulence, biological
molecule movement, noise-contaminated biosignals, and so on; we refer to [3], [9], and [11] for
some recent related works.
(3) Finally, our model (1.1) may be formally seen as a continuous-time analogue to the discrete-time
model
Xj = a(Xj−1 , α) + c(Xj−1 , γ)j ,
j = 1, . . . , n,
where j are i.i.d. random variables. By making use of the locally stable property (1.5), our
model setup enables us to formulate a flexible and unified estimation procedure, which cannot
be shared with the discrete-time counterpart. The bounded-domain asymptotics (1.3) makes it
possible to “localize” the event, sidestepping both stability (such as the ergodicity) and moment
condition on L(J1 ). Instead, in order to deduce the asymptotic mixed normality we need much
more than the (martingale) central limit theorem with Gaussian limit. Fortunately, we have
the very general tool to handle this, that is, Jacod’s characterization of conditionally Gaussian
martingales (see [13] and [20], and also Section 6.1), which in particular can deal with the SDE
(1.1) when J is a pure-jump Lévy process.
The following conventions and basic notations are used throughout this paper. We will largely suppress
the dependence on n from the notations tnj and hn . For any process Y ,
∆j Y = ∆nj Y := Ytj − Ytj−1
denotes the jth increments, and we write
gj−1 (v) = g(Xtj−1 , v)
for a function g having two components, such as aj−1 (α) = a(Xtj−1 , α). For a variable x = {xi }, we write
2
∂
∂x = { ∂x
}i , ∂x2 = { ∂x∂i ∂xj }i,j , and so forth, with omitting the subscript x when there is no confusion;
i
given a function f = f (s1 , . . . , sk ) : S1 × · · · × Sk → Rm with Si ⊂ Rdi , we write ∂sj11 . . . ∂sjkk f for the array
Qk
of partial derivatives of dimension m × ( l=1 dl jl ). The characteristic function of a random variable ξ
is denoted by ϕξ . For any matrix M we let M ⊗2 := M M > (> denotes the transpose). We use C for
a generic positive constant which may vary at each appearance, and write an . bn when an ≤ Cbn for
p
L
every n large enough. Finally, the symbols −
→ and −
→ denote the convergences in P-probability and in
distribution, respectively; all the asymptoics below will be taken for n → ∞ unless otherwise mentioned.
The paper is organized as follows. We first describe the basic model setup in Section 2. The main
results are presented in Section 3, followed by numerical experiments in Section 4. Section 5 presents the
proofs of the criteria for the key assumptions given in Section 2. Finally, Section 6 is devoted to proving
the main results.
2. Basic setup and assumptions
2.1. Locally stable Lévy process.
2.1.1. Definition and criteria. We denote by
g0,β (y) :=

cβ
,
|z|1+β

z 6= 0,

the Lévy density of Sβ , where
(2.1)

cβ :=


 −1
βπ
1 1
Γ(1 − β) cos
2 β
2

with c1 = limβ→1 cβ = π −1 ([48, Lemma 14.11]).

4

HIROKI MASUDA

Assumption 2.1 (Driving noise structure).
(1) The Lévy process J has no drift, no Gaussian component, and a symmetric Lévy measure ν, so that
 Z

(2.2)
ϕJt (u) = exp t (cos(uz) − 1)ν(dz) ,
u ∈ R.
The Lévy measure ν admits a Lebesgue density g of the form
g(z) = g0,β (z) {1 + ρ(z)} ,

z 6= 0,

where ρ : R \ {0} → [−1, ∞) is a measurable symmetric function such that for some constants
δ > 0, cρ ≥ 0, and ρ > 0,
|ρ(z)| ≤ cρ |z|δ ,

|z| ≤ ρ ,

z 6= 0.

(2) Further, ρ is continuously differentiable in R \ {0} and the pair (cρ , β, δ) satisfies either
(a) cρ = 0, or
(b) cρ > 0 and δ > β with
|ρ(z)| + |z∂ρ(z)| ≤ cρ |z|δ ,

z 6= 0.

The function ρ controls the degree of “Sβ -likeness” around the origin. Also, if in particular ρ is
bounded, then E(|J1 |q ) < ∞ for q ∈ (−1, β); see [48, Theorem 25.3].
We denote by φβ the density of Sβ , and by ϕh for the characteristic function of h−1/β Jh :

h
u ∈ R.
ϕh (u) := ϕJ1 (h−1/β u) ,
Lemma 2.2 below, which will play an essential role in the proof of the main results, shows that Assumption
2.1 ensures not only the locally stable property (1.5) but also an L1 -local limit theorem with specific
convergence rate; it is partly a refinement of [32, Lemma 4.4].
Lemma 2.2.
(1) Let Assumption 2.1(1) hold with the function ρ being bounded.
(a) For every C ≥ 0 and s < 1,
Z
(u−s ∨ uC )|ϕh (u) − ϕ0 (u)|du . haν ,
(0,∞)

where the constant aν ∈ (0, 1] is defined by

1
(cρ = 0)
aν =
(δ/β) ∧ 1 (cρ > 0).
In particular, the distribution L(h−1/β Jh ) for h ∈ (0, 1] admits a positive smooth Lebesgue
density, which we denote by fh , such that
sup |fh (y) − φβ (y)| . haν .
y

(2.3)

(b) For each κ ∈ (0, β), we have
Z
|y|κ |fh (y) − φβ (y)| dy → 0.

(2) Let Assumption 2.1 hold, and assume that there exists a constant K > 0 such that g(z) = 0
(equivalently, ρ(z) = −1) for every |z| > K. Then, for any r ∈ (0, 1] we have
Z
β(1∧aν )
(2.4)
|fh (y) − φβ (y)| dy . h β+r .
The additional assumptions on ρ in Lemma 2.2 will not be real restrictions, because in the proof
our main results we will truncate the support of ν in order to deal with possibly heavy-tailed J. The
localization argument is allowed under the bounded-domain asymptotics (1.3); see Section 6.1. Since
1 ∧ aν > 1/2 under Assumption 2.1(2), it follows from (2.4) that under (1.3) we can pick a sufficiently
small r > 0 to ensure nh
(2.5)

2β(1∧aν )
β+r

→ 0, which in turn implies that
Z
√
n |fh (y) − φβ (y)| dy → 0.

As will be seen later, it is the convergences (2.3) and (2.5) that are essential for the proofs of the main
results. Assumption 2.1 (also Assumption 2.4 given below) serves as a set of practical sufficient conditions.

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

5

Remark 2.3. The conclusions of Lemma 2.2 may remain the same even when we add an independent
jump component. For example, we may replace the original J by J ? := J + J [ , where J [ is a pure-jump
Lévy process having a Lévy density g [ ∈ C 1 (R \ {0}) such that g [ ≡ 0 for |z| > K and that


(2.6)
lim sup |z|1+β−δ g [ (z) + |z| ∂g [ (z) < ∞.
|z|→0
?

Then, the Lévy density of J equals



1
g ? (z) = g(z) + g [ (z) = g0,β (z) 1 + ρ(z) + |z|1+β g [ (z) ,
cβ
1+β [
and it can be checked that if the function ρ satisfies Assumption 2.1, then so is z 7→ ρ(z) + c−1
g (z).
β |z|
[
In particular, (2.6) is fulfilled if jumps of J are of compound-Poisson type.


Assumption 2.1 is designed to give conditions only in terms of the Lévy density g. However, Assumption
2.1(2) excludes cases where ρ ∈ C 1 (R \ {0}) with |∂ρ(0+)| > 0 (hence δ = 1) and β ≤ 1; note that we do
not explicitly impose that β ∈ [1, 2) in Lemma 2.2 (and also in Lemma 2.5 below). It is possible to give
another set of conditions. Let
ψh (u) := log ϕh (u).
Assumption 2.4 (Driving noise structure). Assumption 2.1(1) holds with the function ρ being bounded,
ψh ∈ C 1 (R\{0}), and there exist constants cψ ≥ 0 and r ∈ [0, 1] and a function ψ (h) such that ψ (h) → 0
as h → 0 and that
1
(2.7)
u > 0,
|∂u ψh (u)| . ∨ ucψ ,
u
Z
ur ϕ0 (u) ∂u ψh (u) + βuβ−1 du ≤ ψ (h).

(2.8)
(0,∞)

Lemma 2.5. Under Assumption 2.4, we have
Z
β
|fh (y) − φβ (y)| dy . (ψ (h) ∨ haν ) β+r
for the constant aν given in Lemma 2.2(1). In particular, (2.5) holds if

√

β

n(ψ (h) ∨ haν ) β+r → 0.

Lemmas 2.2(2) and 2.5 have no inclusion relation with different domains of the applicability. Although
we do not look at a truncated support of ν in Lemma 2.5, it will be needed that the tail of ν is light
enough: for details, see Theorem 3.5 given later.
2.1.2. Examples.
Example 2.6. Trivially, Assumption 2.1 is satisfied by the β-stable driven case (L(J1 ) = Sβ ) and the
whole class of the driving Lévy process considered in [7] and [8], where cρ = 0 (equivalently ρ ≡ 0). See
also Remark 3.4.

In the next two concrete examples, Assumption 2.4 is helpful for verification of (2.5) while Assumption
2.1(2) may not.
Example 2.7 (Symmetric tempered β-stable Lévy process with β ∈ [1, 2)). The symmetric exponentially
tempered β-stable Lévy process, which we denote by T Sβ (λ) for λ > 0, is defined through the Lévy density
z 7→ g0,β (z) exp(−λ|z|);
we refer to [26] and the references therein for details of general tempered stable distributions. When
L(J1 ) = T Sβ (λ), then L(h−1/β Jh ) = T Sβ (λh) ⇒ Sβ as h → 0. Assumption 2.1(1) is satisfied with
ρ(z) = exp(−λ|z|) − 1, hence δ = 1 for any β < 2, and aν = 1 ∧ (1/β). However, Assumption 2.1(2)
requires β < 1, which conflicts with the case β ∈ [1, 2) of our interest here. We will instead verify
Assumption 2.4 for β ∈ [1, 2); the function ψh is explicitly given by


 
 
1
u2
u


λh
log
1
+
−
2u
arctan
(β = 1)
π
2
2
λ h
λh





ψh (u) =

u

2cβ Γ(−β) (λ2 h2/β + u2 )β/2 cos β arctan
− λβ h
(β ∈ (1, 2))
λh1/β

6

HIROKI MASUDA

u
First we consider β = 1, where ∂u ψh (u) = − π2 arctan( λh
). Using the estimate

(2.9)

sup
y≥0
u
| arctan( λh
)

we have |∂u ψh (u) + 1| .
Z

arctan y − (π/2)
< ∞,
−(1/y)

− π2 | . uh . This gives

ur ϕ0 (u)|∂u ψh (u) + 1|du . h

(0,∞)

Z

ur−1 e−u du . h

(0,∞)

√ R
2
Lemma 2.5 ensures that n |fh (y) − φβ (y)|dy . (nh 1+r )1/2 → 0, with r > 0 being small enough; even
when Tn → ∞, it suffices for the last convergence to suppose that nh2−1 → 0 some 1 > 0.
Next we consider β ∈ (1, 2) with r = 0; we may control r ≥ 0 independently of the case β = 1.
Substituting the expression (2.1) we see that ∂u ψh (u) + βuβ−1 quals the sum of three terms ∆h,k (u)
(k = 1, 2, 3), where


−1 


u
βπ
βπ
u
,
∆h,1 (u) := β cos
cos
− cos β arctan
2
2
λh1/β
(λ2 h2/β + u2 )1−β/2


1−β/2 
u2
∆h,2 (u) := β uβ−1 −
,
λ2 h2/β + u2
and ∆h,3 (u) satisfies that |∆h,3 (u)| . h1/β (λ2 h2/β + u2 )β/2−1 . h1/β uβ−2 . By the mean-value theorem
u
together with (2.9), we derive |∆h,1 (u)| . u2(1−β/2)
· (h−1/β u)−1 = h1/β uβ−2 . Hence, for β > 1,
Z
Z
β
e−u uβ−2 du . h1/β .
ϕ0 (u) ∆h,1 (u) + ∆h,3 (u) du ≤ h1/β
(0,∞)

(0,∞)

Further, we have
Z

Z
ϕ0 (u)|∆h,2 (u)|du ≤

(0,∞)

∆h,2 (u)du
(0,∞)

Z
=


β u

β−1


−

u2

1−β/2 

λ2 h2/β + u2
h
i∞
= − (u2 + λ2 h2/β )β/2 − (u2 )β/2
0+

∞
2 2/β Z 1
λ h
=−
(u2 + sλ2 h2/β )β/2−1 ds
2
0
0+
Z 1
1 β
= λ h
sβ/2−1 ds . h.
2
0

du

(0,∞)

Combining these estimates yields that
Z
ϕ0 (u)|∂u ψh (u) + βuβ−1 |du . h1∧(1/β) = h1/β .
(0,∞)

Again Lemma 2.5 concludes that
automatic under (1.3).

√ R
n |fh (y) − φβ (y)|dy . (nhβ/2 )1/2 → 0 if nh2/β → 0, which is


Example 2.8 (Symmetric generalized hyperbolic Lévy process). The symmetric generalized hyperbolic
distribution [4], denoted by GH(λ, η, ζ), is infinitely divisible with the characteristic function
p

λ/2
Kλ (ζ η 2 + u2 )
η2
,
u 7→
η 2 + u2
Kλ (ηζ)
where Kλ denotes the modified Bessel function of the third kind with index λ ∈ R. In this example,
we will make extensive use of several exact and/or asymptotic properties of Kλ , without notice in most
places; we refer to [1, Chapter 9] for details. If L(J1 ) = GH(λ, η, ζ), then for each u ∈ R
p


λh/2 
Kλ (ζ/h) (ηh)2 + u2 h
(ηh)2
ih−1 Jh
E(e
)=
→ exp(−ζ|u|), h → 0,
(ηh)2 + u2
Kλ (ηζ)
showing that J is locally Cauchy (for ζ = 1). In the sequel we set L(J1 ) = GH(λ, η, 1). By [46] we know
that the Lévy measure of GH(λ, η, 1) admits the density such that




1
π
1
z 7→
1+
λ+
|z| + o(|z|) ,
|z| → 0.
π|z|2
2
2

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

7

Hence Assumption 2.1(2) fails to hold since β = δ = 1 here (except for the case of λ = −1/2, corresponding to a symmetric normal inverse Gaussian Lévy process [5]). We will observe that J instead meets
Assumption 2.4 for any (λ, η) ∈ R × (0, ∞). In this case, E(|J1 |q ) < ∞ for any q > 0.
Direct computations give


u
Kλ+1 1 p
(2.10)
∂u ψh (u) + 1 = 1 − p
(ηh)2 + u2 .
h
(ηh)2 + u2 Kλ
The right-hand side is essentially bounded, hence in particular (2.7)
holds.
R
We will verify (2.8) with r = 0. Given (2.10), we see that (0,∞) ϕ0 (u) |∂u ψh (u) + 1| du ≤ Ih0 + Ih00 ,
where


Z
u
du,
Ih0 :=
ϕ0 (u) 1 − p
(ηh)2 + u2
(0,∞)


Z
u
Kλ+1 1 p
00
2
2
p
Ih :=
ϕ0 (u)
(ηh) + u − 1 du.
h
(ηh)2 + u2 Kλ
(0,∞)
For Ih0 , we divide the domain of integration into (0, 1] and (1, ∞) and then derive the following estimates.
• The (0, 1]-part can be bounded by


Z
hp
i1
2ηh
u
p
du = 1 −
. h.
1− p
=
(ηh)2 + u2
2
2
0+
(ηh) + u
1 + ηh + (ηh)2 + 1
(0,1]
• The (1, ∞)-part equals
Z
Z
(ηh)2
p
ϕ0 (u)
du . h2
ϕ0 (u) . h2 .
p
u + (ηh)2 + u2
(ηh)2 + u2
(1,∞)
(1,∞)
Hence we obtain Ih0 . h. As for Ih00 , we first make the change of variables:
Z

Kλ+1 p 2
vh
η + v 2 − 1 dv.
Ih00 =
ϕ0 (vh) p
η 2 + v 2 Kλ
(0,∞)
Just like the case of Ih0 , we look at the (0, 1]-part and the (1, ∞)-part separately.
• Since we are supposing that η > 0, the (0, 1]-part trivially equals O(h).
• The (1, ∞)-part is somewhat more delicate. It can be written as


Z
vh
1
1
00
p
Ih,1 :=
ϕ0 (vh) p
λ+
+ σ(v) dv,
2
2
2
2
η +v
η + v2
(1,∞)
where
σ(v) :=




Kλ+1 p 2
1
1
p
η + v2 − 1 − λ +
,
2
Kλ
2
η + v2

which satisfies the property supv≥1 (η 2 + v 2 )σ(v) < ∞. We then observe that


Z
Z
1
v
v
00
p
ϕ0 (vh) 2
dv
+
ϕ
(vh)
Ih,1
≤h λ+
|σ(v)|dv
0
2 (1,∞)
η + v2
η2 + v2
(1,∞)


Z
Z
1
v
.h λ+
ϕ0 (vh) 2
dv +
v −2 dv
2 (1,∞)
η + v2
(1,∞)


Z
1
v
.h λ+
ϕ0 (vh) 2
dv
+
1
.
2 (1,∞)
η + v2
Further, using the integration by parts and the change of variables we derive
Z
Z
h
v
1 −h
2
e−vh log(η 2 + v 2 )dv
ϕ0 (vh) 2
dv = − e log(η + 1) +
η + v2
2
2 (1,∞)
(1,∞)

 2 
Z
x
−x
2
.1+
e log η +
dx . 1 + log(1/h).
h
(h,∞)
Summarizing the above computations we conclude that
(
Z
h log(1/h) (λ 6= −1/2)
ϕ0 (u) |∂u ψh (u) + 1| du .
h
(λ = −1/2)
(0,∞)

8

HIROKI MASUDA

verifying (2.8) with r = 0. By Lemma 2.5 we obtain (now aν = 1)
Z
0
|fh (y) − φβ (y)| dy . h log(1/h) . ha
for any a0 ∈ (0, 1).



2.2. Locally stable stochastic differential equation. Now let us recall the underlying SDE model
(1.1). Denote by Θ the closure of Θ = Θα × Θγ .
Assumption 2.9 (Regularity of the coefficients).
(1) The functions a(·, α0 ) and c(·, γ0 ) are globally
Lipschitz and of class C 2 (R), and c(x, γ) > 0 for every (x, γ).
3
(2) a(x,
·) ∈ C 3 (Θα ) and
 c(x, ·) ∈ C (Θγ ) for each x ∈R.

(3) sup
θ∈Θ

max max

0≤k≤3 0≤l≤2

∂αk ∂xl a(x, α) + ∂γk ∂xl c(x, γ)

+ c−1 (x, γ)

. 1 + |x|C .

The standard theory (for example [23, III §2c.]) ensures that the SDE admits a unique strong solution
as a functional of X0 and the Poisson random measure driving J; in particular, each Xt is Ft -measurable.
Assumption 2.10 (Identifiability). The random functions t 7→ (a(Xt , α), c(Xt , γ)) and t 7→ (a(Xt , α0 ), c(Xt , γ0 ))
on [0, T ] a.s. coincide if and only if θ = θ0 .
3. Stable quasi-likelihood estimation
3.1. Heuristic for construction. To motivate
our
R tj
R quasi-likelihood, we here present a formal heuristic
argument. In what follows we abbreviate tj−1
as j . For a moment, we write Pθ for the image measures
of X given by (1.1). In view of the Euler approximation under Pθ ,
Z
Z
Xtj = Xtj−1 + a(Xs , α)ds + c(Xs− , γ)dJs
j

j

≈ Xtj−1 + aj−1 (α)h + cj−1 (γ)∆j J,
from which we may expect that
j (θ) = n,j (θ) :=

∆j X − haj−1 (α)
≈ h−1/β ∆j J
h1/β cj−1 (γ)

in an appropriate sense. It follows from the locally stable property (1.5) that for each n the random
variables 1 (θ), . . . , n (θ) will be approximately i.i.d. with common distribution Sβ .
Now assume that the process X admits a (time-homogeneous) transition Lebesgue density under
Pθ , say ph (x, y; θ)dy = Pθ (Xh ∈ dy|X0 = x), and let Eθj−1 denote the expectation operator under Pθ
conditional on Ftj−1 . Then, we may consider the following twofold approximation of the conditional
distribution L(Xtj |Xtj−1 ):
Z
1
ph (Xtj−1 , Xtj ; θ) =
exp(−iuXtj )Eθj−1 {exp(iuXtj )}du
2π
Z



1
≈
exp(−iuXtj )Eθj−1 exp iu(Xtj−1 + aj−1 (α)h + cj−1 (γ)∆j J) du
2π
(Euler approximation)
Z

1
=
exp {−iu(∆j X − aj−1 (α)h)} ϕh cj−1 (γ)h1/β u du
2π
Z
1
1
=
exp{−iuj (θ)}ϕh (v)dv
cj−1 (γ)h1/β 2π
1
=
fh (j (θ))
cj−1 (γ)h1/β
1
≈
φβ (j (θ))
(Locally stable approximation).
cj−1 (γ)h1/β
This formal observation suggests to estimate θ0 by a maximizer of the random function


n
X
1
(3.1)
Hn (θ) :=
log
φβ (j (θ)) ,
cj−1 (γ)h1/β
j=1

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

9

which we call the stable quasi-likelihood. We then define the stable quasi-maximum likelihood estimator
(SQMLE) by any element θ̂n = (α̂n , γ̂n ) such that

n 
X
(3.2)
− log cj−1 (γ) + log φβ (j (θ)) .
θ̂n ∈ argmax Hn (θ) = argmax
θ∈Θ

θ∈Θ

j=1

Since we are assuming that Θ is compact, there always exists at least one such θ̂n . The heuristic argument
for the SQMLE will be verified in Section 3.2. The SQMLE is the non-Gaussian-stable counterpart to
the Gaussian quasi-likelihood previously studied by [27] and [36] for diffusion and Lévy driven SDE,
respectively.
Remark 3.1. It may happen, though very rarely, that the density fh of L(h−1/β Jh ) is explicit for each
h > 0. The normal-inverse Gaussian J [5], which we will use for simulations in Section 4.1, is such an
example. In that case, the approximation
1
ph (Xtj−1 , Xtj ; θ) ≈
fh (j (θ))
cj−1 (γ)h1/β
may result in a better quasi-likelihood since it precisely incorporates information of the driving noise.
Nevertheless and obviously, such an “exact L(h−1/β Jh )” consideration much diminishes the target class
of J, and going in this direction entails individual case studies.

3.2. Main result: Asymptotic mixed normality of SQMLE. For F-measurable random variables
µ = µ(ω) ∈ Rp and a.s. nonnegative definite Σ = Σ(ω) ∈ Rp ⊗ Rp , we denote by M Np (µ, Σ) the
p-dimensional mixed normal distribution corresponding to the characteristic function



1
v 7→ E exp iµ · v − v · Σv
.
2
That is to say, when Y ∼ M Np (µ, Σ), Y is defined on an extension of the original probability space
(Ω, F, P), and is equivalent in distribution to a random variable µ+Σ1/2 Z for Z ∼ Np (0, Ip ) independent of
F, where Ip denotes the p-dimensional identity matrix. Such an (orthogonal) extension of the underlying
probability space is always possible.
We introduce the two bounded smooth continuous functions:
∂φβ
gβ (y) := ∂y log φβ (y) =
(y),
kβ (y) := 1 + ygβ (y).
φβ
R
R
R
R
We see that gβ (y)φβ (y)dy = kβ (y)φβ (y)dy = 0, and that gβ (y)fh (y)dy = 0 as soon as |gβ (y)|fh (y)dy <
∞ because fh is symmetric. We also write
Z
Z
Cα (β) = gβ2 (y)φβ (y)dy,
Cγ (β) = kβ2 (y)φβ (y)dy,
Z
Z
1 T {∂α a(Xt , α0 )}⊗2
1 T {∂γ c(Xt , γ0 )}⊗2
ΣT,α (θ0 ) =
dt,
Σ
(γ
)
=
dt.
T,γ 0
T 0
c2 (Xt , γ0 )
T 0
c2 (Xt , γ0 )
The asymptotic behavior of the SQMLE defined through (3.1) and (3.2) is given in the next theorem,
which is the main result of this paper.
Theorem 3.2. Suppose that Assumptions 2.1 with β ∈ [1, 2), 2.9, and 2.10 hold. Then we have

√

√
L
nh1−1/β
(α̂n − α0 ), n(γ̂n − γ0 ) −
→ M Np 0, ΓT (θ0 ; β)−1 ,
(3.3)
n
where

ΓT (θ0 ; β) :=


Cα (β)ΣT,α (θ0 )
0
.
0
Cγ (β)ΣT,γ (γ0 )

In Section 3.3, we will deduce the large-time counterpart to Theorem 3.2 under the the ergodicity. In
that case the asymptotic distribution is not mixed normal but normal, with the asymptotic covariance
matrix taking a completely analogous form.
Below we list some immediate consequences of Theorem 3.2 and some related remarks worth being
mentioned.

10

HIROKI MASUDA
∂ c(x,γ )

a(x,α0 )
(1) The asymptotic distribution of θ̂n is normal if both x 7→ γc(x,γ0 )0 and x 7→ ∂αc(x,γ
are non0)
random; this is the case if X is a Lévy process.
(2) The estimators α̂n and γ̂n are asymptotically orthogonal, whereas not necessarily independent
due to possible non-Gaussianity in the limit.
(3) For β ∈ (1, 2), we can rewrite (3.3) as (recall (1.3))


√
n1/β−1/2 (α̂n − α0 ), n(γ̂n − γ0 )


L
−
→ M Np 0, diag T −2(1−1/β) {Cα (β)ΣT,α (θ0 )}−1 , {Cγ (β)ΣT,γ (γ0 )}−1 .

If fluctuation of X is virtually stable in the sense that both of the random time averages ΣT,α (θ0 )
and ΣT,γ (γ0 ) do not vary so much with the terminal sampling time T , then, due to the factor
“T −2(1−1/β) ”, the asymptotic covariance matrix of α̂n would tend to get smaller (resp. larger) in
magnitude for a larger (resp. smaller) T . This feature with respect to T is non-asymptotic.
(4) Of special interest is the locally Cauchy case (β = 1), where Hn is fully explicit:

n 
X

2
Hn (θ) = −
log(πh) + log cj−1 (γ) + log 1 + j (θ) .
j=1

In this case,

√
√
n(α̂n − α0 ), n(γ̂n − γ0 )
−1 
−1 


Z T
Z T
{∂α a(Xt , α0 )}⊗2
{∂γ c(Xt , γ0 )}⊗2
1
1
L
dt
dt
.
−
→ M Np 0, diag
,
2T 0
c(Xt , γ0 )2
2T 0
c(Xt , γ0 )2
This formally extends the i.i.d. model from the location-scale Cauchy population, where we have
√
n-asymptotic normality for the maximum-likelihood estimator. The Cauchy quasi-likelihood
has been also investigated in the robust-regression literature; see [41] and [42] for a breakdownpoint result in some relevant models. It would be interesting to study their SDE-model counterparts.
Remark 3.3. Based on the general criterion for the stable convergence, we could deduce a slightly more
general statement where the SQMLE has non-trivial asymptotic bias. In general, it is however impossible
to make an explicit bias correction in a unified manner without specific information of fh (hence of the
Lévy measure ν). Even when we have a full parametric form of ν, it may contain a parameter which
cannot be consistently estimated unless Tn → ∞; see [38] for specific examples.

Remark 3.4. The asymptotic efficiency in the sense of Hajék-Le Cam-Jeganathan is of primary theoretical importance (see [51]). Compared with the diffusion case studied in [14] and [15], asymptotic-efficiency
phenomena for the Lévy driven SDE (1.1) when observing (1.4) have been less well-known. Nevertheless,
for the classical local asymptotic normality property results when X is a Lévy process, one can consult
[38] for several explicit case studies, and to [18] for a general locally stable Lévy processes. Moreover,
[7] and the recent preprint [8] proved the local asymptotic mixed normality property about the unknown
parameters especially when c(x, γ) is a constant and the Lévy measure ν has a bounded support with
a stable-like behavior near the origin. Importantly, the model settings of [7] and [8] can be covered by
ours (Example 2.6), so that the asymptotic efficiency of our SQMLE is assured. In view of their result
and just like the fact that the Gaussian QMLE is asymptotically efficient for diffusions, it seems quite
promising that the proposed SQMLE is asymptotically efficient for the general class of SDE (1.1) driven
by a locally β-stable Lévy process.

Here is a variant of Theorem 3.2.
Theorem 3.5. Suppose that Assumptions 2.4 holds with β ∈ [1, 2) and
√
β
n(ψ (h) ∨ haν ) β+r → 0
for the constant aν given in Lemma 2.2(1). Suppose also that Assumptions 2.9 and 2.10 hold, and that
Z
|z|q ν(dz) < ∞
|z|>1

for every q > 0. Then we have (3.3).

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

11

To state a corollary to Theorems 3.2 and 3.5, we introduce the following statistics:
n
n
1 X {∂γ cj−1 (γ̂n )}⊗2
1 X {∂α aj−1 (α̂n )}⊗2
Σ̂T,α,n :=
,
Σ̂
:=
.
T,γ,n
n j=1
c2j−1 (γ̂n )
n j=1
c2j−1 (γ̂n )

√
It turns out in the proof that the quantity ( nh1−1/β )−1 ∂α Hn (θ0 ), n−1/2 ∂γ Hn (θ0 ) , the normalized
quasi-score, F-stably converges in distribution (Section 6.4.2), from which the Studentization via the
continuous-mapping theorem is straightforward:
Corollary 3.6. Under the assumptions of either Theorem 3.2 or Theorem 3.5, we have


1/2 √
1/2 √ 1−1/β
L
nhn
(µ̂n − µ0 ), Cγ (β)Σ̂T,γ,n
n(σ̂n − σ0 ) −
→ Np (0, Ip ).
(3.4)
Cα (β)Σ̂T,α,n

Table 1 summarizes the rates of convergence of the β-stable maximum quasi-likelihood estimators with
β ≤ 2, when the target SDE model is
(3.5)

dXt = a(Xt , α)dt + c(Xt− , γ)dZt

for a driving Lévy process Z with the correctly specified coefficient (a, c); again, note that the Gaussian
QMLE requires Tn → ∞, which is not necessary for the SQMLE with β < 2. We refer to [35] for a handy
statistic for testing the case (i) against the case (ii) based on the Gaussian QMLE.
Quasi-likelihood

Driving Lévy process Z

(i) Gauss

Wiener process

(ii) Gauss

Lévy process with jumps

(iii) Non-Gaussian stable

Locally β-stable Lévy process

Rates of convergence
α̂n
γ̂n
√
√
nhn
n
√
√
nhn
nhn
√ 1−1/β
√
nhn
n

Ref.
[27]
[33], [36]

Table 1. Comparison of the Gaussian (β = 2) and non-Gaussian stable (β ∈ [1, 2))
QMLE for the SDE (3.5), where the coefficient (a, c) is correctly specified: Case (iii) is
the contribution of this paper.

Remark 3.7. We have been focusing on β ≥ 1. For β ∈ (0, 1), direct use of the Euler scheme would
spoil the proofs in Section 6 because small-time variation of X by the noise term is dominated by that of
the trend coefficient a(x, α). In this case, direct use of the present stable quasi-likelihood based on the
mere Euler scheme would be inadequate. It would be necessary to take the drift structure into account
more precisely, as in the trajectory-fitting estimator studied in [31].

3.3. Ergodic case under long-time asymptotics. In this section, instead of the bounded-domain
asymptotics (1.3) we consider the sampling design
√ 2−1/β
(3.6)
Tn → ∞ and
nh
→ 0,
√ 1−1/β
which still implies that nh
→ ∞ when β ∈ [1, 2); for example, it suffices to have Tn → ∞ and
nh2 → 0. Theorem 3.11 below shows that under the ergodicity of X the asymptotic normality of the
SQMLE (3.2) holds. The logic of construction of the stable quasi-likelihood is completely the same as in
Section 3.1.
We will adopt Assumption 2.4 for the structural assumptions on J, and impose Assumption 2.9 without
any change.
Assumption 3.8 (Stability).
1
T

(3.7)

Z

(1) There exists a unique invariant measure π0 such that
Z
p
g(Xt )dt −
→ g(x)π0 (dx),
T → ∞,

T

0

for every measurable function g of at most polynomial growth.
(2) sup E(|Xt |q ) < ∞ for every q > 0.
t∈R+

12

HIROKI MASUDA

The property (3.7) follows from the convergence kPt (x, ·) − π0 (·)kT V → 0 as t → ∞ for each x ∈ R,
where Pt (x, dy) denotes the transition function of X under the true measure and kµkT V the total variation
norm of a signed measure µ. The next lemma, which directly follows from [36, Proposition 5.4], provides
a set of sufficient conditions for Assumption 3.8.
Lemma 3.9. Let X be given by (1.1) and suppose that ν({z 6= 0; |z| ≤ }) > 0 for every  > 0. Further,
assume the following conditions.
(1) Both a(·, α0 ) and c(·, γ0 ) are of class C 1 (R) and globally Lipschitz, and c is bounded.
(2) c(x, γ0 ) 6= 0 for every x.
(3) E(J1 ) = 0 and either one
R of the following conditions holds:
• E(|X0 |q ) < ∞ and |z|>1 |z|q ν(dz) < ∞ for every q > 0, and
lim sup
|x|→∞

• E(eq|X0 | ) < ∞ and

R
|z|>1

a(x, α0 )
< 0.
x

eq|z| ν(dz) < ∞ for some q > 0, and
lim sup sgn(x)a(x, α0 ) < 0.
|x|→∞

Then Assumption 3.8 holds.
We also need a variant of Assumption 2.10.
Assumption 3.10 (Model identifiability). The functions x 7→ (a(x, α), c(x, γ)) and x 7→ (a(x, α0 ), c(x, γ0 ))
are coincide π0 -a.e. if and only if θ = θ0 .
Theorem 3.11. Suppose that Assumptions 2.4 holds with
√
β
n(ψ (h) ∨ haν ) β+r → 0
for the constant aν given in Lemma 2.2(1). Suppose also that Assumptions 2.9, 3.8, and 3.10 hold. Then,
under (3.6) we have

√

√
L
nh1−1/β (α̂n − α0 ), n(γ̂n − γ0 ) −
→ Np 0, diag Vα (θ0 ; β)−1 , Vγ (θ0 ; β)−1 ,
where
{∂α a(x, α0 )}⊗2
π0 (dx),
c(x, γ0 )2
Z
{∂γ c(x, γ0 )}⊗2
Vγ (θ0 ; β) := Cγ (β)
π0 (dx).
c(x, γ0 )2
Z

Vα (θ0 ; β) := Cα (β)

The proof of Theorem 3.11 will be sketched in Section 6.7. Obviously, Studentization is possible just
the same as in Corollary 3.6. Again we remarks that Assumption 2.4 could be replaced with any other
one implying the convergences (2.3) and (2.5).
Remark 3.12. We have Cα (2) = 1 and Cγ (2) = 2, hence taking β = 2 in the expressions of Vα (θ0 )
and Vγ (θ0 ) formally results in the asymptotic Fisher information matrices for the diffusion case [27] (also
[50]).

4. Numerical experiments
For simulations, we use the nonlinear data-generating SDE



α2
dt + exp γ1 cos(Xt ) + γ2 sin(Xt ) dJt ,
dXt = α1 Xt +
2
1 + Xt
with θ = (α1 , α2 , γ1 , γ2 ) and J being either:
• The normal inverse Gaussian Lévy process (Example 2.8);
• The 1.5-stable Lévy processes (Example 2.6).

X0 = 0,

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

13

The setting is a special case of a(x, α) = α1 a1 (x) + α2 a2 (x) and c(x, γ) = exp{γ1 c1 (x) + γ2 c2 (x)}, for
√
√
which the asymptotic covariances of the nh1−1/β (α̂k,n − αk,0 ) and n(γ̂l,n − γl,0 ) are given by the
inverses of
Z
Z
1 T a2k (Xt )
1 T 2
Cα (β)
c (Xt )dt,
dt
and
C
(β)
γ
T 0 c2 (Xt , γ0 )
T 0 l
respectively.
4.1. Normal inverse Gaussian driver. Let J be an normal inverse Gaussian (NIG) Lévy process such
that
L(Jt ) = N IG(η, 0, t, 0),
where η > 0 may be unknown. This is a special case of the generalized hyperbolic Lévy process considered
in Example 2.8 with λ = −1/2. The numerical results below show that the SQMLE effectively works.
We set η = 5 or 10; the bigger η leads to a lighter tail of ∆j J, hence a seemingly more “diffusion-like”
sample-path behavior. Also, we set the terminal time T = 1 or 5. For each pair (η, T ), we proceed as
follows.
• First we apply the Euler scheme for the true model with discretization step size being ∆ :=
T /(3000 × 50).
3000×50
• Then we thin generated single path (Xk∆ )k=0
to pick up (Xjh )nj=0 with h = ∆ × 50 × 6,
∆ × 50 × 3 and ∆ × 50 for n = 500, 1000 and 3000, respectively.
Here, the number “50” of generation over each sub-periods (tj−1 , tj ] reflects that X virtually continuously
evolves as time goes along, though not observable. We independently repeat the above procedures for
L = 1000 times to get 1000 independent estimates θ̂n = (α̂n , γ̂n ), based on which boxplots and histograms
for Studentized versions are computed (Corollary 3.6). We used the function optim in R [44], and in each
optimization for l = 1, . . . , L we generated independent uniform random numbers Unif(αk,0 −10, αk,0 +10)
and Unif(γ0,l − 10, γ0,l + 10) for initial values for searching αk and γl , respectively.
The two cases are conducted:
(i) We know a priori that α2,0 = γ2,0 = 0, and the estimation target is θ0 = (α1,0 , γ1,0 ) = (−1, 1.5);
(ii) The estimation target is θ0 = (α1,0 , α2,0 , γ1,0 , γ2,0 ) = (−1, 1, 1.5, 0.5).
From the obtained simulation results, we observed the following.
• Figures 1 and 2: case of (i).
– The boxplots show the clear tendency that estimation accuracy for each T gets better for
larger n.
– The histograms show overall good standard normal approximations; the straight line in red
is the target standard normal density. It is observed is that the estimation performance of
γ̂n gets worse if the nuisance parameter η gets larger from 5 to 10. In particular, for the
cases where η = 10 we can see downward bias of the Studentized γ̂n , although it disappears
as n increases.
Overall, we see very good finite-sample performance of α̂n , while that of γ̂n may be affected to
some extent by the value of (T, η). As in the case of estimation of the diffusion coefficient for a
diffusion type processes, for better estimation of γ the value T should not be so large, equivalently
h should not be so large.
• Figures 3 and Figures 4–5: case of (ii).
– General tendencies are the same as in the previous case: for each T , estimate accuracy gets
better for larger n, while the gain of estimation accuracy for larger n is somewhat smaller
compared with the previous case.
– The histograms show that, compared with the previous case, the Studentized estimators
are of heavier tails and asymptotic bias associated with γ̂n severely remains, especially for
(T, η) = (5, 10) (Figure 5), unless n is large enough.

2
1
0
−1
−2

−2

−1

0

1

2

3

HIROKI MASUDA

3

14

a_3000

g_500

g_1000

g_3000

a_500

a_1000

a_3000

g_500

g_1000

g_3000

a_500

a_1000

a_3000

g_500

g_1000

g_3000

a_500

a_1000

a_3000

g_500

g_1000

g_3000

2
1
0
−1
−2

−2

−1

0

1

2

3

a_1000

3

a_500

Figure 1. NIG-J example. Boxplots of 1000 independent estimates α̂n (green) and γ̂n
(blue) for n = 500, 1000, 3000; (T, η) = (1, 5) (upper left), (T, η) = (1, 10) (upper right),
(T, η) = (5, 5) (lower left), and (T, η) = (5, 10) (lower right).

4

0.4
Density

0.1
0.0
0.4
Density

0.4
0.3
0.2

Density
0

2

4

−4

−2

0

2

4

2

4

Density

0.0

0.1

0.2

0.3

0.4

g.3000−CQMLE

0.4
4

0

0.1
−2

g.1000−CQMLE

2

4

0.3

−2

a.3000−CQMLE

0.3
0

2

0.2
−4

0.0
−4

Density
−2

4

0.1
4

0.1
−4

2

0.0
2

0.4
4

0.4
4

0

0.3
2

0.3
2

0

a.1000−CQMLE

Density
0

0.2

0.3

0.4
0.1
0.0
0.4
Density

0.3

−2

0.1
−2

−2

g.3000−CQMLE

0.2
−4

g.500−CQMLE

Density
0

−4

0.1
4

0.1
−2

4

0.0
2

0.0
−4

2

0.0
−4

0.4
Density
2

0.2

Density

0.3

0.4
Density

0.2
0.1
0.4

Density

0.1
4

0.1
0

0

0.4
Density
2

0.0
−2

−2

0.1
0

0

g.1000−CQMLE

0.3

0.4

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Figure 2. NIG-J example. Histograms of 1000 independent Studentized estimates of α
(green) and γ (blue) for n = 500, 1000, 3000; (T, η) = (1, 5) (upper left 2 × 3 submatrix),
(T, η) = (1, 10) (upper right 2 × 3 submatrix), (T, η) = (5, 5) (lower left 2 × 3 submatrix),
and (T, η) = (5, 10) (lower right 2 × 3 submatrix).

15

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ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

a1_1000

a1_3000

a2_500

a2_1000

a2_3000

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g1_1000

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g2_1000

g2_3000

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a1_1000

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1

2

3

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0

1

2

3

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1

2

3

a1_500

Figure 3. NIG-J example. Boxplots of 1000 independent estimates α̂1,n (green), α̂2,n
(blue), γ̂1,n (pink) and γ̂2,n (red) for n = 500, 1000, 3000; (T, η) = (1, 5) (top), (T, η) =
(1, 10) (second from the top), (T, η) = (5, 5) (second from the bottom), and (T, η) =
(5, 10) (bottom).

HIROKI MASUDA

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16

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0

Figure 4. NIG-J example. Histograms of 1000 independent Studentized estimates of
α1 (green), α2 (blue), γ1 (cream) and γ2 (red) for n = 500, 1000, 3000; (T, η) = (1, 5)
(left 4 × 3 submatrix) and (T, η) = (1, 10) (right 4 × 3 submatrix).
4.2. Genuine β-stable driver. Next we set L(J1 ) = Sβ with β = 1.5. Given a realization (xtj )nj=0 of
(Xtj )nj=0 we have to repeatedly evaluate


n 
X
xtj − xtj−1 − a(xtj−1 , α)h
− log[h1/β c(xtj−1 , γ)] + log φβ
(α, γ) 7→
.
h1/β c(xtj−1 , γ)
j=1
The stable density φβ is no longer explicit while we can resort to numerically integration. Here we used
the function dstable in the R package stabledist. As in the previous example, we give simulation
results for (pα , pγ ) = (1, 1) and (2, 2), with using uniformly distributed initial values for optim search.
In order to observe effect of the terminal-time value T we conduct the cases of T = 5 and T = 10, for
n = 100, 200, and 500. For Studentization, we used the values Cα (1.5) = 0.4281 and Cγ (1.5) = 0.9556
borrowed from [40, Table 6].
• Figures 6 and 7 show the boxplots and the histograms when pα = pγ = 1 for T = 5 and 10. As is
expected, we observe much better estimation accuracy compared with the previous NIG-driven
case. The figures reveal that the estimation accuracy of α are overall better for larger T , while
at the same time a larger h may lead to a more biased α̂n . Different from the NIG driven case
there is no severe bias in estimating γ. Somewhat surprisingly, the accuracy of Studentization
especially for the scale parameters may be good enough even for much smaller n compared with
the NIG driven case: the standard normality is well achieved even for n = 100.
• Figures 8 and 9 show the results for pα = pγ = 2 with T = 5 or 10. The observed tendencies,
including those compared with the NIG driven cases, are almost analogous to the case where
pα = pγ = 1.
In sum, our stable quasi-likelihood works quite well especially when J is standard 1.5-stable, although
so small T should be avoided for good estimation accuracy of α.

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

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Figure 5. NIG-J example. Histograms of 1000 independent Studentized estimates of
α1 (green), α2 (blue), γ1 (cream) and γ2 (red) for n = 500, 1000, 3000; (T, η) = (5, 5)
(left 4 × 3 submatrix) and (T, η) = (5, 10) (right 4 × 3 submatrix).

a_100

a_200

a_500

g_100

g_200

g_500

a_100

a_200

a_500

g_100

g_200

g_500

Figure 6. S1.5 -J example. Boxplots of 1000 independent estimates α̂n (green) and γ̂n
(blue) for n = 100, 200, 500; T = 5 (left) and T = 10 (right).
5. Proofs of Lemmas 2.2 and 2.5
This section presents the proofs of the L1 -local limit theorems given in Section 2.1.
5.1. Proof of Lemma 2.2. (1) We begin with the proof of (a). By the expression (2.2) we have
Z

ϕh (u) = exp
(cos(uz) − 1)h1+1/β g(h1/β z)dz
Z

= ϕ0 (u) exp
(cos(uz) − 1)ρ(h1/β z)g0,β (z)dz =: ϕ0 (u) exp {χh (u)} .
Pick a small 0ρ > 0 such that sup|y|≤0ρ |ρ(y)| ≤ 1/2. We will make use of the following two different
bounds for the function χh : on the one hand, we have
Z
|χh (u)| ≤ (1 − cos(uz))|ρ(h1/β z)|g0,β (z)dz

HIROKI MASUDA

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Figure 7. S1.5 -J example. Histograms of 1000 independent Studentized estimates of α
(green) and γ (blue) for n = 100, 200, 500; T = 5 (left 2 × 3 submatrix) and T = 10
(right 2 × 3 submatrix).
Z

1/β

(1 − cos(uz))g0,β (z)|ρ(h

=

Z

≤

1
2

(1 − cos(uz))g0,β (z)|ρ(h1/β z)|dz

z)|dz +

|z|≤0ρ h−1/β

Z

Z

|z|>0ρ h−1/β
∞

(1 − cos(uz))g0,β (z)dz + 4kρk∞

g0,β (z)dz

|z|≤0ρ h−1/β

0ρ h−1/β

Z
1
1
(cos(uz) − 1)g0,β (z)dz + Ch = |u|β + Ch;
≤−
2
2
on the other hand,
Z
(1 − cos(uz))g0,β (z)|ρ(h1/β z)|dz
|z|≤0ρ h−1/β
δ/β

. cρ h

Z

2

g0,β (z)|z| dz +
|z|≤0ρ h−1/β , |z|>1

Z

sin
|z|≤0ρ h−1/β , |z|≤1

0ρ h−1/β

z −1−β+δ dz + u2
1

. cρ h(δ/β)∧1 + u2 hδ/β ,
. cρ hδ/β

Z

δ

Z

1

z 1−β+δ dz





uz
δ
g0,β (z)|z| dz
2



0

where we used the fact supy | siny y | < ∞ in the second step, so that

|χh (u)| . cρ h(δ/β)∧1 + u2 hδ/β + h.
It follows from these estimates for χh with the mean-value theorem that for every s < 1 and C ≥ 0 we
have
Z
Z
(u−s ∨ uC )|ϕh (u) − ϕ0 (u)|du .
(u−s ∨ uC )ϕ0 (u) |exp{χh (u)} − 1| du
(0,∞)

(0,∞)

Z
≤

−s

(u



C

∨ u )ϕ0 (u)

0≤s≤1

(0,∞)

Z
.


sup exp(sχh (u)) |χh (u)|du

β

(u−s ∨ uC )e−|u|

/2




cρ h(δ/β)∧1 + u2 hδ/β + h du

(0,∞)

. cρ h(δ/β)∧1 + h . haν .
This prove the first half in (a). Since suph∈(0,1] ϕh (u) . exp(−C|u|β ) from the above argument, the
existence of the positive smooth density fh follows from the same argument as in the proof of [32, Lemma
4.4(a)]. The latter half is a direct consequence of the Fourier inversion:
Z
Z
1
e−iuy (ϕh (u) − ϕ0 (u)) du . |ϕh (u) − ϕ0 (u)|du . haν .
sup |fh (y) − φβ (y)| = sup
2π
y
y
This completes the proof of (a).
Next we prove (b). Because of the boundedness of ρ, the Lévy density of L(h−1/β Jh ) is bounded by a
constant multiple of g0,β (z). Invoking [48, Theorem 25.3], we see that the tail of fh is bounded by that

19

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ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

a1_200

a1_500

a2_100

a2_200

a2_500

g1_100

g1_200

g1_500

g2_100

g2_200

g2_500

a1_100

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g1_200

g1_500

g2_100

g2_200

g2_500

−6

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6

a1_100

Figure 8. S1.5 -J example. Boxplots of 1000 independent estimates α̂1,n (green), α̂2,n
(blue), γ̂1,n (pink) and γ̂2,n (red) for n = 100, 200, 500; T = 5 (top) and 10 (bottom).
of φβ uniformly in h ∈ (0, 1]: for each κ < β,
sup sup M β−κ

(5.1)

h∈(0,1] M >0

Z

|y|κ fh (y)dy < ∞.

|y|>M

R
Then, for any positive sequence bn ↑ ∞ the quantity |y|κ |fh (y) − φβ (y)| dy is bounded by the sum of
the two terms
Z
Z
|y|κ fh (y)dy +
|y|κ φβ (y)dy . bκ−β
→0
n
|y|≥bn

and

|y|≥bn



 Z
sup |fh (y) − φβ (y)|
y

|y|κ dy



aν
≤ b1+κ
n h .

|y|≤bn

aν
The convergence b1+κ
→ 0 follows on taking any bn = o(h−aν /(1+κ) ).
n h

(2) Again pick a positive real sequence bn → ∞. Then
Z
Z
Z
(5.2)
|fh (y) − φβ (y)|dy .
|fh (y) − φβ (y)|dy +
(bn ,∞)

(0,bn ]

|fh (y) − φβ (y)|dy =: δn0 + δn00 .

HIROKI MASUDA

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−4

0.2

Density

0.3

0.4
Density

0.2
0.1
0.4

Density

0.1

−4

0.4
4

0.4
4

4

0.3
Density
2

0.3
Density

Density

0

0.1
2

2

g1.100−SQMLE

0.0
0

0

0.1
−2

−2

a2.200−SQMLE

0.0
−4

0.4

−2

−2

g2.500−SQMLE

0.2
−4

−4

0.1

4

0.1
4

−4

0.4
Density
2

0.0
2

4

0.1
0

0.3

0.4
0.3
0.2
0.1

0

2

0.0
−2

g2.200−SQMLE

0.0

−2

0

0.3

0.4
Density

0.2
−4

g2.100−SQMLE

−4

−2

g1.500−SQMLE

0.1
4

4

0.0
−4

0.0
2

2

0.0

4

0

0.3

0.4
2

0.3

0.4
0.3
0.2

0

0.2

Density
0

−2

a2.100−SQMLE

0.1
−2

g1.200−SQMLE

0.1

−2

−4

0.0
−4

0.0
−4

4

0.1

4

2

0.0

2

0

0.3

0.4
Density

0.2
0.1

0

−2

a2.500−SQMLE

0.0
−2

0.0
−4

a1.500−SQMLE

0.2

4

0.3

0.4
0.3
0.2
0.0

0.1

Density

2

a2.200−SQMLE

g1.100−SQMLE

Density

0

0.2

−2

a2.100−SQMLE

−4

0.3

0.4
0.3
Density

0.1
0.0
−4

0.2

4

a1.200−SQMLE

0.2

2

0.2

0

0.2

−2

a1.100−SQMLE

0.2

0.3
Density

0.0

0.1

0.2

0.3
Density

0.2
0.1
0.0
−4

Density

a1.500−SQMLE

0.4

a1.200−SQMLE

0.4

a1.100−SQMLE

0.2

20

−4

−2

0

2

4

−4

−2

0

Figure 9. S1.5 -J example. Histograms of 1000 independent Studentized estimates of
α1 (green), α2 (blue), γ1 (cream) and γ2 (red) for n = 100, 200, 500; T = 5 (left 4 × 3
submatrix) and T = 10 (right 4 × 3 submatrix).
By (5.1) with κ = 0 we have
δn0 . b−β
n .

(5.3)

Recalling that ψh (u) := log ϕh (u) and that we are assuming that g ≡ 0 on {|z| > K}, we have ∂u ϕh (u) =
y|
ϕh (u)∂u ψh (u) for u > 0. Using the Fourier inversion, the integration by parts, and the fact supy∈R | sin
|y|r <
∞ for any r ∈ [0, 1], we can bound δn00 as follows:
Z
Z
e−iuy (ϕh (u) − ϕ0 (u)) du dy
δn00 .
(0,bn ]

Z

Z

cos(uy) (ϕh (u) − ϕ0 (u)) du dy

.
(0,bn ]

Z
.
(0,bn ]

Z

(0,∞)

1
y

sin(uy) (∂u ϕh (u) − ∂u ϕ0 (u)) du dy
(0,∞)

y r−1

.
(0,bn ]

. brn

Z

Z

Z

ur |∂u ϕh (u) − ∂u ϕ0 (u)| dudy

(0,∞)

ur |∂u ϕh (u) − ∂u ϕ0 (u)| du

(0,∞)

(5.4)

.

brn

Z

ur |ϕh (u) − ϕ0 (u)||∂u ψh (u)|du + brn

(0,∞)

Z

ur ϕ0 (u)|∂u ψh (u) + βuβ−1 |du.

(0,∞)

Suppose for a moment that
h
,
u > 0.
u
Then |∂u ψh (u)| . (1 + uβ )/u and it follows from (5.4) and the statement (1)(a) that
Z
Z
00
r
r−1
β
r
δn . bn
u (1 + u )|ϕh (u) − ϕ0 (u)|du + bn h
ur−1 ϕ0 (u)du
|∂u ψh (u) + βuβ−1 | .

(5.5)

(0,∞)

(0,∞)

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

21

. brn haν + brn h . brn h1∧aν

(5.6)

if r ∈ (0, 1]. By (5.3) and (5.6) we obtain
r 1∧aν
δn . b−β
.
n + bn h
1∧aν

Optimizing the upper bound with respect to bn results in the choice bn ∼ h− β+r , with which we conclude
(2.4). We note that introducing the parameter r > 0 is essential in the above estimates.
It remains to prove (5.5). Partially differentiating with respect to u under the integral sign, we obtain
Z
Z
sin(uy)
sin(uy)
1/β
ρ(h
y)dy
−
2c
dy
∂u ψh (u) = −2cβ
β
β
y
yβ
(0,Kh−1/β ]
(0,Kh−1/β ]
=: Rh (u) + Ah (u).
It suffices to show that |Rh (u)| . h/u and |Ah (u) + βuβ−1 | . h/u for u > 0. Write ξβ (y) = y −β ρ(y). We
have Rh ≡ 0 if cρ = 0. In case where cρ > 0, thanks to Assumption 2.1(2)(b), the change of variables
and the integration by parts yield that
Z
1−1/β
|Rh (u)| . h
sin(uh−1/β x)ξβ (x)dx
(0,K]


cos(uh−1/β x)
ξβ (x)dx
uh−1/β
(0,K]


Z
h
h
.
1 + |ξβ (0+)| +
|∂x ξβ (x)|dx . .
u
u
(0,K]
= h1−1/β



Z

∂x

Turning to Ah (u), we need the following specific identity from the Lebesgue integration theory [19]: for
r > 0 and β ∈ (0, 2), we have
 
Z
Z
sin x
βπ
e−ry y β−1 (cos r + y sin r)
1
(5.7)
dx
−
Γ(1
−
β)
cos
dy.
=
β
2
Γ(β) (0,∞)
1 + y2
(0,r) x
From the definition (2.1) and the property of the gamma function, we have the identity

β
2cβ

= Γ(1 −

β) cos( βπ
2 ).

Applying (5.7) together with the change of variables, we obtain
Z
sin x
β−1
β−1
|Ah (u) + βu
| = − 2cβ u
dx + βuβ−1
β
(0,uKh−1/β ] x
 
Z
βπ
sin x
dx
−
Γ(1
−
β)
cos
. uβ−1
β
2
(0,uKh−1/β ) x
Z

Z
−ry β−1
−ry β
e y
e y
β−1
.u
dy +
dy
2
2
(0,∞) 1 + y
(0,∞) 1 + y
r=uKh−1/β


Z
Z
−x β−1
e
x
e−x xβ
−β−1
= uβ−1 r−β
dx
+
r
dx
2
2
(0,∞) 1 + (x/r)
(0,∞) 1 + (x/r)
r=uKh−1/β

Z

Z
−x β
re x
≤ uβ−1 r−β
e−x xβ−1 dx +
dx
2
2
(0,∞)
(0,∞) r + x
r=uKh−1/β


Z
h
re−x xβ
h
.
1 + sup
dx . .
2
2
u
u
r>0 (0,∞) r + x

Here, in the last step we used that

 

 
Z
Z
x
re−x xβ
x −x β
dx = arctan
e x
−
arctan
(βxβ−1 − xβ )e−x dx
2 + x2
r
r
r
(0,∞)
(0,∞)
(0,∞)
Z
Z
β−1 −x
.
x
e dx +
xβ e−x dx < ∞
(0,∞)

(0,∞)

uniformly in r > 0. Thus we have obtained (5.5), completing the proof of the claim (2).
5.2. Proof of Lemma 2.5. It is enough to notice that combining Lemma 2.2(1), (5.2), (5.3) and (5.4)
leads to
Z
aν r
|fh (y) − φβ (y)| dy . b−β
n + (ψ (h) ∨ h )bn ,
β

and that the upper bound is optimized (with respect to bn ) to be (ψ (h) ∨ haν ) β+r .

22

HIROKI MASUDA

6. Proofs of the main results
This section is devoted to proving Theorems 3.2 and 3.5, Corollary 3.6, and Theorem 3.11.
6.1. Localization: elimination of large jumps.
6.1.1. Preliminaries. Prior to the proofs, we need to introduce a localization of the underlying probability
space by eliminating possible large jumps of J, thus enabling us to proceed as if E(|J1 |q ) < ∞ for every
q > 0. The point here is that, since our main results are concerned with the weak properties over the
fixed period [0, T ], we may conveniently focus on a subset ΩK,T (∈ F) ⊂ Ω on which jumps of J are
bounded by a constant K: supω∈Ω, t≤T |∆Jt (ω)| ≤ K, the probability P(ΩK,T ) being arbitrarily close to
1 for K large enough. This simple yet very powerful “localization” device is standard in the context of
limit theory for statistics based on high-frequency data [21], and has been considered for quite general
semimartingale models; we refer to [22, Section 4.4.1] for a comprehensive account.
Here we take a direct and concise route, without resorting to the general localization result. Recall
the Lévy-Khintchine representation (2.2), and let µ(dt, dz) denote the Poisson random measure having
the
R intensity measure dt ⊗ ν(dz), and µ̃(dt, dz) its compensated version. Fix any K > 1. We have
zν(dz) = 0 since ν is assumed to be symmetric, and the Lévy-Itô decomposition of J takes the
1<|z|≤K
form
Z
Z
Jt =

zµ(ds, dz) =: MtK + AK
t ,

z µ̃(ds, dz) +
|z|≤K

|z|>K

where M K is a purely-discontinuous martingale and AK is a compound-Poisson process independent
of M K . Note that the symmetry assumption of ν makes the parametric form of the drift coefficient
unaffected by elimination of large jumps of J. Since supt |∆MtK | ≤ K, we have E(|MtK |q ) < ∞ for any
t ∈ R+ and q > 0. 3 Further, the event


ΩK,T := µ (0, T ], {z; |z| > K} = 0 ∩ {|X0 | ≤ K} ⊂ Ω
R
has the probability exp{−T |z|>K ν(dz)}P(|X0 | ≤ K), which gets arbitrarily close to 1 by picking a
sufficiently large K. Denote by (XtK )t∈[0,T ] a solution process to the SDE
K
dX K = a(XtK , α0 )dt + c(Xt−
, γ0 )dMtK ,

which obviously admits a strong solution as a functional of (X0 , M K ), and satisfy
Xt (ω) = XtK (ω)
for t ∈ [0, T ] and ω ∈ ΩK,T .
Before proceeding, we recall the notation of the stable convergence in law. Let


Ω̃, F̃, P̃(dω, dω 0 ) := Ω × Ω0 , F ⊗ F 0 , P(dω)Q(ω, dω 0 )
be an extended probability space, with Q denoting a transition probability from (Ω, F) to (Ω0 , F 0 ). For
any random variables Gn defined on (Ω, F, P) and G∞ on (Ω̃, F̃, P̃), all taking their values in some metric
L

s
space E, we say that Gn converges stably in law to G∞ , denoted by Gn −−→
G∞ , if E{f (Gn )U } →
Ẽ{f (G∞ )U } for every bounded F-measurable random variable U ∈ R and every bounded continuous
L
function f : E → R. This mode of convergence entails that (Gn , Hn ) −
→ (G∞ , H∞ ) for every random
p
variables Hn and H∞ such that Hn −
→ H∞ . We refer to [20], [21], [22], [23, Chapters VIII.5c and IX.7],
and [17] for comprehensive accounts of the stable convergence in law.
The stable central limit theorem is a special case, where the limit is a mixed normal distribution, and
plays an essential role in the proof of the asymptotic mixed normality of the SQMLE. In Section 6.4.1, we
will apply the general result due to Jacod [20], one of the crucial findings in which is the characterization
of conditionally Gaussian continuous-time martingales defined on an extended probability space; the
technique essentially dates back to [13]. In most of the other existing stable convergence results, the
nesting condition on the underlying triangular arrays of filtrations is assumed, whereas it fails to hold for
our model. The foremost point of Jacod’s result is that it does not require the nesting condition.

3 More precisely, if K = inf{a > 0; supp(ν) ⊂ {z; |z| ≤ a}}, then by [48, Theorem 26.1] we have E{exp(r|J | log |J |)} <
t
t
∞ for each t > 0 and r ∈ (0, 1/K). The moment estimate for Lévy processes in small time is interesting in its own right.
Several authors have studied asymptotic behavior of the moment E{f (Jh )} as h → 0 for a suitable function f . We refer to
the recent paper [29] as well as the references therein for recent developments on this subject.

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

23

6.1.2. Statements and consequences. We write D([0, T ]; R) for the space of càdlàg processes over [0, T ]
taking values in R.
Lemma 6.1. Let ζn (x, θ) : Θ × D([0, T ]; R) → R, ξn (x) : D([0, T ]; R) → Rk , and ξ0 (x, a) : D([0, T ]; R) ×
Rp → Rk be measurable functions. Let η ∈ Rp be a random variable defined on (Ω0 , F 0 , P0 ). Then we
have the following.
p

p

(i) If supθ |ζn (θ, X K )| −
→ 0 for every K > 0 large enough, then supθ |ζn (θ, X)| −
→ 0.
L

L

s
ξ0 (X K , η) for every K > 0 large enough, then ξn (X) −
→ ξ0 (X, η).
(ii) If ξn (X K ) −−→

Proof. (i) Fix any  > 0 and then pick a K > 0 so large that P(ΩcK,T ) < . Then




K
lim sup P sup |ζn (X, θ)| >  ≤  + lim sup P sup |ζn (X , θ)| >  ≤ .
n

n

θ

θ

(ii) Fix any  > 0 and continuous bounded function f , and then pick a K > 0 for which P(ΩcK,T ) <
/(2kf k∞ ). Then, the term |E{f (ξn (X))} − Ẽ{f (ξ0 (X, η))}| is bounded by
E{f (ξn (X)); ΩK,T } − Ẽ{f (ξ0 (X, η)); ΩK,T × Ω0 }
+ E{f (ξn (X)); ΩcK,T } + Ẽ{f (ξ0 (X, η)); ΩcK,T × Ω0 }
≤ E{f (ξn (X K )); ΩK,T } − Ẽ{f (ξ0 (X K , η)); ΩK,T × Ω0 } + 2kf k∞ P(ΩcK,T ),
hence lim supn |E{f (ξn (X))} − Ẽ{f (ξ0 (X, η))}| ≤ .



The item (ii) will be used only for η ∈ Rp being a standard Gaussian random vector independent of F.
Based on Lemma 6.1, in order to prove Theorems 3.2 and 3.5 and Corollary 3.6, we may and do suppose
that
∃K > 0,

(6.1)

P (∀t ∈ [0, T ], |∆Jt | ≤ K) = 1,

thereby focusing on X K for arbitrarily large (but fixed) K > 0. Specific value of K does not matter in
the proofs below. We keep writing X instead of X K for notational simplicity.
For later use, we mention and recall some important consequences of either Assumption 2.1 with (6.1),
or Assumption 2.4.
• Following the argument [22, Section 2.1.5] together with Gronwall’s inequality under the global
Lipschitz condition of (a(·, α0 ), c(·, γ0 )), we see that


(6.2)
E sup |Xt |q ≤ C,
sup
E (|Xt − Xs |q |Fs ) . h(1 + |Xs |C )
t≤T

t∈[s,s+h]∩[0,T ]

for any q ≥ 2 and s ∈ [0, T ]; in particular,
E (|Xt − Xs |q ) = O(h).

sup
t∈[s,s+h]∩[0,T ]

R
• There exists a constant C0 > 0 such that |z|>y ν(dz) . y −β for y ∈ (0, C0 ], with which [30,
Theorem 2(a) and (c)] gives


κ
(6.3)
E sup |Jt |
. hκ/β
t≤h

for each κ ∈ (0, β).
• The convergences (2.3) and (2.5) hold:
Z
√
n |fh (y) − φβ (y)|dy → 0,
Z
|y|κ |fh (y) − φβ (y)|dy → 0,

κ ∈ [0, β).

24

HIROKI MASUDA

6.2. Preliminary asymptotics. Let us recall the notation j (θ) = {h1/β cj−1 (γ)}−1 (∆j X − haj−1 (α)).
Throughout this section, we look at asymptotic behavior of the auxiliary random function
Un (θ) :=

n
X

πj−1 (θ)η(j (θ)),

j=1

where π : R × Θ → Rk ⊗ Rm and η : R → Rm are measurable functions. This form of Un (θ) will appear in
common in the proofs of the consistency and asymptotic (mixed) normality of the SQMLE. The results
in this section will be repeatedly used in the subsequent sections.
Let Ej−1 (·) be a shorthand for E(·|Ftj−1 ) and write Un (θ) = U1,n (θ) + U2,n (θ), where
U1,n (θ) :=

n
X



πj−1 (θ) η(j (θ)) − Ej−1 {η(j (θ))} ,

j=1

U2,n (θ) :=

n
X

πj−1 (θ)Ej−1 {η(j (θ))}.

j=1

Given doubly indexed random functions Fnj (θ) on Θ, a positive sequence (an ), and a constant q > 0, we
will write


∗
−1
q
Fnj (θ) = OLq (an ) if sup sup E sup |an Fnj (θ)| < ∞.
n j≤n

θ

6.2.1. Uniform estimate of the martingale part U1,n .
Lemma 6.2. Suppose that:
(i) π ∈ C 1 (R × Θ) and supθ {|π(x, θ)| + |∂θ π(x, θ)|} . 1 + |x|C ;
(ii) η ∈ C 1 (R) and |η(y)| + |y||∂η(y)| . 1 + log(1 + |y|).
√
∗
n), hence in particular
Then, for every q > 0 we have U1,n (θ) = OL
q(
√

1
U1,n (θ) = Op ( nh1−1/β )−1 = op (1).
(6.4)
sup
1−1/β
nh
θ
Proof. Since we are assuming that the parameter space Θ is a bounded convex domain, the Sobolev
inequality [2, p.415] is in force: for each q > p,






E sup |n−1/2 U1,n (θ)|q . sup E |n−1/2 U1,n (θ)|q + sup E |n−1/2 ∂θ U1,n (θ)|q .
θ

θ

θ
−1/2

To complete the proof, it therefore suffices to show that both {n
U1,n (θ)} and {n−1/2 ∂θ U1,n (θ)} are
q
L -bounded for each θ and q > p. Fix any q > p ∨ 2 and θ in the rest of this proof.
Pn
Put χj (θ) = πj−1 (θ) η(j (θ)) − Ej−1 {η(j (θ))} , so that U1,n (θ) = j=1 χj (θ). Under the present
regularity conditions we may pass the differentiation with respect to θ through the operator Ej−1 :


∂θ χj (θ) = ∂θ πj−1 (θ) η(j (θ)) − Ej−1 {η(j (θ))}


(6.5)
+ πj−1 (θ) ∂η(j (θ))∂θ j (θ) − Ej−1 {∂η(j (θ))∂θ j (θ)} .
For each n, the sequences {χj (θ)}j and {∂θ χj (θ)}j form martingale difference arrays with respect to
Pn
(Ftj ), hence Burkholder’s inequality gives E{|n−1/2 ∂θk Un (θ)|q } . n−1 j=1 E{|∂θk χj (θ)|q } for k = 0, 1.
The required Lq -boundedness of {n−1/2 ∂θk U1,n (θ)} follows on showing that supj≤n E(|∂θk χj (θ)|q ) . 1.
Observe that for β ≥ 1 and r ∈ (0, β),
r

(6.6)

|j (θ)|r = h−1/β c−1
j−1 (γ){∆j X − haj−1 (α)}
n
o
0
. (1 + |Xtj−1 |C ) |h−1/β ∆j X|r + hr (1−1/β) (1 + |Xtj−1 |C )


. (1 + |Xtj−1 |C ) |h−1/β ∆j X|r + 1 .

Applying the estimate (6.3) together with the linear growth property of a(·, α0 ), the Lipschitz property
of c(·, γ0 ), the estimate (6.2), and Burkholder’s inequality for the stochastic integral with respect to J,

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

25

we derive the chain of inequalities:
 Z
r/2


1
Ej−1 |h−1/β ∆j X|r . hr(1−1/β)
Ej−1 {|a(Xs , α0 )|2 }ds
h j
 Z
r
+ h−r/β Ej−1
(c(Xs , γ0 ) − cj−1 (γ0 ))dJs
j

+ (1 + |Xtj−1 |C )E(|h−1/β Jh |r )
r(1−1/β)

. (1 + h

C

−r/β

Z

)(1 + |Xtj−1 | ) + h

j−1

E

r/2
(|Xs − Xtj−1 | )ds
2

j


. (1 + hr(1−1/β) )(1 + |Xtj−1 |C ) + h−r/β h2 (1 + |Xtj−1 |C )

r/2

. 1 + |Xtj−1 |C .

(6.7)

Using (6.6) and (6.7) with the disintegration, we arrive at the estimate

E (1 + |Xtj−1 |C )|j (θ)|r . 1 + sup E(|Xt |C ) . 1
t≤T

valid for r ∈ (0, β). By means of the condition on η,


E(|χj (θ)|q ) . E (1 + |Xtj−1 |C )Ej−1 {|η(j (θ))|q }


. E (1 + |Xtj−1 |C ) 1 + Ej−1 [{log(1 + |j (θ)|)}q ]


. E (1 + |Xtj−1 |C ) 1 + Ej−1 {|j (θ)|r }
. 1 + sup E(|Xt |C ),
t≤T
q

concluding that supj≤n E(|χj (θ)| ) . 1.
Next we note that
∂α j (θ) = −h1−1/β

∂α aj−1 (α)
,
cj−1 (γ)

∂γ j (θ) = −

∂γ cj−1 (γ)
j (θ).
cj−1 (γ)

By (6.5), the components of ∂θ χj (θ) consists of the terms


(1)
πj−1 (θ) η(j (θ)) − Ej−1 {η(j (θ))} ,


(2)
πj−1 (θ) ∂η(j (θ)) − Ej−1 {∂η(j (θ))} ,


(3)
πj−1 (θ) j (θ)∂η(j (θ)) − Ej−1 {j (θ)∂η(j (θ))}
for some π (i) (x, θ), i = 1, 2, 3, all satisfying the conditions imposed on π(x, θ). Again taking the conditions
on η into account, we can proceed as in the previous paragraph to obtain supj≤n E(|∂θ χj (θ)|q ) . 1. The
proof is complete.

6.2.2. Uniform estimate of the predictable (compensator) part U2,n . Introduce the notation:
δj0 (γ) =

cj−1 (γ0 ) −1/β
h
∆j J,
cj−1 (γ)

b(x, θ) = c−1 (x, γ){a(x, α0 ) − a(x, α)},

a∆
c∆
j−1 (s) = a(Xs , α0 ) − aj−1 (α0 ),
j−1 (s) = c(Xs , γ0 ) − cj−1 (γ0 ),
Z
−1/β
−1/β Z
h
h
rj (γ) =
a∆
c∆ (s)dJs .
j−1 (s)ds +
cj−1 (γ) j
cj−1 (γ) j j−1
Then
j (θ) = δj0 (γ) + h1−1/β bj−1 (θ) + rj (γ).
Expanding η we have
0
0
00
U2,n (θ) = U2,n
(θ) + U2,n
(θ) + U2,n
(θ),

(6.8)
where, with rj (θ; η) :=

R1
0

∂η(δj0 (γ) + h1−1/β bj−1 (θ) + srj (γ))ds and π 0 (x, θ) := π(x, θ)c−1 (x, γ),

0
U2,n
(θ) :=

n
X
j=1

n 
o
πj−1 (θ)Ej−1 η δj0 (γ) + h1−1/β bj−1 (θ) ,

26

HIROKI MASUDA

0
U2,n
(θ) := h−1/β

00
U2,n
(θ)

−1/β

:= h

n
X
j=1
n
X



Z
0
πj−1
(θ)Ej−1 rj (θ; η) a∆
(s)ds
,
j−1
j
0
πj−1
(θ)Ej−1



Z
rj (θ; η)

c∆
j−1 (s−)dJs


.

j

j=1

A uniform law of large numbers for (nh1−1/β )−1 U2,n (θ) will be one of the key ingredients in the proofs.
0
00
Lemma 6.3 below reveals that the terms U2,n
(θ) and U2,n
(θ) have no contribution in the limit; we will
0
deal with the remaining term U2,n
(θ) in Section 6.2.3.
Let us recall Itô’s formula, which is valid for any C β -function4 ψ (see [22, Theorems 3.2.1b) and
3.2.2a)]): for t > s,
Z t
ψ(Xt ) = ψ(Xs ) +
∂ψ(Xu− )dXu
s
Z tZ
{ψ(Xu− + c(Xu− , γ0 )z) − ψ(Xu− ) − ∂ψ(Xu− )c(Xu− , γ0 )z} µ(du, dz).
+
s

Let A denote the formal infinitesimal generator of X:
Z
Aψ(x) = ∂ψ(x)a(x, α0 ) + {ψ(x + c(x, γ0 )z) − ψ(x) − ∂ψ(x)c(x, γ0 )z} ν(dz),
the second term in the right-hand side being assumed well-defined. Then
Z t
Z tZ
(6.9)
ψ(Xt ) = ψ(Xs ) +
Aψ(Xu )du +
{ψ(Xu− + c(Xu− , γ0 )z) − ψ(Xu− )} µ̃(du, dz).
s

s
C

Obviously, we have |Aψ(x)| . 1 + |x| for ψ such that the derivatives ∂ k ψ for k ∈ {0, 1, 2} exist and have
polynomial majorants.
Lemma 6.3. Suppose that:
(i) π ∈ C 1 (R × Θ) and supθ {|π(x, θ)| + |∂θ π(x, θ)|} . 1 + |x|C ;
(ii) η ∈ C 1 (R) with bounded first derivative.
0
∗
2−1/β
00
∗
2−1/β
Then we have U2,n
(θ) = OL
) and U2,n
(θ) = OL
) for every q > 0. In particular, we
q (nh
q (nh
−1/2 0
−1/2 00
have supθ |n
U2,n (θ)| = op (1) and supθ |n
U2,n (θ)| = op (1).
0
Proof. In this proof, q denotes any positive real greater than or equal to 2. We begin with U2,n
(θ).
Applying (6.9) with ψ(x) = a(x, α0 ) and then taking the conditional expectation, we get
Z

Z
j−1
∆
Ej−1 {a∆
E
aj−1 (s)ds =
j−1 (s)}ds
j

j

Z Z

s

≤
j

Z

tj−1
Z s

.
j

Z
(6.10)

tj−1
Z s

.
j

Ej−1 {|Aa(Xu , α0 )|} duds


1 + Ej−1 (|Xu |C ) duds

∗
2
(1 + |Xtj−1 |C )duds = OL
q (h ).

tj−1

j−1

∆
j−1
{a∆
Write mj (θ; η) = rj (θ; η) − E {rj (θ; η)} and ã∆
j−1 (s) = aj−1 (s) − E
j−1 (s)}. Using (6.10) and
noting that rj (θ; η) is essentially bounded, we get


Z
n
X
0
−1/β
0
j−1
∆
∗
2−1/β
U2,n (θ) = h
πj−1 (θ)E
mj (θ; η) ãj−1 (s)ds + OL
)
q (nh

= h−1/β

j=1
n
X
j=1

j
0
πj−1
(θ)

Z


∗
2−1/β
Ej−1 mj (θ; η)ã∆
).
j−1 (s) ds + OLq (nh

j

0
∗
2−1/β
By Jensen’s inequality, the claim U2,n
(θ) = OL
) follows if we show
q (nh

q
1 j−1 
∆
(6.11)
sup sup sup E sup E
mj (θ; η)ãj−1 (s)
. 1.
h
n j≤n s∈[tj−1 ,tj ]
θ
4In case of β ∈ (1, 2), this means that ψ is C 1 and the derivative ∂ψ is locally Hölder continuous with index β − [β].

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

By (6.9) we may express ã∆
j−1 (s) as
Z s
Z
∆
(6.12)
ãj−1 (s) =
f (Xtj−1 , Xu )du +
tj−1

s

27

Z
g(Xu− , z)µ̃(du, dz),

tj−1

where Ej−1 {f (Xtj−1 , Xu )} = 0 with f (x, x0 ) being at most polynomial growth in (x, x0 ), and where
g(x, z) := a(x + zc(x, γ0 ), α0 ) − a(x, α0 ).
Hence, for (6.11) it suffices to prove


Z s Z
q
1 j−1
(6.13)
sup sup sup E sup E
< ∞.
g(Xu− , z)µ̃(du, dz)
mj (θ; η)
h
n j≤n s∈[tj−1 ,tj ]
θ
tj−1
Let Hj,t (θ; η) := E {mj (θ; η)| Ft } for t ∈ [tj−1 , tj ]; then, Hj,tj (θ; η) = mj (θ; η). Recall we are supposing
(1.2): Ft = σ(X0 ) ∨ σ(Js ; s ≤ t). Invoking the self-renewing property of a Lévy process [43, Theorem
I.32], we see that {Hj,t (θ; η), Ftj−1 ∨ σ(Jt ); t ∈ [tj−1 , tj ]} is an essentially bounded martingale. According
to the martingale representation theorem [23, Theorem III.4.34], the process Hj,t (θ) can be represented
as a stochastic integral of the form
Z t Z
ξj (s, z; θ)µ̃(ds, dz),
t ∈ [tj−1 , tj ],
(6.14)
Hj,t (θ; η) =
tj−1

with a bounded predictable process s 7→ ξj (s, z; θ) such that
Z

sup sup sup sup Ej−1
ξj2 (s, z; θ)ν(dz) < ∞.
n j≤n s∈[tj−1 ,tj ]

θ

Now, we look at the quantity inside the absolute value sign | · · · | in the left-hand side of (6.13). By taking
the conditioning with respect to Fs inside the sign “Ej−1 ”, substituting the expression (6.14) with t = s,
and then applying the integration-by-parts formula for martingales, it follows that the quantity equals
Z s Z

1 j−1
E
ξj (u, z; θ)g(Xu− , z)ν(dz)du .
h
tj−1
By the regularity conditions on a(x, α0 ) and c(x, γ0 ) we have |g(x, z)| . |z|(1 + |x|). It follows from this
bound together with (6.2) and Jensen and Cauchy-Schwarz inequalities that

Z s Z
 q
1
E sup Ej−1
ξj (u, z; θ)g(Xu− , z)ν(dz)du
h
θ
tj−1
q 

 Z
Z s
1
j−1
.
du
ξj (u, z; θ)g(Xu− , z)ν(dz)
E sup E
h tj−1
θ

Z
q/2 
Z
q/2 
Z
1 s
j−1
2
j−1
2
.
E sup E
ξj (u, z; θ)ν(dz)
E
g (Xu− , z)ν(dz)
du
h tj−1
θ
Z


1 s
.
E Ej−1 1 + |Xu |C du
h tj−1
Z

1 s
.
E 1 + |Xtj−1 |C du . 1.
h tj−1
0
∗
2−1/β
This proves (6.13), concluding that U2,n
(θ) = OL
).
q (nh
00
Next we consider U2,n
(θ). Using the martingale representation for mj (θ; η) as before, we have


Z
n
X
00
−1/β
0
j−1
∆
U2,n (θ) = h
πj−1 (θ)E
mj (θ; η) cj−1 (s−)dJs
j

j=1

+ h−1/β

n
X

0
πj−1
(θ)Ej−1 {rj (θ; η)}Ej−1

= h−1/β
−1/β

=h

j=1
n
X
j=1

c∆
j−1 (s−)dJs



j

j=1
n
X

Z



0
πj−1
(θ)Ej−1 ∆j Hj (θ; η)

Z

c∆
j−1 (s−)dJs



j
0
πj−1
(θ)Ej−1

Z Z

Z Z
ξj (s, z; θ)µ̃(ds, dz)

j

j



c∆
j−1 (s−)z µ̃(ds, dz)

28

HIROKI MASUDA

= h−1/β

n
X

0
πj−1
(θ)Ej−1

Z Z


ξj (s, z; θ)zc∆
(s)ν(dz)ds
.
j−1

j

j=1

Rs
j−1 ∆
∗
As in the case of a∆
{cj−1 (s)}| ≤ tj−1 Ej−1 {|Ac(Xu , γ0 )|}du = OL
q (h). Hence
j−1 , we have |E
Z
n


X
00
0
2−1/β
∗
(6.15)
U2,n
(θ) = h−1/β
πj−1
(θ) Ej−1 Ξ̃j,s (θ)c̃∆
),
j−1 (s) ds + OLq (nh
j

j=1

∆
where Ξ̃j,s (θ) := ξj (s, z; θ)zν(dz) − Ej−1 { ξj (s, z; θ)zν(dz)} for s ∈ [tj−1 , tj ] and c̃∆
j−1 (s) := cj−1 (s) −
j−1 ∆
j−1
2
∆
E {cj−1 (s)}. We have sups∈[tj−1 ,tj ] supθ E {|Ξ̃j,s (θ)| } . 1 and c̃j−1 (s) admits a similar representation to (6.12). Now we once more apply the martingale representation theorem: for each j and
s ∈ [tj−1 , tj ], the processes Mu0j (θ) := Ej−1 {Ξ̃j,s (θ)|Fu } and Mu00j := Ej−1 {c̃∆
j−1 (s)|Fu } for u ∈ [tj−1 , s]
are martingales with respect to the filtration {Ftj−1 ∨ σ(Ju ) : u ∈ [tj−1 , s]}, hence there correspond
R s R 0j
00j
0j
mu (z; θ)µ̃(du, dz) and Ms00j =
predictable processes m0j
u (z; θ) and mu (z) such that Ms (θ) = tj−1
R s R 00j
R
R
2
j−1
2
mu (z)µ̃(du, dz), and that supθ Ej−1 { (m0j
{ (m00j
u (z; θ)) ν(dz)} ∨ E
u (z)) ν(dz)} . 1. Thus,
tj−1
using the integration by parts formula as before we can rewrite (6.15) as
Z

Z Z s
n
1X 0
1
00
00j
U2,n
(θ) = nh2−1/β ·
Ej−1
m0j
(z;
θ)m
(z)ν(dz)
duds
πj−1 (θ) 2
u
u
n j=1
h j tj−1

R

R

∗
2−1/β
+ OL
).
q (nh

We can apply Cauchy-Schwarz inequality to conclude that the first term in the right-hand side is
∗
2−1/β
00
OL
), hence so is U2,n
(θ).
q (nh
√ 2−1/β
3/2−1/β
.h
→ 0 for β ∈ [1, 2), the last part of the lemma is trivial. The proof is thus
Since nh
complete.

6.2.3. Uniform law of large numbers. Building on Lemmas 6.2 and 6.3, we now turn to the uniform law
of large numbers for Un (θ) = U1,n (θ) + U2,n (θ). First we note the following auxiliary result.
Lemma 6.4. For any measurable function f : R × Θ → R such that
sup {|f (x, θ)| + |∂x f (x, θ)|} . 1 + |x|C ,
θ

we have (h = T /n)
sup sup
θ

t≤T

Z
[t/h]
1 t
1 X
p
f (Xtj−1 , θ) −
→ 0.
f (Xs , θ)ds −
n j=1
T 0

Proof. The target quantity can be bounded by
Z
[t/h]
1 X 1
h
sup
sup |f (Xs , θ) − fj−1 (θ)|ds + sup sup |f (Xt , θ)|
T θ t≤T
t≤T n j=1 h j θ


Z
n
1X1
h
.
(1 + |Xtj−1 | + |Xs |)C |Xs − Xtj−1 |ds +
1 + sup |Xt |C .
n j=1 h j
T
t≤T
By (6.2) the expectation of the upper bound tends to zero, hence the claim.



Proposition 6.5. Assume that the conditions in Lemma 6.2 hold and that supθ |∂x π(x, θ)| . 1 + |x|C .
(1) If β = 1, we have
1
1
sup Un (θ) −
n
T
θ

Z

T

Z
π(Xt , θ)

0




c(Xt , γ0 )
η
z + b(Xt , θ) φ1 (z)dzdt = op (1).
c(Xt , γ)

(2) If β ∈ (1, 2), we have
sup
θ

1
1
Un (θ) −
n
T

T

Z


π(Xt , θ)η

0


c(Xt , γ0 )
z φβ (dz)dzdt = op (1).
c(Xt , γ)

If further η is odd, then
sup
θ

1
Un (θ) = Op (1).
nh1−1/β

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

29

0
Proof. By Lemmas 6.2 and 6.3 it suffices to only look at U2,n
(θ) (recall (6.8)); the assumptions in Lemma
6.3 are implied by those in Lemma 6.2. Let
n
n 
o
1
1X
1
0
0
1−1/β
0
j−1
η
δ
(γ)
+
h
b
(θ)
.
U 2,n (θ) :=
U
(θ)
=
π
(θ)
E
j−1
j−1
j
2,n
n j=1
nh1−1/β
h1−1/β

Pn
Pn
0
1
2
(1) For β = 1, we can write U 2,n (θ) as the sum of n−1 j=1 fj−1
(θ) and n−1 j=1 fj−1
(θ), where

Z 
c(x, γ0 )
z + b(x, θ) φ1 (z)dz,
f 1 (x, θ) := π(x, θ) η
c(x, γ)

Z 
c(x, γ0 )
f 2 (x, θ) := π(x, θ) η
z + b(x, θ) {fh (z) − φ1 (z)}dz.
c(x, γ)
Pick a κ ∈ (0, β) = (0, 1). Since |η(y)| . 1 + |y|κ ,


c(x, γ0 )
sup η
z + b(x, θ) . (1 + |x|C )(1 + |z|κ ).
c(x, γ)
θ
Hence we have the bounds:
Z
1
C
(6.16)
sup |f (x, θ)| . (1 + |x| ) (1 + |z|κ )φ1 (y)dy . 1 + |x|C
θ
2

C

R
and |f (x, θ)| . (1 + |x| ) (1 + |z|κ )|fh (y) − φ1 (y)|dy = (1 + |x|C )o(1); in particular,
n

(6.17)

sup
θ

1X 2
f (θ) = op (1).
n j=1 j−1

Under the conditions on η, simple manipulations lead to

  


c(x, γ0 )
c(x, γ0 )
sup ∂η
z + b(x, θ) · ∂x
z + ∂x b(x, θ)
c(x, γ)
c(x, γ)
θ


 

c(x, γ0 )
c(x, γ0 )
C
. (1 + |x| ) k∂ηk∞ + sup
z + b(x, θ) ∂η
z + b(x, θ)
c(x, γ)
c(x, γ)
θ


Z
. (1 + |x|C ) 1 + (1 + |z|κ )φ1 (y)dy . 1 + |x|C .
Consequently,
sup ∂x f 1 (x, θ) . 1 + |x|C .

(6.18)

θ

The claim follows on applying Lemma 6.4 with (6.16), (6.17), and (6.18).
(2) For β ∈ (1, 2), we have
0

h1−1/β U 2,n (θ)
n
n 
o
1X
πj−1 (θ)Ej−1 η δj0 (γ) + h1−1/β bj−1 (θ)
n j=1

Z 
n
1X
cj−1 (γ0 )
=
πj−1 (θ) η
z fh (z)dz
n j=1
cj−1 (γ)
Z 1
n
n 
o
1X
+ h1−1/β
πj−1 (θ)bj−1 (θ)
Ej−1 ∂η δj0 (γ) + sh1−1/β bj−1 (θ) ds.
n j=1
0

=

(6.19)

As with the case β = 1, the first term in the rightmost side of (6.19) turns out to be equal to
R
c(Xt ,γ0 )
1 T
T 0 π(Xt , θ)η( c(Xt ,γ) z)φβ (dz)dzdt + op (1) uniformly in θ. Moreover, by the boundedness of ∂η and the
estimate |πj−1 (θ)bj−1 (θ)| . 1 + |Xtj−1 |C , the second term is Op (h1−1/β ) = op (1) uniformly in θ. Hence
Lemma 6.4 ends the proof of the first half.
Under the conditions in Lemma 6.2, it follows from (6.8) and Lemma 6.3 that
sup
θ

1
nh1−1/β

0

Un (θ) − U 2,n (θ) = Op (h).

If η is odd, the symmetry of the density fh implies that the first term in the rightmost side of (6.19) a.s.
0
equals 0 for each γ. Then supθ |U 2,n (θ)| = Op (1), hence the latter claim.


30

HIROKI MASUDA

We will also need the next corollary.
Corollary 6.6. Assume that the conditions in Lemma 6.2 hold, let β ∈ (1, 2), and let η : R → R be an
odd function. Then, for every q > 0 we have
n
X

1
nh1−1/β

∗
πj−1 (θ)Ej−1 {η(j (α0 , γ))} = OL
q (h),

j=1

and also
n
X

1
nh1−1/β


√
∗
πj−1 (θ)η(j (α0 , γ)) = OL
( nh1−1/β )−1 .
q

j=1

0
Proof. We have U2,n
(θ) ≡ 0 from the first identity in (6.19) and the fact bj−1 (α0 , γ) ≡ 0. This combined
with (6.8) and Lemmas 6.2 and 6.3 ends the proof.


6.3. Proof of Theorem 3.2: consistency. We first prove the following more or less well-known fact.
Lemma 6.7 (Consistency under possible multi-scaling). Let K1 ⊂ Rp1 and K2 ⊂ Rp2 be compact sets,
and let Hn : K1 × K2 → R be a random function of the form
Hn (u1 , u2 ) = k1,n H1,n (u1 ) + k2,n H2,n (u1 , u2 )
for some positive non-random sequences (k1,n ) and (k2,n ) and some continuous random functions H1,n :
K1 → R and H2,n : K1 × K2 → R. Let (u1,0 , u2,0 ) ∈ K1◦ × K2◦ be a non-random vector. Assume the
following conditions:
• k2,n = o(k1,n );
p
p
• supu1 |H1,n (u1 ) − H1,0 (u1 )| −
→ 0 and sup(u1 ,u2 ) |H2,n (u1 , u2 ) − H2,0 (u1 , u2 )| −
→ 0 for some continuous random functions H1,0 and H2,0 ;
• {u1,0 } = argmax H1,0 and {u2,0 } = argmax H2,0 (u1,0 , ·) a.s.
p

Then, for any (û1,n , û2,n ) ∈ K1 ×K2 such that Hn (û1,n , û2,n ) ≥ sup Hn −op (k2,n ), we have (û1,n , û2,n ) −
→
(u1,0 , u2,0 ).
Proof. The claim is a special case of [45, Theorem 1]. For convenience, we sketch the proof.
L
From the assumption we have (H1,n , H2,n ) −
→ (H1,0 , H2,0 ) in C(K1 × K2 ). Let
−1
−1
Hn0 (u1 ) := k1,n
Hn (u1 , û2,n ) = H1,n (u1 ) + k2,n k1,n
H2,n (u1 , û2,n ).
L

The second term in the rightmost side is op (1) uniformly in u1 ∈ K1 , so that Hn0 (·) −
→ H1,0 (·) in C(K1 )
−1
−1
with the limit a.s. uniquely maximized at û1,0 . Since Hn0 (û1,n ) ≥ supu1 k1,n
Hn (u1 , û2,n ) − op (k2,n k1,n
)=
p

→ u1,0 . We can follow a
sup Hn0 − op (1), the argmax theorem (for example [51]) concludes that û1,n −
p
similar line to deduce û2,n −
→ u2,0 with replacing Hn0 by
−1
Hn00 (u2 ) := k2,n
{Hn (û1,n , u2 ) − Hn (û1,n , u2,0 )} = H2,n (û1,n , u2 ) − H2,n (û1,n , u2,0 ),

which has the continuous limit process H2,0 (u1,0 , ·) − H2,0 (u1,0 , u2,0 ) in C(K2 ), uniquely maximized at
û2,0 a.s.

Returning to our model, we make a few remarks on Assumption 2.10. Recall the notation b(x, θ) =
c (x, γ){a(x, α0 ) − a(x, α)} and gβ (y) = ∂y log φβ (y). For β > 1, we define the random functions
Yβ,1 (·) = Yβ,1 (·; γ0 ) : Θγ → R and Yβ,2 (·) = Yβ,2 (·; θ0 ) : Θ → R by




Z Z 
1 T
c(Xt , γ0 )
c(Xt , γ0 )
Yβ,1 (γ) =
(6.20)
log
φβ
z
− log φβ (z) φβ (z)dzdt,
T 0
c(Xt , γ)
c(Xt , γ)


Z T
Z
1
c(Xt , γ0 )
2
Yβ,2 (θ) =
b (Xt , θ) ∂gβ
z φβ (z)dzdt.
2T 0
c(Xt , γ)
−1

We also define Y1 (·) = Y1 (·; θ0 ) : Θ → R by




Z Z 
1 T
c(Xt , γ0 )
c(Xt , γ0 )
Y1 (θ) =
log
φ1
z + b(Xt , θ)
− log φ1 (z) φ1 (z)dzdt.
T 0
c(Xt , γ)
c(Xt , γ)

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

31

These three functions are a.s. continuous in θ. Since the function z 7→ c1 φβ (c1 z +c2 ) defines a probability
density for every constants c1 > 0 and c2 ∈ R, Jensen’s inequality (applied ω-wise) imply that the dtintegrand in (6.20) is a.s. non-positive. The equality Yβ,1 (γ) = 0 holds only when the dt-integrand is
zero for t ∈ [0, T ] a.s., hence {γ0 } = argmax Yβ,1 a.s. Similarly, {θ0 } = argmax Y1 a.s. Moreover,
Z
Z T
1
Yβ,2 (α, γ0 ) =
b2 (Xt , (α, γ0 )) ∂gβ (z)φβ (z)dzdt
2T 0
Z
Z
1
{∂φβ (z)}2
1 T −2
=−
c (Xt , γ0 ){a(Xt , α0 ) − a(Xt , α)}2 dt ≤ 0,
dz ·
2
φβ (z)
T 0
where the maximum 0 is attained if and only if α = α0 .
6.3.1. Case of β = 1. Let
n

Y1,n (θ) :=


1
1X
Hn (θ) − Hn (θ0 ) =
n
n j=1


log


cj−1 (γ0 )
+ log φ1 (j (θ)) − log φ1 (j (θ0 )) .
cj−1 (γ)

Since {θ0 } = argmax Y1 a.s., by means of Lemma 6.7 the consistency of θ̂n (∈ argmax Y1,n ) is ensured
p
by the uniform convergence supθ |Y1,n (θ) − Y1 (θ)| −
→ 0. This follows from Lemma 6.4 and Proposition
6.5(1) with π(x, θ) ≡ 1 and η = log φ1 .
6.3.2. Case of β ∈ (1, 2). We have
Hn (θ) − Hn (θ0 ) = kn Yβ,1,n (γ) + ln Yβ,2,n (α, γ),
where kn := n, ln := nh2(1−1/β) , and
1
{Hn (α0 , γ) − Hn (α0 , γ0 )},
n
1
Yβ,2,n (α, γ) :=
{Hn (α, γ) − Hn (α0 , γ)}.
2(1−1/β)
nh
Yβ,1,n (γ) :=

By Lemma 6.7, it suffices to prove the uniform convergences:
p

sup |Yβ,1,n (γ) − Yβ,1 (γ)| −
→ 0,

(6.21)

γ

p

sup |Yβ,2,n (θ) − Yβ,2 (θ)| −
→ 0.

(6.22)

θ

The proof of (6.21) is much the same as in the case of β = 1, hence we only prove (6.22). Observe that

n 
X
1
Yβ,2,n (θ) =
log φβ (j (θ)) − log φβ (j (α0 , γ))
nh2(1−1/β) j=1
=

nh1−1/β
+

=:

n
X

1

1
2n

n

bj−1 (θ)gβ (j (α0 , γ)) +

j=1
n
X

1 X 2
b (θ)∂gβ (j (α0 , γ))
2n j=1 j−1

b2j−1 (θ) {∂gβ (˜
j (θ)) − ∂gβ (j (α0 , γ))}

j=1

Y0β,2,n (θ)

+ Y0β,2,n (θ) + Y00β,2,n (θ),

where ˜j (θ) is a random point on the segment connecting j (θ) and j (α0 , γ). Since gβ is odd, by means of
(6.4) and Corollary 6.6 we have supθ |Y0β,2,n (θ)| = op (1). We also get supθ |Y00β,2,n (θ)| = op (1), by noting
that supθ |˜
j (θ) − j (α0 , γ)| ≤ supθ |j (θ) − j (α0 , γ)| . (1 + |Xtj−1 |C )h1−1/β = Op (h1−1/β ) = op (1). It
remains to look at Y0β,2,n . The function gβ is bounded and smooth, and satisfies that
(6.23)

sup |y|k+1 ∂ k gβ (y) < ∞
y

for each non-negative integer k. The convergence supθ |Y0β,2,n (θ) − Yβ,2 (θ)| = op (1) now follows on
applying Proposition 6.5(2) for π(x, θ) = 21 b2 (x, θ) and η = ∂gβ with the trivial modification that inside
the function η we have “j (α0 , γ)” instead of “j (θ)”.

32

HIROKI MASUDA

6.4. Proof of Theorem 3.2: asymptotic mixed normality. We introduce the rate matrix
√

√
Dn = diag(Dn,1 , . . . , Dn,p ) := diag
nh1−1/β Ipα , nIpγ ∈ Rp ⊗ Rp .,
and then denote the normalized SQMLE by
√

√
ûn =
nh1−1/β (α̂n − α0 ), n(γ̂n − γ0 ) := Dn (θ̂n − θ0 ).
The consistency allows us to focus on the event {θ̂n ∈ Θ}, on which we have ∂θ Hn (θ̂n ) = 0 so that the
two-term Taylor expansion gives

(6.24)
−Dn−1 ∂θ2 Hn (θ0 )Dn−1 + r̂n ûn = Dn−1 ∂θ Hn (θ0 ),
where r̂n = {r̂nkl }k,l is a bilinear form such that


p
X
−1 −1
|r̂n | .
Dn,k
Dn,l sup |∂θk ∂θl ∂θm Hn (θ)| θ̂n,m − θ0,m .
θ

k,l,m=1

Here we wrote θ = (θi )pi=1 , and similarly for θ0 and θ̂n . Let
∆n,T := Dn−1 ∂θ Hn (θ0 ),

Γn,T := −Dn−1 ∂θ2 Hn (θ0 )Dn−1 .

If we have
L

(6.25)

(∆n,T , Γn,T ) −
→ (∆T , ΓT (θ0 ; β))

(6.26)

−1 −1
Dn,k
Dn,l sup |∂θk ∂θk ∂θm Hn (θ)| = Op (1),

where

∆T ∼ M Np (0, ΓT (θ0 ; β)) ,
k, l, m ∈ {1, . . . , p},

θ

then r̂n = op (1) and

−1
ûn = ΓT (θ0 ; β) + op (1)
∆n,T
= Γ−1
T (θ0 ; β)∆n,T + op (1)

L
−1
−
→ Γ−1
T (θ0 ; β)∆T ∼ M Np 0, ΓT (θ0 ; β) ,
completing the proof; since ΓT (θ0 ; β) may be random, the appropriate mode of convergence to deduce
(6.25) is the stable convergence (see Section 6.1). Thus, it suffices to prove (6.26) and

Ls
(6.27)
∆n,T −−→
∆T ∼ M Np 0, ΓT (θ0 ; β) ,
p

Γn,T −
→ ΓT (θ0 ; β).

(6.28)

6.4.1. Proof of (6.26). We may and do set pα = pγ = 1. Write R(x, θ) for generic matrix-valued function
on R × Θ such that supθ |R(x, θ)| . 1 + |x|C . By straightforward computations,
1

∂ 3 H (θ)
2(1−1/β) α n

nh

=

1
nh1−1/β

n
X

Rj−1 (θ)gβ (j (θ))

j=1


n 
1X
Rj−1 (θ)∂gβ (j (θ)) + h1−1/β Rj−1 (θ)∂ 2 gβ (j (θ)) ,
n j=1

n 
X
1
1
2
∂ ∂γ Hn (θ) =
Rj−1 (θ)gβ (j (θ)) + Rj−1 (θ)j (θ)∂gβ (j (θ))
nh2(1−1/β) α
nh1−1/β j=1

n 
1X
+
Rj−1 (θ)j (θ)∂ 2 gβ (j (θ)) + Rj−1 (θ)∂gβ (j (θ)) ,
n j=1
n 
1 3
1X
∂γ Hn (θ) =
Rj−1 (θ) + Rj−1 (θ)j (θ)gβ (j (θ))
n
n j=1

+ Rj−1 (θ)2j (θ)∂gβ (j (θ)) + Rj−1 (θ)3j (θ)∂ 2 gβ (j (θ)) ,
+

1

∂ ∂ 2 H (θ)
1−1/β α γ n

nh

n 
1X
=
Rj−1 (θ)gβ (j (θ)) + Rj−1 (θ)j (θ)∂gβ (j (θ))
n j=1

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

33


2
2
+ Rj−1 (θ)j (θ)∂ gβ (j (θ)) .
By (6.23), all the terms having the factor “1/n” in front of the summation sign in the above right-hand
sides are Op (1) uniformly in θ. Since the functions y 7→ gβ (y) and y 7→ y∂gβ (y) are odd, it follows from
Proposition 6.5(2) that both
1
nh1−1/β
1
nh1−1/β

n
X
j=1
n
X

Rj−1 (θ)gβ (j (θ)) = Op (1),
Rj−1 (θ)j (θ)∂gβ (j (θ)) = Op (1),

j=1

hold uniformly in θ. These observations are enough to conclude (6.26).
6.4.2. Proof of (6.27). Let j := j (θ0 ) and observe that


1
1
∆n,T = √ 1−1/β ∂α Hn (θ0 ), √ ∂γ Hn (θ0 )
n
nh


n
n
X
1
∂α aj−1 (α0 )
1 X ∂γ cj−1 (γ0 )
√
√
= −
gβ (j ), −
{1 + j gβ (j )} .
n j=1 cj−1 (γ0 )
n j=1 cj−1 (γ0 )
To apply Jacod’s stable central limit theorem, we introduce the partial sum process in D([0, T ]; Rp ):


[t/h]
[t/h]
1 X ∂γ cj−1 (γ0 )
1 X ∂α aj−1 (α0 )
gβ (j ), − √
{1 + j gβ (j )} , t ∈ [0, T ].
∆n,t := − √
n j=1 cj−1 (γ0 )
n j=1 cj−1 (γ0 )
Let


(6.29)

πj−1

∂α aj−1 (α0 )
∂γ cj−1 (γ0 )
= πj−1 (θ0 ) := diag −
,−
cj−1 (γ0 )
cj−1 (γ0 )



∈ Rp ⊗ R2 ,

η(y) := (gβ (y), 1 + ygβ (y)) = (gβ (y), kβ (y)) ∈ R2

(6.30)

(bounded),

RT
πj−1 η(j ). Write Γt (θ0 ; β) for ΓT (θ0 ; β) with the integral signs “ 0 ” in
Rt
their definitions replaced by “ 0 ”. Then, by means of [20, Theorem 3-2] (or [23, Theorem IX.7.28]), the
stable convergence (6.27) is implied by the following conditions: for each t ∈ [0, T ] and for any bounded
(Ft )-martingale M ,

[t/h]
4
X
1
p
j−1
√ πj−1 η(j )
(6.31)
−
→ 0,
E
n
j=1
so that ∆n,t = n

(6.32)

(6.33)

Pn
−1/2

j=1

[t/h]
n
⊗2 o
1 X
p
πj−1 · Ej−1 η(j ) − Ej−1 {η(j )}
πj−1 −
→ Γt (θ0 ; β),
n j=1

sup
t∈[0,T ]

[t/h]
1 X
p
√
→ 0,
πj−1 Ej−1 {η(j )} −
n j=1

[t/h]

(6.34)

X

j−1

E

j=1



1
√ πj−1 η(j )∆j M
n



p

−
→ 0.

The Lyapunov condition (6.31) trivially holds
since η is bounded and |πj−1 | . 1+|Xtj−1 |C . For (6.32),
R
arguing as in the proof of Lemma 6.3 with η(z)φβ (z)dz = 0 and noting that
Z
Z
η(z){fh (z) − φβ (z)}dz ≤ kηk∞ |fh (z) − φβ (z)|dz = o(n−1/2 ),
Z
Z
gβ (y)kβ (y)φβ (y)dy = {gβ (y) + ygβ2 (y)}φβ (y)dy = 0,
we obtain for each q > 0
(6.35)

Ej−1 {η(j )} =

Z

∗
2−1/β
η(z)fh (z)dz + OL
)
q (h

34

HIROKI MASUDA

Z
=

Ej−1 η ⊗2 (j ) =

Z

−1/2
−1/2
∗
∗
),
) = OL
η(z)φβ (z)dz + OL
q (n
q (n
∗
2−1/β
η ⊗2 (z)fh (z)dz + OL
)
q (h

Z
=

η

⊗2

(z)φβ (z)dz +

∗
−1/2
OL
)
q (n


=


Cα (β)
0
∗
−1/2
+ OL
).
q (n
sym. Cγ (β)

Then the left-hand side of (6.32) equals

Z
[t/h]
1 X
η ⊗2 (z)φβ (z)dz πj−1 + Op (n−1/2 )
πj−1 ·
n j=1


[t/h]
1 X
Cα (β)
0
=
πj−1 ·
π
+ Op (n−1/2 ).
0
Cγ (β) j−1
n j=1
By Lemma 6.4 the first term in the right-hand side converges in probability to Γt (θ0 ; β), hence (6.32) is
verified.
The convergence (6.33) follows on applying (6.35) and Lemma 6.4:


Z
[t/h]
[t/h]
√
√
1 X
1 X
√
πj−1
πj−1 Ej−1 {η(j )} =
n η(z)fh (z)dz + Op ( nh2−1/β )
n j=1
n j=1
= op (1) + Op (h3/2−1/β ) = op (1),
all the order symbols above being uniformly valid in t ∈ [0, T ].
Finally we turn to (6.34). By means of the decomposition theorem for local martingales (see [23,
Theorem I.4.18]), we may write M = M c + M d for the continuous part M c and the associated purely
discontinuous part M d . Our underlying probability space supports no Wiener process, so that in view of
the martingale representation theorem [23, Theorem III.4.34] for M , we may set M c = 0; recall (1.2). To
show (6.34) we will follow an analogous way to [49] with successive use of general theory of martingales
convergence.
It suffices to prove the claim when both π and η are real-valued. The jumps of M over [0, T ] are
P[t/h]
a.s.
bounded, and we have Mtn := j=1 ∆j M −−→ Mt = Mtd in D([0, T ]; R). Let
[t/h]

Ntn

:=

X 1
√ πj−1 η(j ).
n
j=1

For each n, N n is a local martingale with respect to (Ft ), and (6.34) equals that hM n , N n it → 0 for each
t ≤ T . The angle-bracket process
hN n it =

[t/h]
1 X 2
π Ej−1 {η 2 (j )}
n j=1 j−1

is C-tight, that is, it is tight in D([0, T ]; R) and any weak limit process has a.s. continuous sample paths;
this can be deduced as in the proof of (6.32). Hence, by [23, Theorem VI.4.13] the sequence (N n ) is tight
in D([0, T ]; R). Further, for every  > 0, as in the case of (6.31) we have



 X
n
n
X

n
n
n
P sup |∆Nt | >  = P max |∆j N | >  ≤
P (|∆j N | > ) .
E |∆j N n |4 → 0.
t≤T

j≤n

j=1

j=1

We conclude from [23, Theorem VI.3.26] that (N n ) is C-tight.
Fix any {n0 } ⊂ N. By [23, Theorem VI.3.33] the process H n := (M n , N n ) is tight in D([0, T ]; R).
Hence, by Prokhorov’s theorem we can pick a subsequence {n00 } ⊂ {n0 } for which there exists a process
00 L
H = (M d , N ) with N being continuous, such that H n −
→ H along {n00 } in D([0, T ]; R). By (6.2) we
have




1
n
C
sup E max |∆j N | . sup √ E 1 + sup |Xt |
< ∞,
j≤n
n
n
n
t≤T
00

hence it follows from [23, Corollary VI.6.30] that the sequence (H n ) is predictably uniformly tight. In
00
00
L
particular, (H n , [H n ]) −
→ (H, [H]) with the off-diagonal component of the limit quadratic-variation

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

35

P
process being 0 a.s.: [M, N ] = hM c , N c i + s≤· (∆Ms )(∆Ns ) = 0 a.s. identically (see [23, Theorem
I.4.52]). Therefore, given any {n0 } ⊂ N we can find a further subsequence {n00 } ⊂ {n0 } for which
00
00
L
[M n , N n ] −
→ 0. This concludes that
[t/h]
n

(6.36)

n

[M , N ]t =

X 1
p
√ πj−1 η(j )∆j M −
→0
n
j=1

in D([0, T ]; R).
Since [M n , N n ] − hM n , N n i is a martingale, we may write
Z tZ
n
n
n
n
n
χn (s, z)µ̃(ds, dz)
Gt := [M , N ]t − hM , N it =
0

for some predictable process χn (s, z). Using the isometry property and the martingale property of
RtR
RtR
the stochastic integral, we have E{(Gnt )2 } = E{ 0 χn (s, z)2 dsν(dz)} = E{ 0 χn (s, z)2 µ(ds, dz)} =
P
P
E{ 0<s≤t (∆Gns )2 } = E{ 0<s≤t (∆Msn ∆Nsn )2 }. Since


1
n 2
n 2
C
(∆Ns ) ≤ max(∆j N ) .
1 + sup |Xt |
j≤n
n
t≤T
P
n 2
and 0<s≤T (∆Ms ) is essentially bounded (for M is bounded), we obtain


1
sup E{(Gnt )2 } . E 1 + sup |Xt |C → 0,
n
t≤T
t≤T
p

which combined with (6.36) yields (6.34): hM n , N n it −
→ 0. The proof of (6.27) is complete.
Remark 6.8. The setting (1.2) of the underlying filtration is not essential. Even when the underlying
probability space carries a Wiener process, we may still follow the martingale-representation argument
as in [49].

6.4.3. Proof of (6.28). The components of Γn,T consist of
(6.37)

−

n
n
X
1
∂α2 aj−1 (α0 )
1 X {∂α aj−1 (α0 )}⊗2
2
∂
H
(θ
)
=
g
(
)
−
∂gβ (j ),
n
0
β
j
α
n j=1
c2j−1 (γ0 )
nh2(1−1/β)
nh1−1/β j=1 cj−1 (γ0 )

1

n
1
1 X ∂γ2 cj−1 (γ0 )
− ∂γ2 Hn (θ0 ) = −
{1 + j gβ (j )}
n
n j=1 cj−1 (γ0 )

(6.38)

−

n
1 X {∂γ cj−1 (γ0 )}⊗2 
1 + 2j gβ (j ) + 2j ∂gβ (j ) ,
n j=1
c2j−1 (γ0 )
n

1 X {∂α aj−1 (α0 )} ⊗ {∂γ cj−1 (γ0 )}
− 1−1/β ∂α ∂γ Hn (θ0 ) = −
{gβ (j ) + j ∂gβ (j )} .
n j=1
c2j−1 (γ0 )
nh
1

ByR (6.4), with Corollary
R 26.6 when β ∈ (1, 2), the first term in the right-hand side of (6.37) is op (1). Since
− ∂gβ (z)φβ (z)dz = gβ (z)φβ (z)dz = Cα (β), by Proposition 6.5 we derive
1
∂ 2 Hn (θ0 ) = Cα (β)ΣT,α (θ0 ) + op (1).
nh2(1−1/β) α
R
By Proposition 6.5 and kβ (z)φβ (z)dz = 0, the first term in the right-hand side of (6.38)
is op (1). As
R
forR the second term, noting that the function lβ (z) := 1 + 2zgβ (z) + z 2 ∂gβ (z) satisfies lβ (z)φβ (z)dz =
− kβ2 (z)φβ (z)dz = −Cγ (β), we obtain
Z
n
1 2
1 X {∂γ cj−1 (γ0 )}⊗2
− ∂γ Hn (θ0 ) = −
lβ (z)φβ (z)dz + op (1)
n
n j=1
c2j−1 (γ0 )
−

= Cγ (β)ΣT,γ (γ0 ) + op (1).
R

Finally, since {gβ (z) + z∂gβ (z)}φβ (z)dz = 0, Proposition 6.5 concludes that
−
completing the proof of (6.28).

1
nh1−1/β

∂α ∂γ Hn (θ0 ) = op (1),

36

HIROKI MASUDA

R
6.5. Proof of Theorem 3.5. Under the condition |z|>1 |z|q ν(dz) < ∞ for every q > 0, the moment
estimates (6.2) and (6.3) are in force without truncating the support of ν (Section 6.1). Further, it follows
from Lemmas 2.2(1) and 2.5 that we have both (2.3) and (2.5).
6.6. Proof of Corollary 3.6. The random mapping θ 7→ (ΣT,α (θ), ΣT,γ (γ)) is a.s. continuous, hence
applying the uniform law of large numbers presented in Lemma 6.4 we can deduce the convergences
p
p
Σ̂T,α,n −
→ ΣT,α (θ0 ) and Σ̂T,γ,n −
→ ΣT,γ (γ0 ). Then it is straightforward to derive (3.4) from (6.24), (6.25),
and (6.26).
6.7. Proof of Theorem 3.11. Most parts are essentially the same as in the proof of Theorem 3.5 (hence
as in Theorem 3.2). We only sketch a brief outline.
The convergences (2.3) and (2.5) are valid under the present assumptions. As in Theorem 3.5,
the localization introduced in Section 6.1 is not necessary here, since, under the moment boundedness supt E(|Xt |q ) < ∞ for any q > 0 and the global Lipschitz property of (a, c), we can deduce the
large-time version of the latter inequality in (6.2) by the standard argument: for any q ≥ 2 we have
E(|Xt+h − Xt |q |Ft ) . h(1 + |Xt |C ), hence in particular
sup E(|Xt+h − Xt |q ) . h sup E(1 + |Xs |C ) . h.
t∈R+

t∈R+

Obviously, (6.3) remains the same and Lemmas 6.2 and 6.3 stay valid as well.
As for the uniform low of large numbers under Tn → ∞, we have the following ergodic counterpart to
Lemma 6.4:
Lemma 6.9. For any measurable function f : R × Θ → R such that
sup {|f (x, θ)| + |∂x f (x, θ)|} . 1 + |x|C ,
θ

we have

n

1X
fj−1 (θ) −
sup
n j=1
θ

Z

p

→ 0.
f (x, θ)π0 (dx) −

R
Pn
p
→ 0 for each θ, hence
Proof. Write ∆fn (θ) = n−1 j=1 fj−1 (θ)− f (x, θ)π0 (dx). By (3.7) we have ∆fn (θ) −
it suffices to show the tightness of {supθ |∂θ ∆fn (θ)|}n in R, which implies the tightness of {∆fn (·)}n in
C(Θ). But this is obvious since
n

sup |∂θ ∆fn (θ)| .
θ

n

1X
1X
sup |fj−1 (θ) − π0 (f (·, θ))| .
(1 + |Xtj−1 |C ).
n j=1 θ
n j=1


Having Lemma 6.9 in hand, we can follow the contents of Sections 6.3, 6.4.1, and 6.4.3. The proof of
the central limit theorem is much easier than the mixed normal case, for we now have no need for looking
at the step processes introduced in Section 6.4.2 and also for taking care of the asymptotic orthogonality
condition (6.34). By means of the classical central limit theorem for martingale difference arrays [10], it
suffices to show, with the same notation as in (6.29) and (6.30),

n
4
X
1
p
j−1
√
E
πj−1 η(j )
−
→ 0,
n
j=1
n
n
⊗2 o

1X
p
πj−1 · Ej−1 η(j ) − Ej−1 {η(j )}
πj−1 −
→ diag Vα (θ0 ; β), Vγ (θ0 ; β) ,
n j=1
n

1 X
p
√
πj−1 Ej−1 {η(j )} −
→ 0,
n j=1
all of which can be deduced from the same arguments as in Section 6.4.2.
Acknowledgement. This work was partly supported by JSPS KAKENHI Grant Number 26400204 and
JST CREST Grant Number JPMJCR14D7, Japan.

ESTIMATION OF LOCALLY STABLE LÉVY DRIVEN SDE

37

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E-mail address: hiroki@math.kyushu-u.ac.jp

