Central Limit Theorem for Approximate
Quadratic Variations of Pure Jump Itô
Semimartingales∗
Forthcoming in Stochastic Processes and their
Applications
Assane Diop† Jean Jacod‡ Viktor Todorov§
November 5, 2012
Abstract
We derive Central Limit Theorems for the convergence of approximate quadratic
variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case
of an Itô semimartingale having a non-vanishing continuous martingale part. Here we
focus on the case where the continuous martingale part vanishes and find faster rates
of convergence, as well as very different limiting processes.
AMS 2000 subject classifications: 60F05, 60F17, 60G51, 60G07.
Keywords: Quadratic variation, Itô semimartingale, Pure jump processes, Approximate quadratic variation, Central Limit Theorem, Stable convergence in law.
∗
Assane Diop would like to thank the Laboratoire de Probabilités et Modèles Aléatoires (UPMC) for
its hospitality while this paper was completed. Todorov’s work was partially supported by NSF grant
SES-0957330. We would like to thank two anonymous referees for helpful comments.
†
Département de Mathématiques, Université Cheikh Anta Diop, Dakar, Sénégal; email: assane.diop.math@gml.com.
‡
Institut de Mathématiques de Jussieu, CNRS – UMR 7586, Université Pierre et Marie Curie–P6, 4
Place Jussieu, 75252 Paris-Cedex 05; email: jean.jacod@upmc.fr.
§
Department of Finance, Northwestern University, Evanston, IL 60208-2001, email:
vtodorov@northwestern.edu.
1
CLT for Approximate Quadratic Variation
1
2
Introduction
The quadratic variation [X, X] of a semimartingale X is one of the main tools of stochastic
calculus, and it can be defined in two different (and equivalent) ways. An “abstract” one,
which is
∑
[X, X]t = [X c , X c ]t + s≤t (∆Xs )2 ,
where ∆Xs = Xs − Xs− denotes the jump size at time t of the càdlàg process X and where
[X c , X c ] is the continuous increasing process coming in the Doob-Meyer decomposition of
the continuous (local) martingale part X c of X. The other one is that [X, X] is the unique
càdlàg process such that, for any t such that ∆Xt = 0 almost surely, we have the following
convergence in probability:
∑
P
XT (n,i) − XT (n,i−1) 2 −→
(1.1)
[X, X]t ,
i≥1: T (n,i)≤t
where for each n the T (n, i)’s constitute an increasing sequence of finite stopping times,
with T (n, 0) = 0 and limi→∞ T (n, i) = ∞, subject to the condition that the mesh of the
subdivision at stage n and in restriction to [0, t] goes to 0 in probability for all t, that
P
is supi≥1 (T (n, i) ∧ t − T (n, i − 1) ∧ t) −→ 0. This result goes back to Itô when X is a
stochastic integral with respect to Brownian motion, and is due to Meyer [12] in the most
general case. The left side of (1.1) is naturally called “approximate quadratic variation”.
The rate of convergence in (1.1), expressed in terms of the mesh size, and the limit of
the normalized difference between the two sides of (1.1), are of central importance in many
applications. One application is to the Euler approximation for a stochastic differential
equation driven by X, as seen in the paper [11] of Kurtz and Protter for example. Another
application is in finance, where quadratic variation is used as a leading measure of risk, see
e.g., [1] and is further used in designing derivative contracts for trading volatility, see e.g.,
the review article [5]. A Central Limit Theorem for the approximate quadratic variation
is then used to quantify the precision in estimating the unobservable [X, X] from discrete
observations. When X is continuous, this has been first done in the econometric literature
in [2] by Barndorff-Nielsen and Shephard.
Finally, let us also mention that in high-frequency statistics for discretely observed processes, most methods use in some form power variations (or truncated power variations),
which are the same as the approximate quadratic variation above, except that the power
2 is replaced by an arbitrary p > 0. In the case when the process X contains a continuous
martingale, the limit behavior of power variations has been studied in [3] and [9]. When X
is pure jump Itô semimartingale, [13] and [14, 15] have studied the limit behavior of power
variations but only in the case when the power is below the Blumenthal-Getoor index ([4])
of the driving jump process. Here we consider the case when p = 2 which leads to a faster
rate of convergence and a different limiting process than in the above mentioned papers.
The precise description of the asymptotic behavior of the approximate quadratic variation is in a sense similar to the case of the “approximate total variation” when the function
t 7→ Xt is of finite variation: this is the left side of (1.1) with the power p = 1 instead of
p = 2, and the convergence holds toward the total variation of X over (0, t], assumed to
be finite. In this case, no description is possible without very specific assumptions on the
sampling times T (n, i), like they are of the form T (n, i) = i∆n for a sequence ∆n of non
CLT for Approximate Quadratic Variation
3
random positive numbers, going to 0. And, even in this case one needs assumptions on
the function t 7→ Xt , such as it is absolutely continuous.
For the quadratic variation, we also make the (greatly) simplifying assumption that
T (n, i) = i∆n . Therefore, the approximate quadratic variation is simply
∑[t/∆ ]
(1.2)
[X, X]nt = i=1 n (∆ni X)2 , where ∆ni X = Xi∆n − X(i−1)∆n .
Our aim is to determine the behavior as n → ∞ of the error processes
Utn = [X, X]nt − [X, X]∆n [t/∆n ] .
(1.3)
In other words, we need to find a normalizing sequence δn → ∞ such that δn U n converges
to a non-trivial limit, in the “functional” sense (as processes). This is the reason why in
(1.3) we also discretize the limit [X, X].
As for the absolute continuity of the function of finite variation, in the case of quadratic
variation it is replaced by the fact that X should be an Itô semimartingale, which means
that its characteristics are “absolutely continuous” with respect to Lebesgue measure, see
Section 2 for more details. In this case the following result has been proved for a Lévy
process in [6] and further extended to general Itô semimartingales in [10]. We have the
functional convergence in law (and even a stronger form of it, the stable convergence in
law)
L−s
√1
(1.4)
U n =⇒ U,
∆
n
where the limiting process U has, loosely speaking, the form
∫ t
∞
(√
)
∑
√
√
√
Ut = 2
cs dWs + 2
∆XTp
κp cTp − Φp + 1 − κp cTp Φ′p 1{Tp ≤t}
0
p=1
where:
(i) ct is the “volatility”, that is the nonnegative process such that [X c , X c ]t =
∫t
0 cs ds, and which is assumed to be càdlàg; (ii) the Tp ’s are a sequence of stopping times
which exhausts the jumps of X; (iii) W is a Brownian motion, κp is uniform over (0, 1),
Φp , Φ′p are standard normal, and all those are mutually independent and independent of
the process X itself.
Now, when ct vanishes identically (one says that X is a “pure jump” semimartingale,
although it may have a drift and jumps are not necessarily summable), we have
√ a problem:
the limit U above also vanishes identically. Therefore, the normalization 1/ ∆n is not the
proper one, but one may hope for a non trivial convergence with a normalizing sequence
δn going to infinity faster than that. To do this, we basically show two types of results:
∫t
∑
1) When X is of finite variation, with a drift at , so that Xt = 0 as ds + s≤t ∆Xs ,
we have (under some weak conditions, including the càdlàg property of at ) the following
convergence:
∫ t
∞
(
)
∑
1
L
U n −→ U, Ut =
a2s ds + 2
∆XTp κp aTp − + (1 − κp )aTp 1{Tp ≤t} .
∆n
0
p=1
In this case, when the drift vanishes we again have a trivial (vanishing) limit.
CLT for Approximate Quadratic Variation
4
2) When X has infinite variation, or finite variation with vanishing drift, the situation is
quite different. We need a much stronger assumption, which basically says that the small
jumps of X behave more or less like those of a stochastic integral with respect to a (not
necessarily symmetrical) stable process with index β ∈ (0, 2).√ Then, the normalization
becomes δn = (∆n log(1/∆n ))−1/β (going much faster than 1/ ∆n to infinity, and much
faster even than 1/∆n when β < 1), and the limiting process is a stochastic integral with
respect to an independent stable process with the same index β. The results in this case
can be viewed as an extension of those of [8], in which Euler approximation for a Lévy
driven SDE are studied, providing as a corollary (not explicitly stated as such in that
paper) the above limit theorem when the process X is a (homogeneous, pure jump) Lévy
process. The present setting is much more general and asks for more sophisticated and
mostly different proofs; some parts of the proofs, though, are analogous, but nevertheless
reproduced here for the sake of completeness.
The paper is organized as follows. The setting and the assumptions are described in
Section 2, and the results are presented in Section 3. The next sections are devoted to the
proofs.
2
Setting and assumptions
Let us first describe three well known classes of processes. First, we have Lévy processes: a
process V is Lévy if it has stationary independent increments, and its law is characterized
by its characteristic triple (bV , cV , F V ) (bV is the drift, cV the variance of the Gaussian
part, and F V is the Lévy measure), via the Lévy-Khintchine formula:
∫ (
(
)
(
)
)
u2 cV
iuVt
V
+
E e
= exp t iub −
eiux − 1 − iux1{|x|≤1} F V (dx) .
2
Second, a semimartingale V is called an Itô semimartingale if its characteristics (B V , C V , ν V )
(see for example [7] for all unexplained notions about semimartingales, and also about the
“stable convergence in law” to be used later) having the following form:
∫
BtV (ω)
∫
t
bVs (ω) ds,
=
0
CtV (ω)
t
cVs (ω) ds, ν V (ω, dt, dx) = dt FtV (ω, dx).
=
(2.1)
0
Here bV and cV are optional (or predictable, if one wishes) processes, with cV ≥
∫ t 0, and
FtV = FtV (ω, dx) is an optional (or predictable) random measure on R, and 0 |bVs |ds
∫t ∫
∫t
and 0 cVs ds and 0 ds (x2 ∧ 1)FsV (dx) are finite for all t. The triple (bVt , cVt , FtV ) constitutes the spot characteristics of V . A Lévy process is an Itô semimartingale, whose spot
characteristics coincide with the – non random – characteristic triple.
In between these two notions, there are processes which are Itô semimartingales with
independent increments, and which we call non-homogeneous Lévy processes. They are
all Itô semimartingales having a non random version of their spot characteristics. They
also are all processes with independent increments whose (necessarily deterministic) characteristics are “absolutely continuous” in the sense of (2.1).
CLT for Approximate Quadratic Variation
5
Recall that the jumps of an Itô
V are (almost surely) summable on
∫ t semimartingale
∫
each finite interval if and only if 0 ds {|x|≤1} |x| FsV (dx) < ∞ a.s. for all t. When this
∫
is the case it is no restriction to suppose that {|x|≤1} |x| FsV (dx) < ∞ identically, and if
further CtV ≡ 0 we can write
∫
∫ t
∑
V
′V
′V
′V
′V
x FsV (dx) (2.2)
∆Vs , where Bt =
bs ds, bt = bt −
Vt = V0 + Bt +
0
s≤t
{|x|≤1}
(so b′V
t is in a sense the “genuine” drift).
Next, we turn to the assumptions. There are two – very different – situations, expressed
in assumptions 1 and 2 below. Assumption 1 requires the basic Itô semimartingale X
to be of finite variation (hence, with summable jumps), but apart from this it is rather
minimal. Assumption 2 requires much more structure on the process, and basically that it
is a stochastic integral with respect to a non-homogeneous Lévy process, which itself has
“small jumps” behaving like those of a stable process. All our processes below are defined
on some underlying filtered space (Ω, F, (Ft )t≥0 , P).
A common and fundamental feature of our setting, however, is that the second characteristic of X vanishes identically: as said before, when this is not the case, the asymptotic
behavior of the approximate quadratic variation is well known, and very different from
what we find below.
X
Assumption 1: The process
semimartingale with spot characteristics (bX
t , 0, Ft )
∫ X is an Itô
X
′X
such that the process t 7→ ∫ (|x| ∧ 1) Ft (dx) is locally bounded, and the paths t 7→ bt (ω)
X
X
of the process b′X
t = bt − {|x|≤1} x Ft (dx) are càdlàg.
Assumption 2: We have
∫
Xt =
t
σs− dZs + Yt ,
(2.3)
0
where
a) Z is a non-homogeneous Lévy process with (non-random) spot characteristics (bt , 0, Gt ),
and there are a number β ∈ (0, 2) and two functions θt+ , θt− ≥ 0 on R+ such that, for all
T > 0,
±
limx↓0 supt≤T |xβ G±
where
t (x) − θt | = 0,
(2.4)
−
G+
G
(x)
=
G
((x,
∞)),
(x)
=
G
((−∞,
−x)).
t
t
t
t
and the function bt is locally bounded, and the functions θt+ and θt− are Riemann-integrable
over each finite interval.
b) The process σ is an Itô semimartingale,
and its spot characteristics (bσt , cσt , Ftσ ) are
∫ 2
σ
σ
σ
such that the processes bt , ct and (x ∧ 1)Ft (dx) are locally bounded.
c) The process Y is an Itô semimartingale,
and its spot characteristics (bYt , 0, FtY ) are
∫
′
Y
β
Y
such that the processes bt and (|x| ∧1) Ft (dx) are locally bounded, for some β ′ ∈ [0, 1]
satisfying also β ′ < β.
The reader might – rightly – wonder about the requirements on β ′ in (c) above: that
CLT for Approximate Quadratic Variation
6
β ′ < β is absolutely necessary for the forthcoming results. That β ′ ≤ 1 is perhaps not so,
but so far we do not know what happens when 1 < β ′ < β.
Under this assumption, we use the following notation:
θt = θt+ + θt− ,
θt′ = θt+ − θt− ,
−
Gt (x) = G+
t (x) + Gt (x).
(2.5)
In Assumption 2,∫ saying that bYt is locally bounded is equivalent to saying that the true
Y
Y
′
drift b′Y
t = bt − {|x|≤1} x Ft (dx) is locally bounded (since β ≤ 1). Under Assumption 2
′Y
′
and when β < 1,∫the process X is of finite variation, and the true drift is b′X
t = bt + bt σt− ,
where b′t = bt − {|x|≤1} x Gt (dx).
Under some circumstances, we also need the following:
∫1
−
Assumption 3: We have Assumption 2 with β = 1, and supt≤T 0 |G+
t (x) − Gt (x)| dx <
′Y
′Y
∞ for all T . Moreover, the drift process bt of Y is bt = f (t, σ
et− ), where f = f (t, y)
p
is a locally bounded function on R+ × R which is Lipschitz in y, uniformly in (t, y) on
each compact set, and σ
et is a p-dimensional Itô semimartingale
whose spot characteristics
∫
(bσte , cσte , Ftσe ) are such that the processes bσte , cσte and (∥x∥2 ∧ 1)Ftσe (dx) are locally bounded.
Assumption 2 with β = 1, plus the restriction of Gt to [−α, α] being symmetric about
0 for all t, for some α > 0, implies the first part of Assumption 3. On the other hand, if
Assumptions 3 holds, we have θt′ = 0 for all t.
Remark 2.1. When Z is a homogeneous Lévy process the functions θt± are constants
θ± . In this case, Assumption 2 implies that the “degree of activity” of the jumps of X, as
measured by the local Blumenthal-Getoor index at each time t, is the constant β whenever
σt− ̸= 0, and is at most β ′ when σt− = 0.
When Z is a non-homogeneous Lévy process we have much wider flexibility. The
degree of activity at time t is β when σt− θt ̸= 0, but otherwise it can be anything smaller
than β. In particular this accommodates switching regimes: for example, Z varies like a
given Lévy process Z 1 when time runs through a locally finite union of intervals J, and
like another Lévy process Z 2 on the complement J c . If both Z j satisfy (a) of Assumption
2, with two indices β1 ≤ β2 , say, and the constants θj± , then Z satisfies (a) as well with
β = β2 and θt± = θ2± if t ∈ J, and θt± for t ∈ J c equal to 0 when β1 < β2 and to θ1± if
β1 = β2 .
We end this section with some possible extensions of Assumption 2.
1) One possibility is to require that the spot characteristics (bt , 0, Gt ) of Z be random
(instead of non-random) and F0 -measurable. This amounts to saying that Z is a nonhomogeneous Lévy process “conditionally on F0 ”. The results are exactly the same, as
well as the proofs: one can argue “conditionally on F0 ”, that is, the results are proved
under each Qω , where Qω is a regular version of the conditional probability, conditionally
on F0 , and restricted to the σ-field F ′ generated by F0 and G = σ(Zt , σs , Ys : s ≥ 0) (such
a regular version exists, since G is a separable σ-field). In this case, the functions θt± are
also random and F0 -measurable.
CLT for Approximate Quadratic Variation
7
In the same line, one could even state weaker assumptions like that Z is an Itô semimartingale with spot characteristics (bt , 0, Gt ) which are random and Ftj -measurable when
t ∈ (tj , tj+1 ], for a sequence of numbers (or, even, stopping times) increasing to infinity.
2) It would be nice to weaken the assumption even more, by requiring that the spot
Lévy measure FtX satisfies (2.4) for some processes θt± , with some uniformity in (ω, t) in
the convergence, and dropping σ and Y . Such a natural extension is so far not available.
However, in this direction, one can show after some simple manipulations that the following
X
set of assumptions, expressed in terms of the spot characteristics (bX
t , 0, Ft ) only, implies
X
X
Assumption 2. We require b to be locally bounded, and Ft to be of the form
FtX (ω, dx) = at− (ω)f X (t, x)dx + Ft′ (ω, dx)
where Ft′ is a measure satisfying the same conditions as FtY in Assumption 2, and the
1/β
process σt = at is like in Assumption 2, and f is a non-random function vanishing
outside R+ × [−1, 1] and
β θt+ T
⇒ f X (t, x) − x1+β
≤ xK
β ′ +1
β θt− ≤ |x|KβT′ +1 ,
x ∈ [−1, 0), t ≤ T ⇒ f X (t, x) − |x|1+β
x ∈ (0, 1], t ≤ T
for all T , and with again θt± and β ′ as in Assumption 2.
3) Assuming that Z is a symmetrical (homogeneous) Lévy process, (2.4) asserts that
G(x) = G+ (x) + G− (x) behaves as θ/xβ near 0. One could replace this by the fact that G
is regularly varying with index −β. This would modify the rates of convergence below by
a factor which is an appropriate slowly varying function of ∆n (the same kind of extension
also works for non-symmetrical and/or inhomogeneous Lévy processes). This provides a
large gain in terms of generality, and a very slight gain in terms of modeling power, and
we omit this generalization below.
In contrast, one cannot allow for a more general behavior like, say, when both xβn G(xn ) →
θ and ynγ G(yn ) → ζ, for two sequences xn and yn going to 0 and β ̸= γ: this is because
the rate of convergence obtained below depend upon the index β in an essential way.
4) In the non-homogeneous case, there is another property in Assumption 2 which absolutely cannot be relaxed, strong as it is: this is the fact that the index β is a constant.
We cannot take a function βt (unless of course this function reaches its maximum on a set
of positive Lebesgue measure, and then the assumption is satisfied with β equal to this
maximum): this is again because the rates below essentially depend on β.
3
The results
As said before, the results are very different under Assumptions 1 and 2, and we begin
with the result under Assumption 1. We first need to describe the limit. For this, we
consider a sequence Tq of stopping times which weakly exhausts the jumps of X: that
is, we have Tq ̸= Tm when q ̸= m and Tm < ∞, and for all ω and t with ∆Xt (ω) ̸= 0
CLT for Approximate Quadratic Variation
8
then there is a (necessarily unique) q = q(ω) such that Tq (ω) = t. Then, we introduce a
sequence (κq )q≥1 of i.i.d. variables, uniform over [0, 1], on some auxiliary space (Ω′ , F ′ , P′ ),
and we set
e = P ⊗ P′
e = Ω × Ω′ , Fe = F ⊗ F ′ , P
Ω
e such that Ft ⊂ Fet
(3.1)
(Fet ) = the smallest filtration on Ω
e
and each κq is FTq -measurable.
e is a very good extension of (Ω, F, (Ft )t≥0 , P), in the sense of [10].
e F,
e (Fet )t≥0 , P)
Thus (Ω,
Theorem 3.1. Under Assumption 1, we have stable convergence in law
1 n L−s
U =⇒ U
∆n
(3.2)
where U is the process defined on the above extension by
∫ t
∑
(
)
2
′X
Ut =
(b′X
)
ds
+
2
∆XTq b′X
Tq − κp + bTq (1 − κp ) .
s
0
(3.3)
q: Tq ≤t
In this case, the driving term
∑ of the error process is the drift. The sum in (3.3) is
absolutely convergent, because s≤t |∆Xs | < ∞ by hypothesis.
When the drift b′X
t vanishes identically, the limit vanishes as well, and the normalization
1/∆n is not appropriate. The next theorem shows the proper normalization in that case.
Theorem 3.2. Under Assumption 2, and either one of the following four additional assumptions:
• case I: β > 1,
• case II: β = 1 and Assumption 3 holds,
′Y
′
• case III: β < 1 and b′X
t (ω) = bt (ω) + bt σt− (ω) = 0 for all (t, ω),
• case IV: β = 1,
then:
a) In cases I, II and III, we have the following (functional) stable convergence in law:
(
1
)1/β
∆n log(1/∆n )
∫
L−s
U n =⇒ U,
with Ut =
t
2
σs−
dUs′ ,
(3.4)
0
where U ′ is defined on an extension of the space (Ω, F, (Ft )t≥0 , P) and is a non-homogeneous
′
U′
Lévy process, independent of F, with spot characteristics (bU
t , 0, Ft ) given by
′
β θ ′2
t
bU
t = − 21−β (β−1)
)
′
β ( 2
FtU (dx) = 22−β
(θt + θt′2 ) 1{x>0} + (θt2 − θt′2 ) 1{x<0}
1
|x|1+β
dx.
(3.5)
in cases I and III, whereas in case II it is
bU
t
′
= 0,
′
FtU (dx) =
θt2
dx.
2 x2
(3.6)
CLT for Approximate Quadratic Variation
9
b) In case IV, we have
1
1
n u.c.p.
(
)2 Ut =⇒ Ut = −
2
∆n log(1/∆n )
∫
t
θs′2 |σs |2 ds.
(3.7)
0
′
The measures FtU are the same in (3.5) and (3.6), because θt′ = 0 and β = 1 in case
II. Note that in (a) above, U ′ is a non-homogeneous stable process with index β, and is a
stable process when Z is a Lévy process.
′
Remark 3.3. Note that |θt′ | ≤ θt , so the measures FtU are positive. It may happen
that θt′ = θt (then the positive jumps of X are “more active” than the negative jumps),
or θt′ = −θt (then the negative jumps are more active). In these two cases, U has only
′
positive jumps. When θt′ ̸= 0, then FtU puts more mass on the positive side, whereas the
′
drift bU
t is negative when β > 1 and positive when β < 1. If θt ≡ 0, the drift vanishes and
′
U
′
Ft is symmetric, so U is a symmetrical process.
Remark 3.4. The two claims in cases II and IV are compatible, because if β = 1 and
Assumption 3 holds, then θt′ ≡ 0 and the limit in (3.7) vanishes.
Remark 3.5. In (3.7) we have convergence in probability, so one might expect an associated (second order type) CLT, but this is not the case. For example assume that Z is
a Lévy process and the convergence in (2.4) is fast enough (faster than 1/ log(1/x)), or
that Z is a non-homogeneous Lévy process, provided its characteristics are not varying
too wildly as functions of time. Then the second order term is still a drift, namely
∫
∫ t
)
1 t ′2
log(1/∆n ) (
1
u.c.p.
n
2
θ |σs | ds =⇒
θs′2 |σs |2 ds.
(
) U +
log(log(1/∆n )) ∆n log(1/∆n ) 2 t
2 0 s
0
Remark 3.6. As said before, in the definition (1.3) of U n we must take the discretized
version [X, X]∆n [t/∆n ] of the quadratic variation if we want the functional convergence,
in both theorems above. If however we are interested in the convergence at a particular
time t, we can replace this discretized version by the genuine quadratic variation [X, X]t
at time t in (1.3): the convergence results (for the processes evaluated at time t, and also
evaluated at any finite number of times tj ), are then valid.
Remark 3.7. One could wonder whether the convergence in the two theorems above
implies the convergence of moments, in the sense that for any t the sequence E(|δn Utn |p )
e t |p ). The
(where δn is the normalization factor) is bounded, or even converges to E(|U
answer is no, for several reasons: the first – obvious – reason is that in general |Utn |p is not
integrable. But even when the jumps of X are, for example, bounded (so Utn has moments
of all orders), the answer is still no, because in the situation of Theorem 3.2, and except in
e t |p ) = ∞ when p ≥ β. Hence, by a uniform integrability argument,
case IV, we have E(|U
the sequence E(|δn Utn |p ) cannot be bounded when p > β.
For example if X is a subordinator (case III with X = Z), it is easily shown that U n
is an increasing process and E(Utn ) = Ct∆n for a suitable constant C > 0. Hence the
“scaling factor” in L1 should be 1/∆n instead of the 1/(∆n log(1/∆n ))1/β found for the
convergence in law.
CLT for Approximate Quadratic Variation
4
10
Localization
Our first task is to reduce the problem to a situation where X satisfies some strengthened
versions of our assumptions. Those are as follows.
Assumption S1: We have Assumption 1, and the processes
bounded by a constant.
∫
|x| Ft (dx) and |b′X
t | are
Assumption S2: We have Assumption 2, and moreover
• we have |∆Zt | ≤ 1 (hence the measures Gt are supported by [−1, 1]), and |∆Yt | and
|σt | are bounded;
• we have supx>0, t≥0 xβ Gt (x) < ∞, and bt = 0 when β ≥ 1, and b′t = 0 when β < 1;
)
(
+
−
β −
• φ(z) = supt≥0 |z β G+
t (z) − θt | + |z Gt (z) − θt | goes to 0 as z ↓ 0 (hence the
functions θt+ and θt− are also bounded);
∫ β′ Y
• the processes |b′Y
|x| Ft (dx) are bounded, and b′Y
t | and
t = 0 when β < 1;
∫ 2 σ
σ
σ
• the processes |bt | and ct and x Ft (dx) are bounded.
∫1
Assumption S3: We have Assumption S2 with β = 1, and supt≥0 0 |Gt+ (x)−G− (x)t | dx <
∞, and the function f (t,
∫ y) is bounded and Lipschitz in y uniformly in (t, y), and the processes σ
et , |bσte |, cσte and x2 Ftσe (dx) are bounded.
In the remainder of the paper, K denotes a constant which may change from line to
line, and may depend on the characteristics of the processes and other parameters without
special mention, but it will never depend on the indices n and i used later on, neither on
time t, nor on ω.
Lemma 4.1. In Theorems 3.1 and 3.2 one can replace Assumptions 1 or 2 or 3 with
Assumptions S1 or S2 or S3.
Proof. We only give the proof for Theorem 3.2, since for Theorem 3.1 it is analogous (and
much simpler) and also classical.
L−s
We suppose that δn U n =⇒ U if X satisfies Assumption S2, and Assumption S3 in case
II, with the appropriate normalizing sequence δn . We take a process X satisfying only
Assumption 2, and Assumption 3 in case II, and b′X
t = 0 in case III, and we wish to prove
L−s
that δn U n =⇒ U again.
1) We first show that one can additionally suppose the following properties:
|∆Zt | ≤ 1,
β ≥ 1 ⇒ bt = 0,
β < 1 ⇒ b′t = b′Y
t = 0.
(4.1)
To see this, we replace Z and Y by
{
∫t
∑
Zbt = Zt − 0 bs ds − s≤t ∆Zs 1{|∆Zs |>1}
∫t
∑
β≥1 ⇒
Ybt = Yt + 0 σs− bs ds + s≤t σs− ∆Zs 1{|∆Zs |>1}
)
∑ (
bt = ∑ ∆Zs 1{∆Z |≤1} ,
β<1 ⇒
Z
Ybt = s≤t ∆Ys + σs− ∆Zs 1{∆Zs |>1} .
s
s≤t
CLT for Approximate Quadratic Variation
11
∫t
b Yb ) satisfies Assumption
Therefore Xt = 0 σs− dZbs + Ybt and it remains to prove that (Z,
bt | ≤ 1
2 or 3 (note that σt is not modified), with the same β and θt± , and with further |∆Z
(this is evident by construction) and, with obvious notation, that bbt = 0 when β ≥ 1 and
bb′ = b′Yb = 0 when β < 1.
t
t
b is a non-homogeneous Lévy process without Gaussian part and a first spot
Clearly, Z
b t are the restrictions of
characteristics satisfying bbt = 0 when β ≥ 1. The Lévy measures G
±
Gt to [−1, 1], so they satisfy (2.4) with the same β and θt , and also Assumption 3 when Z
∫
b t (dx)
does. When β < 1 the genuine drift bb′t vanishes by construction again, thus bbt = x G
b satisfies all required conditions.
is locally bounded. Therefore Z
b
Next, Yb∫ is an Itô semimartingale of finite variation. Its Lévy measures satisfy FtY (A) ≤
FtY (A) + {|x|>1} 1A (σt− x) Gt (dx), the latter term being locally bounded by (2.4), so
∫
′
b
b
(|x|β ∧ 1) FtY (dx) is locally bounded, as well as bt′Y = b′Y
t + σt− bt . When β < 1, we have
b
b
′X
′Y
′
′
Y
′ ), where
bt = bt + σt− bt = bt = 0. Finally if Y satisfies Assumption 3, bt′Y = f ′ (t, σ
et−
′
σ
e = (e
σ , σ) is a (p + 1)-dimensional Itô semimartingale with locally bounded characteristics, and f ′ (t, (y, z)) = f (t, y) + bt z is a function on R+ × Rp+1 satisfying the requirements
of Assumption 3.
b Yb ), we may assume (4.1).
In other words, upon substituting (Z, Y ) with (Z,
(
)
+
−
β −
2) Below, we assume (4.1). We write φT (x) = supt≤T |xβ G+
t (x) − θt | + |x Gt (x) − θt | ,
which goes to 0 as x ↓ 0 by hypothesis, for each T > 0. We have a sequence of stopping
times (τq )q≥1 , and a sequence of non-random times (tq )q≥1 , with τq ≤ tq , and τq ↑ ∞ as
q → ∞, and such that
|bt | ≤ q
sup
xβ Gt (x) ≤ q,
{ σx∈(0,1]
σ ≤ q,
′Y | ≤ q,
|b
|
≤
q,
c
|b
|σt | ≤ 2q , |∆Yt | ≤ q
t
t ∫
∫t 2
⇒
′
(x ∧ 1)Ftσ (dx) ≤ q,
(|x|β ∧ 1)FtY (dx) ≤ q
t ≤ tq ⇒
t ≤ τq
(4.2)
and also, in case II,
∫1
−
t ≤ tq ⇒ 0 |G+
t (x) − Gt (x)| dx ≤ q
σ
e
t ≤ τq ⇒ |e
σt− | ≤ q, |bt | ≤ q, cσte ≤ q,
∫
(x2 ∧ 1)Ftσe (dx) ≤ q.
In this case, we can also find a function fq (t, y) which is bounded by some constant Cq ≥ q
and Lipschitz in y, uniformly in (t, y), and such that f (t, y) = fq (t, y) when t ≤ tq and
∥y∥ ≤ 2q.
Then, for q ≥ 1, we set
∑
σ(q)t = σt∧τq − s≤t∧τq ∆σs 1{|∆σs |>q} − σ0 1{|σ0 |>q}
∑
σ
e(q)t = σ
et∧τq − s≤t∧τq ∆e
σs 1{∥∆eσs ∥>q} − σ
e0 1{∥eσ0 ∥>q}
{ ′Y
in cases I, III, IV
bt 1{t≤τq }
b′ (q)t =
fq (t, σ
e(q)t− )
in case II
∫t
∑
Y (q)t = 0 b′ (q)s ds + s≤t∧τq ∆Ys 1{|∆Ys |≤q}
∫t
Z(q)t = Zt∧tq ,
X(q)t = 0 σ(q)s− dZ(q)s + Y (q)t .
in case II
By construction |σ(q)t | ≤ 2q, and ∥e
σ (q)t ∥ ≤ 2q in case II, hence
{
σ(q)t = σt , σ
e(q)t = σ
et , Z(q)t = Zt
t < τq ⇒
,
Y
(q)
=
Yt , X(q)t = Xt
b′ (q)t = b′Y
t
t
(4.3)
CLT for Approximate Quadratic Variation
12
The process X(q) is a non-homogeneous Lévy process, with spot characteristics (bt , 0, Gt )
for t ≤ tp , and (0, 0, 0) for t > tq . These characteristics satisfy all requirements of As±
sumption S2 or S3, with the same β and the functions θ(q)±
t = θt 1{t≤tp } . The process
σ(q) is an Itô semimartingale, with spot characteristics
σ(q)
σ(q)
bt = bσt 1{t≤τq } ,
ct = cσt 1{t≤τq }
∫
σ(q)
Ft (A) = 1{t≤τq } {|x|≤q} 1A (x) Ftσ (dx),
∫
σ(q)
σ(q)
σ(q)
hence |bt | and ct
are smaller than q, and ∥x∥2 Ft (dx) ≤ q 3 , so σ(q) satisfies the
requirements of Assumption S2, and analogously σ
e(q) satisfies those of Assumption S3.
The process Y (q) is also an Itô semimartingale, with
∫
′Y (q)
Y (q)
′
bt
= b (q)t ,
Ft
(A) = 1{t≤τq }
1A (y) FtY (dy).
{|y|≤q}
∫
′
′
′Y (q)
Y (q)
Hence |bt
| ≤ Cq and |x|β Ft
(dx) ≤ q 1+β and Y (q) satisfies the requirements of
′Y (q)
Assumption S2. Finally, in case II, bt
= fq (t, σ
e(q)t− ), thus ending the proof that each
process X(q) satisfies Assumptions S2 or S3.
3) Our hypothesis now implies that the normalized error processes δn U n (q) associated
with X(q) converge stably in law to the limit U (q), as specified in Theorem 3.2, and with
±
±
±
θ(q)±
t and σ(q)t instead of θt and σt . Since θ(q)t = θt and σ(q)t = σt when t < τq , we
also have Ut = U (q)t when t < τq .
Therefore, if Φ is a variable on (Ω, F, P) and H is a continuous function on the Skorokhod space D(R+ , R), which satisfies H(x) = H(y) as soon as x(t) = y(t) for all t ≤ T ,
for some T , and if both Φ and H are bounded by 1, we have
E(Φ H(δn U n )) − E(Φ H(δn U ′n (q))) ≤ P(τq ≤ T )
e H(U ′ (q))) ≤ P(τq ≤ T )
e
E(Φ H(U )) − E(Φ
e H(U ′ (q))) as n → ∞, for all q ≥ 1.
E(Φ H(δn U ′n (q))) → E(Φ
L−s
e H(U )), hence δn U n =⇒
Since P(τq ≤ T ) → 0 as q → ∞, we obtain E(Φ H(δn U n )) → E(Φ
U.
We end this section with two useful facts, for which we use the notation I(n, i) =
((i−1)∆n , i∆n ]. First, If f and f ′ are two locally bounded and locally Riemann-integrable
functions on R+ , then the following convergence holds locally uniformly in t (as n → ∞,
recall ∆n → 0):
∫
∫ s
[t/∆n ] ∫
1 ∑
1 t
′
f (s) f ′ (s) ds.
f (s) ds
f (r) dr →
∆n
2
0
I(n,i)
(i−1)∆n
(4.4)
i=1
Second, by Itô’s formula the error process U n can be expressed as
∑
∫
[t/∆n ]
Utn
= 2
i=1
ζin ,
where
ζin
=
I(n,i)
(
)
Xs− − X(i−1)∆n dXs
(4.5)
CLT for Approximate Quadratic Variation
5
13
Proof of Theorem 3.1
1) By Lemma 4.1 it is no restriction to assume that, for some constant K,
∫
|x| Ft (dx) ≤ K,
|b′X
t | ≤ K.
(5.1)
Since X is an Itô semimartingale, we can represent its jumps by a Poisson measure
(through its Grigelionis representation, see e.g., Theorem 2.1.2 in [10]): up to enlarging
the space if necessary, we have a Poisson random measure p on R+ × R+ with intensity
measure q(dt, dz) = dt ⊗ dz and a predictable function ϕ on Ω × R+ × R+ such that
∫
∆Xt ̸= 0 ⇒ ∆Xt =
ϕ(t, z) p({t}, dz).
(5.2)
If m ≥ 1 we call (T (m, q) : q ≥ 1) the successive (finite) jump times of the Poisson process
t 7→ p([0, t] × [m − 1, m)), and we relabel the double sequence (T (m, q) : m, q ≥ 1) as to
form a single sequence Tq of stopping times, which by construction have pairwise disjoint
graphs, and weakly exhaust the jumps of X by (5.2): this means that, outside a null set
of ω’s, we have the (possibly strict) inclusion {t > 0 : ∆Xt (ω) ̸= 0} ⊂ {Tq (ω) : q ≥ 1}.
We also denote by Qm the set of all q such that Tq = T (m′ , q ′ ) for some q ′ ≥ 1 and some
m′ = 1, · · · , m.
Now we recall a result from [10]. For any q, n we set
κ(n, q) =
1
(Tq − (i − 1)∆n )
∆n
on the set
{Tq ∈ I(n, i)}.
Then we have the following stable convergence in law, where the κp ’s are defined as before
Theorem 3.1 on the extended space:
L−s
(κ(n, q) : q ≥ 1) −→ (κq : q ≥ 1).
2) In this step we prove the result in the special situation where, for some m,
∫ t
∑
Xt = X0 +
b′X
ds
+
∆XTq 1{Tq ≤t} .
s
0
(5.3)
(5.4)
q∈Qm
Letting ΩnT be the set on which |Tq − Tl | > ∆n for all q, l distinct and in Qm and such that
Tq ≤ T , where T > 0 is given, we observe that ΩnT ↑ Ω as n increase to ∞. When ω ∈ ΩnT ,
each interval I(n, i) for i ≤ [T /∆n ] contains at most one Tq , hence (4.5) yields on ΩnT :
∫
∫s
∫
′X
′X
′X

 I(n,i) bs ds (i−1)∆n br dr + ∆XTq I(n,i) bs ds if Tq ∈ I(n, i)
for some q ∈ Qm
(5.5)
ζin =

∫s
∫
′X
′X
otherwise
I(n,i) bs ds (i−1)∆n br dr
We have
∫ s
∫ t
[t/∆n ] ∫
1 ∑
u.c.p. 1
′X
′X
bs ds
br dr =⇒
(b′X )2 ds
∆n
2 0 s
I(n,i)
(i−1)∆n
i=1
(5.6)
CLT for Approximate Quadratic Variation
14
by (4.4). On the other hand, since b′X
t is càdlàg, for any given q and ω we have
∑ 1 ∫
( ′X
)
′X
b′X
ds
−
b
κ(n,
q)
+
b
(1
−
κ(n,
q))
1{Tq ∈I(n,i)} → 0,
Tq −
Tq
∆n I(n,i) s
[T /∆n ] i=1
This, plus (5.6), (5.3), (5.5), and the facts that the number of q ∈ Qm with Tq ≤ T is
finite, whereas ΩnT ↑ Ω, yield
1
∆n
L−s
U n =⇒ U with U given by (3.3).
4) Now we remove the assumption (5.4). The process X satisfies (5.1) and (5.2). For any
m, set
∫t
∑
X(m)t = X0 + 0 b′X
∆XTq 1{Tq ≤t}
s ds +
∑ q∈Qm
Y (m)t = Xt − X(m)t = q∈Q
/ m ∆XTq 1{Tq ≤t} .
The previous step shows that the error process U n (m) associated with X(m) satisfies
L−s
1
∆n
U n (m) =⇒ U (m), where U (m) is given by (3.3), except that the sum is extended over
all p in Qm only. Since by the dominated convergence theorem we have U (m) → U (for
each ω, locally uniformly in time), and using a classical argument (see e.g. [10]), it thus
remains to show that
)
1 (
E sup |Usn − U n (m)s | = 0.
(5.7)
lim lim sup
m→∞
∆n
n
s≤t
We associate ζin (m) with X(m) by (4.5), hence
(
)
∫
ζin − ζin (m) = I(n,i) Y (m)s− − Y (m)(i−1)∆n dXs
(
)
∫
+ I(n,i) X(m)s− − X(m)(i−1)∆n dY (m)s .
Standard estimates using (5.1) and (5.2) give us for any predictable process H:
)
( ∫
(∫
)
∫∞
s
s
E t Hr dY (m)r ≤ E t |Hr | dr m |ϕ(r, z)| dz
)
( ∫
(
)
∫s
s
E t Hr dXr ≤ K t E(|Hr |) dr, E |X(m)s− − X(m)t | ≤ K(s − t).
Plugging this repeatedly in the previous expression gives
∫
(∫
(
)
E |ζin − ζin (m)| ≤ K∆n E
dr
I(n,i)
∞
)
|ϕ(r, z)| dz ,
m
)
(∫ t ∫ ∞
)
1 (
n
n
E sup |Us − U (m)s | ≤ KE
ds
|ϕ(s, z)| dz .
∆n
s≤t
0
m
∫∞
∫∞
∫
Finally, m |ϕ(s, z)| dz is smaller than 0 |ϕ(s, z)| dz = |x| Fs (dx) and goes to 0 by the
dominated convergence theorem, and another application of the same and (5.1) allow to
deduce (5.7). Thus the proof of Theorem 3.1 is complete.
and thus
CLT for Approximate Quadratic Variation
6
15
Some technical tools for Theorem 3.2
We establish a number of technical results, to be used for the proof of Theorem 3.2. By
Lemma 4.1, we can assume the strengthened assumption S2 or S3, hence
•
•
•
•
•
′Y
σ
σ
|∆Z
K
∫ t |β ≤ Y1, |σt | +∫ |bt2| +σ|bt | + |bt | + ct ≤
β
|x| Ft (dx) + x Ft (dx) + supx>0 x Gt (x) ≤ K
−
|xβ G+
(x) − θt+ | + |xβ G−
t (x) − θt | ≤ φ(x)
∫ 1 t+
−
0 |Gt (x) − G∫ (x)t | dx ≤ K in case II
σ
e
σ
e
∥bt ∥ + ∥ct ∥ + ∥x∥2 Ftσe (dx) ≤ K in case II
(6.1)
where φ is increasing continuous on R+ with φ(0) = 0. These conditions will be in force
throughout all the rest of the paper, without special mention, and will not be repeated in
the statements below.
6.1
Moments of the Lévy measures
In this subsection, we give estimates on moments of the measures Gt . They are based on
the following - classical - identities: let H be a (possibly infinite) measure on (0, ∞), such
that H(x) = H((x, ∞)) is finite for all x > 0. Then
∫
∫ b χ−1
∨
χ
(H(y a) − H(b)) dy
{a<x≤b} x H(dx) = χ 0 y
∫
∫b
(6.2)
∨
a) − H(b)) dy
{a<x≤b} (x log x) H(dx) = 0 (1 + log y)(H(y
for 0 ≤ a < b ≤ ∞ and χ > 0. Similar equalities hold on the negative side if H is a
measure on (−∞, 0).
When f is a positive function on some (possibly infinite) interval I and (gj ) and (hj )
are two families of functions on I, indexed by an arbitrary set J (such as J = R+ or
J = N), we write
gj (y) = hj (y) + ou,y→y0 (f (y)) if
1
sup |gj (y) − hj (y)| → 0 as y → y0 .
f (y) j∈J
For χ > 0 and y ∈ [0, 1] we set (recall that
∫1 χ
d+
t (χ, y) = y x Gt (dx),
−
dt (χ, y) = d+
t∫ (χ, y) − dt (χ, y),
y χ
′+
dt (χ, y) = 0 x Gt (dx),
′−
d′+
d′t (χ, y) =
t (χ, y) − dt (χ, y),
∫
1
d′′t (y) = {x:|x|>y} x log |x|
Gt (dx).
Gt only charges [−1, 1]):
∫ −y χ
d−
t (χ, y) = −1 |x| Gt (dx)
−
dt (χ, y) = d+
t∫ (χ, y) + dt (χ, y)
0
′−
χ
dt (χ, y) = −y |x| Gt (dx)
′−
d′t (χ, y) = d′+
t (χ, y) + dt (χ, y)
(6.3)
(6.4)
±
′±
Note that d′±
t (χ, y) may take the value +∞ when χ ≤ β, and dt (χ, 0) = dt (χ, 1) is
±
infinite if χ ≤ β and θt > 0. Then (6.1) and (6.2) yield
χ>β
χ=β
χ<β
χ = β, y ≤ z ≤ 1
β = 1, y ≤ z ≤ 1
⇒
⇒
⇒
⇒
⇒
dt (χ, y) ≤ K,(
d′t (χ, y) ≤ Ky χ−β
)
dt (χ, y) ≤ K 1 + log y1
K
dt (χ, y) ≤ yβ−χ
(
)
±
dt (χ, y) − d±
K 1 + log yz
t (χ, z) ≤
(
)(
)
|d′′t (y) − d′′t (z)| ≤ K 1 + log y1 1 + log yz ,
(6.5)
CLT for Approximate Quadratic Variation
16
as well as the following (uniform) asymptotic behavior:
β θ±
χ−β + o
χ−β
t
χ > β ⇒ d′±
u,y→0 y
t (χ, y) = χ−β y
(
)
±
±
1
χ = β ⇒ dt (χ, y) = βθt log y + ou,y→0 log y1
( 1 )
β θt±
1
χ < β ⇒ d±
t (χ, y) = β−χ y β−χ + ou,y→0 y β−χ
)2
(
)2
θ′ (
β = 1, y ≤ z ≤ 1 ⇒ d′′t (y) = 2t log y1 + ou,y→0 log y1 .
(6.6)
Another application of (6.1) gives us, where →u means convergence as y ↓ 0, uniformly in
t (recall b′t ≡ 0 in case III, under Assumption S2):
)
∫1( +
−
in case II:
dt (1, y) ≤ K, dt (1, y) →u 0 Gt (z) − Gt (z) dz
(6.7)
in case III: dt (1, y) ≤ K, dt (1, y) →u bt .
6.2
Estimates for Z and Y
We start with some estimates for the process Y , which come from (6.1) and also from
(2.1.41) of [10] when χ = 2 below: for any finite (Ft )-stopping time T and any predictable
process H, we have
( ∫
(∫
)
χ )
T +t
T +t
χ = 1, 2 ⇒ E T Hs dYs ≤ K(1 + t) E T |Hs |χ ds
( ∫
(∫
)
(6.8)
)
T +t
′ ≤ χ ≤ 1 ⇒ E T +t H dY χ ≤ KE
χ ds .
b′Y
≡
0,
β
|H
|
s
s
s
t
T
T
Next, we turn to the process Z, which under Assumption S2 is a martingale when
β ≥ 1. For any v ∈ (0, 1] we have the decomposition
A(v)t =
Z = B(v) + M (v) + A(v),
where
{ ∫t
− 0 ds (1, v) ds if β ≥ 1
s≤t ∆Zs 1{|∆Zs |>v} , B(v)t =
0
if β < 1
∑
(6.9)
(so A(1) = B(1) = 0 and M (1) = Z), and we associate the two filtrations (Gtv ) and Htv ,
which are the smallest ones containing (Ft ), and such that:
v
• A(v)
for all t
∑ t is G0 -measurable
• s≤t 1{|∆Zs |>v} is H0v -measurable for all t.
(6.10)
Then Ft ⊂ Htv ⊂ Gtv and the process∑
M (v) is an (Gtv )- martingale when β ≥ 1, and when
′
β < 1 we have bt ≡ 0 and M (v)t = s≤t ∆Zs 1{|∆Zs |≤v} . (6.5) and Proposition 2.1.10 of
[10] yield, for χ ∈ (β, 2] and T a finite (Gtv )-stopping time,
either
1 ≤ β < χ ≤ 2, or β < χ )
≤1 ⇒
(
(∫
)
∫ T +s
χ
T +t
v
E sups≤t T Hr dM (v)r | GT ≤ Kv χ−β E T |Hs |χ ds | GTv .
(6.11)
We also have, by Lemma 2.1.6 of [10], and for any finite (Ft )-stopping time T :
(
)
∫ T +s
χ
E sups≤t T Hr dA(v)r | FT

(∫
)
T +t
χ ds | F

KE
|H
|
if χ > β, β < 1

s
T

 ( T ) (∫
)
(6.12)
T
+t
if χ = β ≤ 1
≤
K log v1 E T |Hs |χ ds | FT

(∫
)


T +t
K
χ ds | F
 β−χ
E
|H
|
if χ < β, χ ≤ 1.
s
T
T
v
CLT for Approximate Quadratic Variation
17
The following properties are classical for processes with independent increments and
no fixed times of discontinuity, hence for non-homogeneous Lévy processes: with S(v)j
denoting the successive jump times of Z with size bigger than v (which are H0v -measurable),
we have
• the times S(v)j form a non-homogeneous Poisson
process with intensity Gt (v)
• conditionally on H0v and S(v)q < ∞, the variables
(6.13)
(∆ZS(v)j )1≤j≤q are independent,
with respective laws G 1 (v) GS(v)j (dx) 1{|x|>v} .
S(v)j
(note that GS(v)j (v) > 0 almost surely on the set {S(v)j < ∞}).
6.3
Estimates for σ and σ
e
Since σ is an Itô semimartingale with bounded spot characteristics, in the sense of (6.1),
for any finite (Ft )-stopping time T and χ ∈ (0, 2], we have
(
)
E
sup |σs − σT |χ | FT ≤ K tχ/2 .
(6.14)
s∈[T,T +t]
Analogously, in case II, and for T and χ as above, we have
(
)
E
sup ∥e
σsj − σ
eTj ∥χ | FT ≤ K tχ/2 .
(6.15)
s∈[T,T +t]
We deduce from (6.11) and (6.14) that
1 (≤ β < χ ≤ 2 ⇒
)
∫ T +s
χ
E sups≤t T (σr− − σT ) dM (v)r | FT ≤ Kv χ−β t1+χ/2
⇒
)
∫ T +s
E sups≤t T (σr− − σT ) dM (v)r | FT ≤ Kv 1−β t3/2 .
β (< 1
(6.16)
We indeed have finer estimates than (6.14), in connection with the decomposition
(6.9). The process σ may jump together with Z, and also at some times where Z is
continuous, but it is always possible to find a “Grigelionis representation” for σ having
the following form: there is a Poisson random measure p on R+ × R+ , with compensator
q(dt, dz) = dt⊗dx and independent of Z (hence p({t}×R) = 0 for all t with ∆Zt ̸= 0), and
also a Brownian motion W , a predictable process σ
e, and two predictable functions ψ and
′
Z
ψ on Ω×R+ ×R, such that (where µ is the jump measure of Z and ν Z (dt, dz) = dt Gt (dz)
is its – non-random – compensator; the formula below is a combination of the “projection”
of σt onto the compensated Poisson measure µZ − ν Z , which gives the last term below,
and of the Grigelionis representation of the remaining part):
∫
∫
t
bσs
σt = σ0 +
0
ds +
0
t
σ
es dWs + ψ ∗ (p − q)t + ψ ′ ∗ (µZ − ν Z )t .
CLT for Approximate Quadratic Variation
18
The spot Lévy measure of σ is Ftσ = Ft′σ + Ft′′σ , where Ft′σ is the restriction to
R\{0}
of the image of Lebesgue measure by the map x 7→ ψ(., t, x) and Ft′′σ (A) =
∫
1A (ψ ′ (., t, x)) Gt (dx). Thus (6.1) yields, when v ∈ (0, 1/2]:
∫ ′
ψ (., t, x)2 Gt (dx) ≤ K
√
√
∫
(6.17)
dt (β,v)
log(1/v)
′
≤
K
,
|ψ
(.,
t,
x)|
G
(dx)
≤
K
t
β
β
{|x|>v}
v
v
the second ∫property following from the Cauchy-Schwarz inequality and (6.5). Moreover, bσ
and σ
e and ψ(., s, x)2 dx are bounded, by (6.1) again. Hence we further get the following
decomposition:
σ = σ(v) + σ ′ (v), where
∫t
∫t
σ(v)t = σ0 + 0 b(v)σs ds + 0 σ
es dWs
+ψ ∗ (p − q)t + (ψ ′ 1{|x|≤v} ) ∗ (µZ − ν Z )t
(6.18)
∑
′
′
σ (v)t = s≤t∫ ψ (s, ∆Zs ) 1{|∆Zs |>v}
b(v)σt = bσt − {|x|>v} ψ ′ (t, x) Gt (dx).
√
Since |b(v)σt | ≤ K log(1/v)/v β by (6.17), and σ(v) is an Itô semimartingale relative to
the filtration (Gtv ), we deduce the following version of (6.14) for this process: If T is a
finite (Gtv )-stopping time and χ ∈ (0, 2], then
(
E
|σ(v)s − σ(v)T |χ | GTv
sup
)
s∈[T,T +t]
6.4
(
( t log(1/v) )χ/2 )
≤ Ktχ/2 1 +
.
vβ
(6.19)
About the jumps of Z - Part I
Below we use the decomposition (6.9), for v = vn a suitably chosen sequence, in connection
with the rate of convergence in our theorem, and this rate is denoted as δn . That is, we
set, with the simplifying notation ln = log(1/∆n ):
Case I :
δn =
Cases II and III : δn =
Case IV :
δn =
1
,
(∆n ln )1/β
1
,
(∆n ln )1/β
1
2 ,
∆n ln
1/2β
v n = ∆n
ln
vn = (∆n ln )1/β
v n = ∆n l n .
(6.20)
Note that in all cases we have
vn → 0,
∆n /vnβ
→ 0,
1
log
∼
vn
{
ln /2β in case I
ln /β in cases II, III, IV.
Therefore, in all the sequel we will assume that n is large enough to have vn ≤
1
∆n /vnβ ≤ 2C
, where C = supt≥0,x>0 xβ Gt (x).
(6.21)
1
2
and
Now we collect some estimates on the “big” (meaning, bigger than vn ) jumps of the
process Z, that is the jumps of the process A(vn ) of (6.9). We set
T (n, i)0 = (i − 1)∆n ,
T (n, i)j = inf(t > T (n, i)j−1 : |∆Zt | > vn )
τ (n, i) = sup(j ≥ 0 : T (n, i)j ≤ i∆n ).
(6.22)
CLT for Approximate Quadratic Variation
19
We will use very often the notation, with s ≤ t:
( ∫
)
n = exp − t G (v ) dr ,
αs,t
Dn = {t : Gt (vn ) > 0}
r n
s
Dn = {(s, t) : s < t, s ∈ Dn , t ∈ Dn }.
(6.23)
Observe that, by (6.1),
n
≤K
s ≤ t ≤ s + ∆n ⇒ 0 ≤ 1 − αs,t
∆n
vnβ
→ 0.
(6.24)
For the next lemma, we recall that T (n, i)j ∈ Dn almost surely on the set {T (n, i)j <
∞}.
Lemma 6.1. Let χ > 0 and y > 0. Then on the set {T (n, i)j < ∞} we have
 K yχ−β

if χ > β


 GT (n,i)j (vn )
(
)
K ln
if χ = β
E (|∆ZT (n,i)j | ∧ y)χ | HTvn(n,i)j − ≤
GT (n,i)j (vn )


K


if χ < β.
(6.25)
β−χ
vn
GT (n,i)j (vn )
(
E ∆ZT (n,i) 1{|∆Z
j
T (n,i)
|≤y}
j
)
| HTvn(n,i)j − ≤
(
)
E |∆σT (n,i)j ∆ZT (n,i)j |χ | HTvn(n,i)j − ≤
K
GT (n,i)j (vn )
K
GT (n,i)j (vn )
in case II
(6.26)
2β
.
1+β
(6.27)
if χ >
Proof. The key point is the following consequence of (6.13): for any bounded Borel function
g on R we have, on the set {T (n, i)j < ∞}:
∫
(
)
1
vn
g(z) GT (n,i)j (dz).
E g(∆ZT (n,i)j ) | HT (n,i)j − =
(6.28)
GT (n,i)j (vn ) {|x|>vn }
This extends to any bounded function g on Ω × R which is HTvn(n,i)j − ⊗ R-measurable.
Applying this to g(z) = (|z| ∧ y)χ , we get with the notation (6.4):
∫
χ
{|x|>vn{
} (|x| ∧ y) Gt (dx)
(6.29)
dt (χ, vn ) − dt (χ, vn ∨ y) + y χ Gt (vn ∨ y) if (χ ≤ β)
=
d′t (χ, vn ∨ y) − d′t (χ, vn ) + y χ Gt (vn ∨ y) if (χ > β).
Then (6.25) follows from Gt (x) ≤ K/xβ , (6.5) and log(1/vn ) ≤ Kln .
Next, (6.28) with g(x) = x 1{|x|≤y} yields (recall that Gt ([−1, 1]c ) = 0)
∫
x Gt (dz) = |dt (1, vn ) − dt (1, y ∨ vn )| ≤ K
{vn <|x|≤y}
′
χ
in case II, by (6.7). This yields (6.26). Finally we apply
∫ (6.28) with g(ω, x) = |ψ (ω, T (n, i)j (ω), x) x|
and deduce from (6.17) and Hölder’s inequality that {|x|>vn } g(., x) GT (n,i)j (dx) ≤ K when
χ>
2β
2+β ,
and (6.27) follows.
CLT for Approximate Quadratic Variation
20
Lemma 6.2. Recalling C = supt≥0,x>0 xβ Gt (x), for all 1 ≤ j ≤ k ≤ m and y, χ > 0 we
have on the set {τ (n, i) ≥ j − 1}:
(
)m−j+1
P(τ (n, i) ≥ m | FT (n,i)j−1 ) ≤ C∆n /vnβ
(
)
E (|∆ZT (n,i)k | ∧ y)χ 1{τ (n,i)≥m} | FT (n,i)j−1

)m−j χ−β
(
y
if χ > β
K∆n C∆β n


vn )

( C∆
m−j
K∆n β n
ln
if χ = β
≤
vn )

(

m−j
χ−β
 K∆ C∆n
v
if χ < β
n
(6.30)
(6.31)
n
β
vn
(
)
E ∆ZT (n,i) 1{|∆Z
≤ K∆n in case II
|
F
|≤y,
τ
(n,i)≥j}
T
(n,i)
j
j−1
T (n,i)j
(
)
E |∆σT (n,i)k ∆ZT (n,i)k |χ 1{τ (n,i)≥m} | FT (n,i)j−1
(
)m−j
2β
≤ K∆n C∆β n
if χ > 1+β
.
(6.32)
(6.33)
vn
Proof. (6.13) yields that τ (n, i) follows a Poisson law with parameter smaller than C∆n /vnβ ≤
1
2 . Then (6.30) readily follows.
Let Yn be nonnegative random variables such that
E(Yn | HTvn(n,i)k − ) ≤
a
GT (n,i)k (vn )
(6.34)
for some constant a. A combination of (6.28) and, recalling (6.13), of a classical property
of non-homogeneous Poisson process, yields that on the set {τ (n, i) ≥ j − 1} and with
sj−1 = T (n, i)j−1 we have:
(
)
(
)
E Yn 1{τ (n,i)≥m} | FT (n,i)j−1 ≤ E G a (v ) 1{τ (n,i)≥m} | FT (n,i)j−1
T (n,i)k n
∫
∏
∏
n
= a {sj−1 <sj <···<sm <i∆n } m
l=j αsl−1 ,sl
j≤l≤m, l̸=k Gsl (vn ) dsj · · · dsm
(
)m−j
≤ a∆n C∆β n
,
vn
n ≤ 1. In view of (6.25) and (6.27), we deduce (6.31) and (6.33).
because αs,t
Finally, if Yn are random variables such that, instead of (6.34), |E(Yn | HTvn(n,i)k − )| ≤
a
, and when m = k = j, we can repeat the same argument to deduce (6.32) from
(v )
GT (n,i)k
n
(6.26).
Recall that |∆Zt | ≤ 1. Then (6.31) applied twice with y = 1 and a successive conditioning procedure yields, for all 1 ≤ j ≤ k < r ≤ m and on the set {τ (n, i) ≥ j − 1},
(
)
E |∆ZT (n,i)k ∆ZT (n,i)r |χ 1{ τ (n,i)≥m} | FT (n,i)j−1

(
)m−j−1

K∆2n C∆β n
if χ > β

vn


(
)
(6.35)
m−j−1
K∆2n ln2 C∆β n
if χ = β
≤
v
n



 K∆2 vn2(χ−β) ( C∆n )m−j−1 if χ < β
β
n
vn
CLT for Approximate Quadratic Variation
21
Lemma 6.3. For all 1 ≤ j ≤ k < r ≤ m and y > 0 we have on the set {τ (n, i) ≥ j − 1}:
)
(
χ
E (|∆ZT (n,i)k ∆ZT (n,i)
| FT (n,i)j−1
r | ∧ y) 1({τ (n,i)≥m}
)m−j−1


K∆2n ln2 C∆β n
if χ = β
(6.36)
vn (
)m−j−1
≤

C∆
2
χ−β
n
 K∆n ln y
if χ > β.
β
vn
Proof. As in the previous proof, and if sj−1 = T (n, i)j−1 , the left side of (6.36) in restriction
to the set {τ (n, i) ≥ j − 1} is equal to
(∏
∫
∏
m
n
l=j αsl−1 ,sl
j≤l≤m, l̸=k,r Gsl (vn )
{sj−1 <sj <···<sm <i∆n }
)
∫
∫
χ G (dz) ds · · · ds ,
(dx)
(|xz|
∧
y)
G
s
j
m
s
r
k
{|z|>vn }
{|x|>vn }
(
)m−j−1
n and
which is smaller than an ∆2n C∆β n
, where an = sups,t γs,t
vn
∫
∫
n
γs,t
= {|x|>vn } Gs (dx) {|z|>vn } (|xz| ∧ y)χ Gt (dz)
(
(
(
))
(y
))
∫
χ d (χ, v ) − d χ, y ∨ v
χG
=
G
(dx)
|x|
+
y
∨
v
s
t
n
t
n
t |x|
n
{|x|>v
|x|
{n }
∫
β
2
Kln {|x|>vn } |x| Gs (dx)
≤ Kln
if χ = β
∫
≤
χ−β
β
χ−β
Ky
if χ > β,
{|x|>vn } |x| Gs (dx) ≤ Kln y
where we have used (6.1) and twice (6.5). This yields (6.36).
Next, we give some estimates for the same jumps of Z as above, together with the
increments of the process σ.
Lemma 6.4. On the set {τ (n, i) ≥ j − 1}, we have for all y > 0 and if χ ≥ β:
((
)χ
E |(σT (n,i)j − − σT (n,i)j−1 )∆ZT (n,i)j | ∧ y 1{τ (n,i)≥j} | FT (n,i)j−1 )
1+β/2
≤ K∆n
ln y χ−β .
(6.37)
Proof. For simpler notation we write Φn (s, t) = supr∈(s,t] |σ(vn )r − σ(vn )s | and also τ =
τ (n, i) and Tk = T (n, i)k . We have
(
)χ
|(σTj − − σTj−1 )∆ZTj | ∧ y (1{τ ≥j}
)χ
y
1{τ ≥j} .
≤ ηn := Φn (Tj−1 , Tj )χ |∆ZT2 | ∧ Φn (Tj−1
,Tj )
Observing that Φn (Tj−1 , Tj ) and τ are HTvnj − -measurable (note that σ(vn )Tj = σ(vn )Tj − ),
we deduce from (6.25) yield
E(ηn | HTvnj − ) ≤ Kln
Φn (Tj−1 , Tj )β y χ−β
1{τ ≥j} .
GTj (vn )
Next, Tj is HTvnj−1 -measurable, so (6.19) and ∆n log(1/vn )/vnβ ≤ K that
E(ηn | HTvnj−1 ) ≤ Kln ∆β/2
n
y χ−β
1{τ ≥j} .
GTj (vn )
Then arguing as in the proof of Lemma 6.2, with k = m = j, yields (6.37).
CLT for Approximate Quadratic Variation
6.5
22
About the jumps of Z - Part II
This subsection is devoted to determining the precise asymptotic behavior of the jumps
of Z with size bigger than vn , as n → ∞ and thus vn → 0. We use the notation of the
previous subsection, and add some more: below, ε and ε′ are two numbers taking the
′
values ±1; we write θtε for θt+ when ε = 1 and for θt− when ε = −1, and also θtεε is, for
example, θt− , when εε′ = −1.
In the next lemma, when (s, t) ∈ Dn (recall (6.23)), we let Vsn and Vtn be two inde1
1
G (dx)1{|x|>vn } and G (v
G (dx)1{|x|>vn } .
pendent variables with respective laws G (v
) s
) t
s
n
t
n
Note that Dn increases as n increases. We then introduce the following functions on Dn ,
where w > 0 and ε, ε′ are arbitrary:
n n
n
′ n
fw,ε,ε
′ (s, t) = Gs (vn ) Gt (vn ) P(εδn Vs Vt > w, ε Vs > 0)
)
((
)
2
fw′n (s, t) = Gs (vn ) Gt (vn ) E δn Vsn Vtn 1{|δn Vsn Vtn |≤w}
)
(
fw′′n (s, t) = Gs (vn ) Gt (vn ) E δn Vsn Vtn 1{|δn Vsn Vtn |≤w}
(6.38)
Lemma 6.5. In the previous setting, we have as n → ∞:
′
′
f n ′ (s, t)
β θsε θtεε w,ε,ε
sup β
−
→0
wβ
(s,t)∈Dn δn log(1/vn )
sup fw′n (s, t)
−
in cases II and III
β 2 θs θt w2−β →0
2−β
in cases II and III
δnβ log(1/vn )
f ′′n (s, t)
β 2 θs′ θt′ w1−β sup β w
−
→ 0 in case III
1−β
(s,t)∈Dn δn log(1/vn )
f ′′n (s, t) sup β w
→ 0 in case II
(s,t)∈Dn δn log(1/vn )
fw′′n (s, t)
θs′ θt′ sup −
→ 0 in case IV.
2
2
(s,t)∈D n δn (log(1/vn ))
(s,t)∈D n
(6.39)
(6.40)
(6.41)
(6.42)
(6.43)
Proof. a) Note that Vsn ̸= 0 a.s. because we take s ∈ Dn . First we prove (6.39) when, say,
ε = 1 and ε′ = −1, the other cases being similar. We have
an (s, t) :=
n
fw,ε,ε
′ (s, t)
δnβ log(1/vn )
=
Gs (vn )
δnβ log(1/vn )
( ( ∨
E G−
t vn
)
w )
n
1
.
δn |Vsn | {Vs <0}
With the function φ appearing in (6.1), we have when Vsn < 0:
−(
Gt
w ) θt− δnβ |Vsn |β δnβ |Vsn |β ( w )
−
φ
.
≤
δn |Vsn |
wβ
wβ
δn |Vsn |
Therefore, recalling δn vn = 1 in cases II and III,
an (s, t) −
( nβ
)
θt− Gs (vn )
n
E
|V
|
1
≤ a′n (s, t) + a′′n (s),
{V
<−w}
s
s
wβ log(1/vn )
CLT for Approximate Quadratic Variation
where
23
Gs (vn ) G−
t (vn )
P(Vsn < −w)
β
δn
log(1/vn )(
)
( w )
Gs (vn )
n |β φ
n <0} .
E
|V
1
n
β
{V
s
δn |Vs |
s
w log(1/vn )
a′n (s, t) =
a′′n (s) =
(
)
−
We have Gs (vn ) E |Vsn |β 1{Vsn <−w} = d−
s (β, vn )−ds (β, w ∨vn ), which by (6.6) is equal
−
to βθs log(1/vn )+ ou(s),n→∞ (log(1/vn )). We are thus left to prove that a′n and a′′n go to 0
uniformly in s, t ∈ Dn .
We have Gs (vn ) P(Vsn < −w) ≤ G−
s (w), hence (6.1) yields
a′n (s, t) ≤ K/(wvn δn )β log(1/vn ) = K/wβ log(1/vn ),
implying the result for a′n . Next, since φ is increasing and may be chosen bounded, for
any ρ ∈ (0, w) we have
(
)
( w )
(
)
n
E |Vsn |β φ δn |V
1
≤ φ(ρ) E |Vsn |β 1{Vsn <−w/ρδn }
n|
{V
<0}
s
s
( nβ
)
+K
E
|V
|
1
{−w/ρδ
≤V
<0}
s
n
n
(
( w )
1
≤ G (v
φ(ρ)d−
s β, ρδn
s n)
)))
(
( w
−
+K d−
s (β, vn ) − ds β, ρδn ∨ vn
)
(
ρδn
w
≤ GK
+
log
1
+
φ(ρ)
log
w
ρ
(v )
s
n
√
by (6.5) and vn δn = 1. Taking ρ = ρn = e− ln , we see that ρn → 0 and log ρ1n / log v1n → 0.
Then the previous estimate and the definition of a′′n , plus the fact that w is fixed, yield
(
)
1
K
1
+
φ(ρ
)
log(1/v
)
+
log
a′′n (s, t) ≤ log(1/v
n
n
ρ
n
( n)
)
n)
≤ K φ(ρn ) + 1+log(1/ρ
.
log(1/vn )
We deduce a′′n (s, t) → 0 uniformly in s, t ∈ Dn , and the proof of (6.39) is complete.
b) The proof of (6.40) is similar. We have
an (s, t) :=
=
′n (s,t)
fw
log(1/vn ) (
( (
2−β
∨ w )
Gs (vn ) δn
n )2 d′ 2, v
E
(V
n
s
t
log(1/vn )
δn |Vsn |
β
δn
))
− d′t (2, vn ) .
1 ′
β θr decreases to 0 as x ↓ 0 by (6.6),
The function γ(x) = sup0≤y≤x,r≥0 y2−β
dr (2, y) − 2−β
hence
(
( w )
β θt w2−β
w2−β
w )
′
−
≤
γ
.
dt 2,
n
n
2−β
n
2−β
δn |Vs |
(2 − β)(δn |Vs |)
(δn |Vs |)
δn |Vsn |
w
s
We also have d′s (2, vn ) ≤ Kvn2−β , and vn ≤ δn |V
n is the same as |Vn | ≤ w because δn vn = 1.
s |
Hence
)
β θt w2−β Gs (vn ) ( n β
′′
′
n |≤w} ≤ an (s) + an (s),
where
|
1
E
|V
an (s, t) − (2−β)
{|V
s
log(1/vn )
s
(
)
(
(
))
KG
KG
(v
)
(v
)
s n
s n
w
′′ (s) =
n )2 ,
n |β γ
a′n (s) = log(1/v
a
E
(V
E
|V
.
n
n
s
s
log(1/vn )
δn |V |
n)
s
CLT for Approximate Quadratic Variation
24
)
(
Exactly as in the previous step, Gs (vn ) E |Vsn |β 1{|Vsn |≤w} is equivalent to βθs log(1/vn ),
uniformly in s, hence it remains to prove that a′n → 0 and a′′n → 0 uniformly in s ∈ Dn .
By (6.5) we have a′n (s) ≤ K/ log(1/vn ), implying the result for a′n . On the other hand, a′′n
here is the same as in (a), upon substituting φ with Kwβ γ. Therefore sups∈Dn a′′n (s) → 0.
c) Now we prove (6.41) and (6.42) at the same time: with the convention
θt′
1−β
0
0
= 0,
according to which
= 0 in case II, the second convergence is indeed the same as the
first one. We again have β ≤ 1 and δn vn = 1, and now
( (
1−β
(
′′n
∨ w )))
s (vn ) δn
n d (1, v ) − d 1, v
.
an (s, t) := βfw (s,t) = Glog(1/v
E
V
t
n
t
n
s
δn |V n |
n)
δn log(1/vn )
s
By (6.7) the functions dt (1, y) converge to a limit bbt as y → 0, uniformly in t; in case III
we have dt (1, y) = bbt − d′t (1, y) (recall bbt = bt in this case), whereas θt′ = 0 in case II. Thus
b r (1,y)
′ − β θr decreases to 0 as x ↓ 0
in both cases the function γ(x) = sup0≤y≤x,r≥0 br −d1−β
1−β
y
by (6.6), and we have
(
)
b
w
bt − dt 1, δn |V
n| −
s
β θt′ w1−β
(1−β)( δn |Vsn |)1−β ≤
w1−β
(δn |Vsn |)1−β
γ
(
w
δn |Vn |
)
.
Therefore, since |bbt − dt (1, vn )| ≤ Kvn1−β in both cases, and as in step (b),
)
β θt′ w1−β Gs (vn ) ( n β
E |Vs | sign(Vsn ) 1{|Vsn |≤w} ≤ a′n (s) + a′′n (s),
an (s, t) −
(1 − β) log(1/vn )
where
a′n (s) =
a′′n (s) =
KGs (vn )
log(1/vn )
KGs (vn )
log(1/vn )
( n
)
E Vs 1{|V n |≤w} s
(
( w ))
E |Vsn |β γ δn |V
.
n|
s
(
)
We have Gs (vn ) E Vsn 1{|Vsn |≤w} = ds (1, vn )−ds (1, w∨vn ), which is bounded uniformly
in (n, s) in both cases by (6.7), so sups∈Dn a′n (s) → 0. Since a′′n (s)(is as in the previous steps)
we have sups∈Dn a′′n (s) → 0. In case III (6.6) yields that Gs (vn ) E |Vsn |β sign(Vn ) 1{|Vsn |≤w}
is equivalent to βθs′ log(1/vn ), uniformly in s ∈ Dn . Recalling
obtain both (6.41) and (6.42).
θt′
1−β
= 0 in case II, we thus
d) It remains to prove (6.43), so β = 1 and δn vn = 1/ln . Since |Vsn | ≤ 1, we then have
vn < w/δn |Vsn | as soon as ln ≥ 1/w. In this case we have
an (s, t) :=
=
′′n (s,t)
fw
β
δn
(log(1/vn ))2 (
(
Gs (vn )
n d (1, v )
E
V
t
n
2
s
(log(1/vn ))
))
)
(
w
.
− dt 1, δn |V
n | 1{|Vsn |>w/δn }
s
Thus an = cn − c′n , where, since Gs (vn ) E(Vsn ) = ds (1, vn ),
cn (s, t) =
c′n (s, t) =
ds (1,vn ) dt (1,vn )
(log(1/vn ))2 (
)
(
)
Gs (vn )
w
n d 1,
n
E
V
1
{|Vs |>w/δn } .
s t
δn |Vsn |
(log(1/vn ))2
CLT for Approximate Quadratic Variation
25
1
The function γ(x) = sup0≤y≤x,r≥0 log(1/y)
dr (1, y) − θr′ is bounded and decreases to 0
as x ↓ 0 by (6.6). Thus, if |Vsn | > w/δn , and since |Vsn | ≤ 1, we have
(
( w )
)
n
δn |Vsn |
w
′ log δn |Vs | ≤ Kγ
d
1,
−
θ
t
t
w w
δn |Vsn |
δn |Vsn | log
( w )
δn
≤ Kγ δn |V n | log w .
s
Hence c′n = a′n + a′′n , where, with the notation d′′t (y) of (6.4),
)
ds (1, w/δn ) log(δn /w) − d′′s (w/δn )
(
( w ))
s (vn ) log(δn /w)
n| γ
|a′′n (s, t)| ≤ KG(log(1/v
.
E
|V
2
s
δn |V n |
n ))
a′n (s, t) =
θt′
(log(1/vn ))2
(
s
n /w)
′′
Since log(δ
log(1/vn ) is bounded by a constant (depending on w), an (s, t) here is smaller than
′′
′′
Kan (s) where an (s) is like in Steps (b) or (c) (with β = 1), then sups,t a′′n (s, t) → 0.
Finally, in view of (6.5) and (6.20), we easily see that cn (s, t) → θs′ θt′ and an (s, t) →
uniformly in s, t. Since an = cn − a′n − a′′n , (6.43) follows.
θs′ θt′ /2
6.6
Stable convergence in law
In this subsection we provide some useful results about the convergence of triangular
arrays. First, we have several criteria for “asymptotic negligibility” of a triangular array
of random variables (the first one is nearly trivial, the other two ones, well known, follow
for example from Theorem IX.7.19 of [7]):
∑[t/∆ ]
u.c.p.
Lemma 6.6. Let ξin be Fi∆n -measurable variables. Then i=1 n ξin =⇒ 0 under either
one of the following sets of conditions (for all t)
∑
[t/∆n ]
E(|ξin | ∧ 1) → 0
(6.44)
i=1
∑
[t/∆n ]
∑
[t/∆n ]
u.c.p.
E(ξin | F(i−1)∆n ) =⇒ 0
i=1
E((ξin )2 ) → 0.
(6.45)
i=1

) u.c.p.
∑[t/∆n ] ( n

n |≤1} | F(i−1)∆
E
ξ
1
=⇒
0
{|ξ
i
n
i=1
i
)
∑[t/∆n ] ( n 2
∑[t/∆n ]
E (ξi ) 1{|ξin |≤1} → 0,
P(|ξin | > 1) → 0. 
i=1
i=1
(6.46)
u.c.p.
Second, and still writing =⇒ for the local (in time) uniform convergence although no
randomness is involved, we have the following (since supi≤[t/∆n ] |ani | → 0 is implied by
(6.47), it readily follows by taking the logarithms in (6.48)):
Lemma 6.7. Let ani be complex numbers, satisfying
∑
[t/∆n ]
i=1
u.c.p.
ani =⇒ At ,
(6.47)
CLT for Approximate Quadratic Variation
26
where A is a complex-valued continuous function on R+ . Then we have
∏
[t/∆n ]
u.c.p.
(1 + ani ) =⇒ eAt .
(6.48)
i=i
Next, we recall some criteria for the convergence in law of rowwise independent trian∑[t/∆ ]
gular arrays. Let Γn = i=1 n ξin , where for each n the d-dimensional variables (ξin )i≥1
are i.i.d. The putative limit Γ is a non-homogeneous Lévy process with no Gaussian part
and such that none of its components has any jump of absolute size equal to 1 (these two
properties will be satisfied in the applications below, but of course similar criteria can be
given in a totally general situation).
We denote by (bΓt , 0, FtΓ ) the spot characteristics of Γ, where to compute bΓ we use the
truncation function with jth component equal to y 7→ y j 1{|yj |≤1} . We also denote by A an
arbitrary convergence-determining
family of Borel subsets A of Rd , at a positive distance
∫∞ Γ
of 0 and satisfying 0 Ft (∂A) dt = 0. The criteria make use of one or several conditions:
∏[t/∆n ] ( iu∗ ξn ) u.c.p.
k
∀u ∈ R(d ,
=⇒
k=1 (E e
)
)
∫t
∫
∑
∗
exp 0 iu∗ bΓs +
eiu y − 1 − i dj=1 uj y j 1{|yj |≤1} FsΓ (dy) ds
∑
[t/∆n ]
j≤d ⇒
∑
[t/∆n ]
j ≤ d ⇒ lim lim sup
n
∑
(6.50)
0
(
)
E |ξin,j |2 1{|ξn,j |≤ε} = 0
(6.51)
i
i=1
∫
[t/∆n ]
A∈A ⇒
t
bΓ,j
s ds
i
i=1
ε→0
∫
(
) u.c.p.
E ξin,j 1{|ξn,j |≤1} =⇒
(6.49)
P(ξin ∈ A) →
t
FsΓ (A) ds
(6.52)
0
i=1
L
Lemma 6.8. The convergence in law Γn =⇒ Γ is equivalent to each one of the following
two conditions:
a) We have (6.49).
b) We have (6.50), and also (6.51) and (6.52) for all t ≥ 0.
Proof. The equivalence with (a) is Corollary VII.4.43 of [7]. The equivalence with (b)
follows from Theorem VII.3.4 in the same book, upon noticing the following, with the
notation of that theorem: (6.50) is [Sup-β3 ] and (6.52) for all t is [δ3,1 − R+ ]. The latter
)
)
∫t(∫
∑[t/∆ ] (
implies i=1 n E h(ξin,j )2 1{|ξn,j |>ε} → 0 {|yj |>ε} h(y j )2 FsΓ (dy) ds (where h is a coni
tinuous truncation function on R), for Lebesgue-almost all
under
) (6.50) and
∫ tε(>
∫ 0. Hence
jj
j
2
Γ
e
(6.52), we see that (6.51) for all t is [γ3 − R+ ] with Ct = 0
h(y ) Fs (dy) ds. Finally,
this for all j amounts to saying that the Gaussian part of Γ vanishes.
Finally we turn to the stable convergence in law. We have our non-homogeneous Lévy
process Z, and a rowwise independent array (ξin ) of 1-dimensional variables. We suppose
CLT for Approximate Quadratic Variation
27
Z -measurable, where (F Z )
that each ξin is Fi∆
t t≥0 is the filtration generated by the process
n
Z. Finally we consider a bounded càdlàg process H, adapted to the filtration (Ft ) (not
necessarily (FtZ )-adapted). We set
∑
[t/∆n ]
Γ′n
t
=
∑
[t/∆n ]
ξin ,
Γnt
=
i=1
H(i−1)∆n ξin .
(6.53)
i=1
We also consider another non-homogeneous Lévy process Γ′ , defined on an extension
e of the original space (Ω, F, (Ft )t≥0 , P) and independent of F. Below, we
e F,
e (Fet )t≥0 , P)
(Ω,
∫t
consider the stochastic integral process Γt = 0 Hs− dΓ′s . The discretized version of Z at
∑[t/∆ ]
(n)
stage n is denoted as Z (n) , and given by Zt = Z∆n [t/∆n ] = i=1 n ∆ni Z. Then we have:
Lemma 6.9. With the previous notation, and if we have the joint convergence in law
L
(Z (n) , Γ′n ) =⇒ (Z, Γ′ ),
(6.54)
∫t
L−s
we also have the stable convergence in law Γn =⇒ Γ. When further Γ′t = 0 as ds is a
pure drift (here at is a non-random function), we can replace (6.54) by the local uniform
u.c.p.
u.c.p.
convergence in probability Γ′n =⇒ Γ′ , and we obtain Γn =⇒ Γ.
Proof. 1) The second claim is a particular case of the first one: on the one hand, we
have Z (n) → Z (pathwise) for the Skorokhod topology, whereas Γ′t is continuous, so if
L−s
Γ′n =⇒ Γ′ we do have (6.54). On the other hand, if Γn =⇒ Γ with Γ being continuous
u.c.p.
and F-measurable, then Γn =⇒ Γ.
u.c.p.
2) Now we start proving the first claim. We can realize Γ′ on an auxiliary space
e of (Ω, F, (Ft )t≥0 , P) defined
e F,
e (Fet )t≥0 , P)
(Ω′ , F ′ , P′ ), and use the very good extension (Ω,
by
e = P ⊗ P′
e = Ω × Ω′ , Fe = F ⊗ F ′ , P
Ω
e with Ft ⊂ Fet and Γ′ adapted to (Fet ).
(Fet ) = the smallest filtration on Ω
We can rewrite (6.54) as
(
)
(
)
e f (Z, Γ′ )
E f (Z (n) , Γ′n ) → E
(6.55)
for any bounded continuous function f on the Skorokhod space D(R+ , R2 ).
L−s
We begin by proving that (6.55) implies the stable convergence in law Γ′n =⇒ Γ′ ,
which amounts to have
(
)
(
)
e Φ g(Γ′ )
E Φ g(Γ′n ) → E
(6.56)
for any bounded measurable Φ on (Ω, F, P) and any (bounded )Lipschitz
( ′function
) g on
′n
′n
D(R+ , R). First, if H = σ(Zs , s ≥ 0), we have E Φ f (Γ )) = E Φ f (Γ )) and
(
)
(
)
e Φ g(Γ′ ) = E
e Φ′ g(Γ′ ) , where Φ′ = E(Φ | H). In other words, it is enough to prove
E
(6.56) when Φ is H-measurable.
Next, if (6.56) holds for each member Φq of a sequence of variables converging in L1 (P)
to a bounded limit Φ, then it also holds for Φ. Hence it is enough to prove (6.56) for Φ
CLT for Approximate Quadratic Variation
28
in a dense subset of L1 (Ω, H, P), for example the set of Φ = h(Z) where h ranges through
all bounded Lipschitz functions on D(R+ , R).
One last reduction can be done. Since Z (n) → Z for the Skorokhod topology, if h is
as above, then |h(Z (n) ) − h(Z)| ≤ Kζn where ζn is the Skorokhod distance between Z (n)
and Z. Hence for any ε > 0 we have
(
)
(
)
E h(Z) g(Γ′n ) − E h(Z (n) ) g(Γ′n ) ≤ Kε + KP(ζn > ε)
(recall that h and g are bounded), with K not depending on ε. It remains to apply (6.55)
to f (y, z) = h(y)g(z), and we deduce
(
)
(
)
e Φ g(Γ′ ) ≤ Kε,
lim sup E h(Z) g(Γ′n ) − E
n
because P(ζn > ε) → 0. Since ε is arbitrarily small, we deduce (6.56) for Φ = h(Z), hence
for any bounded F-measurable Φ.
L−s
L−s
3) Now we deduce from Γ′n =⇒ Γ′ that Γn =⇒ Γ. Consider the discretized processes
(n)
Ht = H∆n [t/∆n ] . The stable convergence in law (6.56) implies that the pair (H, Γ′n ) converges stably in law as well, for the product Skorokhod topology on D(R+ , R) × D(R+ , R),
and the same is true for (H (n) , Γ′n ) (arguing as in Step 2). Since the limits H and Γ′ have
no common jumps (because Γ′ is independent of H and with no fixed times of discontinuL−s
ity), it follows that indeed (H (n) , Γ′n ) =⇒ (H, Γ′ ), but now for the Skorokhod topology on
D(R+ , R2 ) (this follows from Propositions VI.2.1-(a) and VI.2.2-(b) of [7], for example).
Take Φ nonnegative with expectation 1, and call Q the measure with Radon-Nikodym
L
derivative Φ with respect to P. What precedes gives us (H (n) , Γ′n ) =⇒ (H, Γ′ ), under
the measure Q. Since Γ′n and Γ′ have independent increments and Γ′ has no fixed times
of discontinuity, by combining Theorems VI.6.21 and VII.3.4 of [7] we obtain that the
L
convergence in law Γ′n =⇒ Γ′ implies that the sequence Γ′n satisfies the so-called P-UT
property. By Theorem VI.6.22 of the same reference, this property and the convergence in
L
(n)
law (H (n) , Γ′n ) =⇒ (H, Γ′ ) (under Q) imply that the stochastic integral processes H− •
(n)
Γ′n = Γn of H− converge in law (under Q again) to the integral process H− • Γ′ = Γ.
(
)
(
)
e Φ g(Γ) for any continuous bounded function g, and for the
Therefore E Φ g(Γn ) → E
variable Φ associated with Q, that is in fact for any nonnegative Φ with E(Φ) = 1. By
linearity, this extends to any bounded Φ, and the claim is proved.
7
7.1
Proof of Theorem 3.2
Elimination of Y
As said before, we assume (6.1) and (4.1). The theorem would of course be much easier
to prove, if X = Z were itself a non-homogeneous Lévy process, and in the next
∫ t two subsections we proceed to reduce the general problem to this case. Letting Xt′ = 0 σs− dZs ,
so X = X ′ + Y , we first eliminate the process Y .
CLT for Approximate Quadratic Variation
29
We have seen that the key formula for evaluating the error process U n is (4.5). For X ′
it becomes
∫
[t/∆n ]
∑
′n
′n
′n
′
′
Ut = 2
ζi ,
ζi =
(Xs−
− X(i−1)∆
) dXs′ .
(7.1)
n
I(n,i)
i=1
The next proposition shows that the problem reduces to the case X = X ′ :
Proposition 7.1. We have δn (U n − U ′n ) =⇒ 0.
u.c.p.
Proof. 1) We begin with some preliminaries. We have δn (ζin − ζi′n ) =
∑3
n
j=1 ηi (j),
where
∫
ηin (1) = δn I(n,i) (Ys− − Y(i−1)∆n ) dYs
∫
ηin (2) = δn I(n,i) (Ys− − Y(i−1)∆n ) dXs′
∫
′ − X′
ηin (3) = δn I(n,i) (Xs−
(i−1)∆n ) dYs .
Therefore it is enough to prove that, for j = 1, 2, 3, we have
∑[t/∆n ]
i=1
u.c.p.
ηin (j) =⇒ 0.
2) Applying (6.8) twice (once with Ht = Yt− − Y(i−1)∆n for t ≥ (i − 1)∆n , once with
Ht = 1), we get for χ = 1 in cases I, II and IV and χ = β ′ in case III that
E(|ηin (1)|χ ) ≤ Kδnχ ∆2n .
In all cases δnχ ∆n → 0, so we have
∑[t/∆n ]
i=1
u.c.p.
ηin (1) =⇒ 0 by (6.44).
3) We treat the cases j = 2 and j = 3 simultaneously. We use (6.9) with v = wn =
and write ηin (j) = ξin (j) + ξi′n (j) + ξi′′n (j) for j = 3, 4, where
∫
ξin (2) = δn I(n,i) (Ys− − Y(i−1)∆n )σs− dB(wn )s
(∫
)
∫
s
ξin (3) = δn I(n,i) (i−1)∆n σr− dB(wn )r dYs
∫
ξi′n (2) = δn I(n,i) (Ys− − Y(i−1)∆n )σs− dM (wn )s
(∫
)
∫
s
ξi′n (3) = δn I(n,i) (i−1)∆n σr− dM (wn )r dYs
∫
ξi′′n (2) = δn I(n,i) (Ys− − Y(i−1)∆n )σs− dA(wn )s
(∫
)
∫
s
ξi′′n (3) = δn I(n,i) (i−1)∆n σr− dA(wn )r dYs .
1/β
∆n ,
∑[t/∆ ]
∑[t/∆ ]
∑[t/∆ ]
u.c.p.
u.c.p.
u.c.p.
It is then enough to prove that i=1 n ξin (j) =⇒ 0, i=1 n ξi′n (j) =⇒ 0 and i=1 n ξi′′n (j) =⇒
0 for j = 2, 3, and before starting the proof we recall the following properties:

Kwn1−β in case I


∫ t

Kln
in case IV
B(wn )t =
b(wn )s ds with |b(wn )t | ≤
(7.2)

K
in case II
0


0
in case III,
4) By (7.2) the numbers E(|ξin (j)|) vanish in case III, and by (6.8) they are smaller
than Kδn ∆2n wn1−β in case I, than Kδn ∆2n ln in case IV, and than Kδn ∆2n in cases II. So in
∑[t/∆ ]
u.c.p.
all cases the sequences (ξin (j)) satisfy (6.44), hence i=1 n ξin (j) =⇒ 0, for j = 2, 3.
CLT for Approximate Quadratic Variation
30
In cases I, II, IV we have E(ξi′n (2) | F(i−1)∆n ) = 0 and, applying successively (6.11)
∑[t/∆ ]
u.c.p.
and (6.8) with χ = 2, we obtain E(ξi′n (2)2 ) ≤ Kδn2 wn2−β ∆2n , implying i=1 n ξi′n (2) =⇒ 0
by (6.45) in those cases. On the other hand, in the same cases, if we apply first (6.8)
with χ = 1, and then (6.11) with χ = 2 and the Cauchy-Schwarz inequality, we get that
∑[t/∆ ]
u.c.p.
1−β/2 3/2
E(|ξi′n (3)|) ≤ Kδn wn
∆n , implying i=1 n ξi′n (3) =⇒ 0 by (6.44).
For case III, we apply (6.11) and (6.8) with χ = 1, in this order when j = 2 and in the
∑[t/∆ ]
u.c.p.
reverse order when j = 3, to get E(|ξi′n (j)|) ≤ Kδn wn1−β ∆2n , implying i=1 n ξi′n (j) =⇒ 0
by (6.45).
5) In cases I and IV we use (6.12) and (6.8) with χ = 1, in this order when j = 2 and
in the reverse order when j = 3 to get that E(|ξi′′n (j)|) is smaller than Kδn ∆2n wn1−β in
∑[t/∆ ]
u.c.p.
case I and than Kδn ln ∆2n in case IV, hence the i=1 n ξi′′n (j) =⇒ 0 for j = 2, 3 by (6.44)
in these cases.
In case III we use (6.12) and (6.8) plus the property B ′Y = 0 (see (2.2) for the definition
of B ′Y ), in this order for j = 2 and in the reverse order for j = 3, for some χ ∈ (β ′ , β),
∑[t/∆ ]
u.c.p.
to get E(|ξi′′n (j)|χ ) ≤ Kδnχ wnχ−β ∆2n , hence since χ < 1 we deduce i=1 n ξi′′n (j) =⇒ 0 for
j = 2, 3 from (6.44) in case III.
∑[t/∆n ] ′′n
u.c.p.
6) It remains to show
ξi (j) =⇒ 0 in case II, so β = 1 and b′Y
et− ). We
t = f (t, σ
i=1
∑
set Yt′ = Yt − Bt′Y = s≤t ∆Ys , and we can write ξi′′n (j) = ρni (j) + ρ′n
(j),
where
i
∫
′ −Y′
)
(i−1)∆n )σs− dA(w
I(n,i) (Y
(∫s−
) n s
s
n
ρi (3) = δn I(n,i) (i−1)∆n σr− dA(wn )r dYs′
∫
′Y
′Y
ρ′n
)
s− − B(i−1)∆n )σs− dA(w
i (2) = δn I(n,i) (B
(
) n s
∫
∫
s
′Y
ρ′n
i (3) = δn I(n,i)
(i−1)∆n σr− dA(wn )r dBs .
ρni (2) = δn
∫
As in the previous step, we use (6.12) and (6.8) (with Y ′ instead of Y ), in this order
for j = 2 and in the reverse order for j = 3, for some χ ∈ (β ′ , 1), to get E(|ρni (j)|χ ) ≤
∑[t/∆ ]
u.c.p.
Kδnχ wnχ−β ∆2n , hence i=1 n ρni (j) =⇒ 0 for j = 2, 3 by (6.44).
n
Next, we consider ρ′n
i (2). The process M = A(wn ) + B(wn ) is a martingale, and we
′n
n
′n
have ρi (2) = ξbi + ξbi , where
ξbin = −δn
∫
ξb′n = δn
i
∫
′Y
′Y
I(n,i) (Bs− − B(i−1)∆n )σs− dB(wn )s
′Y
n
′Y
I(n,i) (Bs− − B(i−1)∆n )σs− dMs .
′Y − B ′Y
bn
We have (7.2) and |Bs−
(i−1)∆n | ≤ K∆n when s ∈ I(n, i). We first deduce |ξi | ≤
∫
∑[t/∆ ] u.c.p.
t
Kδn ∆2n , which implies i=1 n ξbin =⇒ 0. Next, since ⟨M n , M n ⟩t = 0 ans ds with |ant | ≤ K,
we deduce that E((ξbi′n )2 ) ≤ Kδn2 ∆3n , whereas E(ξbi′n | F(i−1)∆n ) = 0: so the sequence (ξbi′n )
∑[t/∆ ]
u.c.p.
satisfies (6.45), and all these properties imply i=1 n ρ′n
i (2) =⇒ 0.
′Y =
The case of ρ′n
i (3) is more complicated, and this is where the special form bt
bn b′n b′′n
f (t, σ
et− ) comes into the picture. We now use the decomposition ρ′n
i (3) = ξi + ξi + ξi ,
CLT for Approximate Quadratic Variation
31
(∫
)
∫
s
ξbin = −δn I(n,i) (i−1)∆n σr− dB(wn )r dBs′Y
(∫
)
∫
s
ξbi′n = δn I(n,i) (i−1)∆n σr− dMrn f (s, σ
e(i−1)∆n ) ds
(∫
)(
)
∫
s
ξbi′′n = δn I(n,i) (i−1)∆n σr− dMrn f (s, σ
es− ) − f (s, σ
e(i−1)∆n ) ds.
∑[t/∆n ] bn u.c.p.
As above, we get |ξbin | ≤ Kδn ∆2n , so
ξi =⇒ 0. Since σ
e(i−1)∆n is F(i−1)∆n i=1
′n
b
measurable, we still have E(ξi | F(i−1)∆n ) = 0, and using the Cauchy-Schwarz inequality for the integral on I(n, i) yields E((ξbi′n )2 ) ≤ Kδn2 ∆3n again, so by (6.45) we have
∑[t/∆n ] b′n u.c.p.
ξi =⇒ 0.
i=1
∑[t/∆ ]
u.c.p.
It remains to prove that i=1 n ξbi′′n =⇒ 0. To this end we use (6.15) and the uniform
Lipschitz property of y 7→ f (t, y) and the Cauchy-Schwarz inequality, to get E(|ξbi′′n |) ≤
Kδn ∆2n , so the sequence (ξbi′′n ) satisfies (6.44). This completes the proof.
where
7.2
Discretization of σ
So far we have eliminated the process Y , and here we show that it is in fact enough to
prove the result for the process Z. The error process when X = Z is
∑[t/∆ ]
U nt = [Z, Z]n − [Z,
Z]∆n [t/∆n ] = 2 i=1 n ζ ni , where
∫
(7.3)
ζ ni = I(n,i) (Zs− − Z(i−1)∆n ) dZs .
With the help of Lemma 6.9, Theorem 3.2 is a consequence of the following two propositions:
∑[t/∆ ]
u.c.p.
2
Proposition 7.2. We have δn i=1 n (ζi′n − σ(i−1)∆
ζ n ) =⇒ 0 as n → ∞ (recall (7.1) for
n i
ζi′n ).
Proposition 7.3. In cases I, II and III we have, as n → ∞:
L
(Z (n) , δn U n ) =⇒ (Z, U ′ ),
and in case IV we have:
δn U nt
1
=⇒ −
2
∫
u.c.p.
t
(7.4)
θs′2 ds.
(7.5)
0
This subsection is devoted to proving Proposition 7.2, together with some additional
results to be used later. We begin with some notation. We set, with vn defined in (6.20):
An = A(vn ),
∫t
n
A′n
t = 0 σs− dAs ,
M n = M (vn ),
∫t
Mt′n = 0 σs− dMsn ,
implying
Z = An + M n + B n ,
B n = B(vn )
∫t
Bt′n = 0 σs− dBsn ,
(7.6)
X ′ = A′n + M ′n + B ′n .
By virtue of (6.1), (6.5) and (6.7) we have
∫
Btn =
t
bns ds,
0
Bt′n =
∫
0
t
b′n
s ds,

K/vnβ−1



Kln
where |bnt | + |b′n
t |≤

K


0
in
in
in
in
case
case
case
case
I
IV
II
III.
(7.7)
CLT for Approximate Quadratic Variation
32
Next, we set
ζin (1) = δn ∫∆ni B n ∆ni M n
n − Mn
n
ζin (2) = δn I(n,i) (Ms−
(i−1)∆n ) dMs
ζin (3) = δn ∆ZT (n,i)1 ∆Z
∫ T (n,i)2 1{τ (n,i)=2}
ζin (4) = δn 1{τ (n,i)≥3} I(n,i) (Ans− − An(i−1)∆n ) dAns
∫
n − Bn
n
ζin (5) = δn I(n,i) (Bs−
(i−1)∆n ) dBs
ζin (6) = δn ∆ni B n ∆ZT (n,i)1 1{τ (n,i)=1}
ζin (7) = δn ∆ni B n ∆ni An 1{τ (n,i)≥2}
ζin (8) = δn ∆ni M n ∆ZT (n,i)1 1{τ (n,i)=1}
ζin (9) = δn ∆ni M n ∆ni An 1{τ (n,i)≥2}
and ζi′n (j) for j = 1, · · · , 9 is defined in the same way, upon substituting (An , B n , M n ) and
∆ZT (n,i)j with (A′n , B ′n , M ′n ) and ∆XT′ (n,i)j = σT (n,i)j − ∆ZT (n,i)j , respectively. A simple
(although tedious) computation, based on the integration by parts formula, gives
δn ζ ni
=
9
∑
ζin (j),
δn ζ ′n
i
=
j=1
9
∑
ζi′n (j).
(7.8)
j=1
We also need a decomposition of the increments ∆ni Z:
n
∆ni Z = ζin (10)
where
{ +nζi (11),
n + ∆ n An 1
∆
M
if j = 10
{τ (n,i)≥2}
i
i
ζin (j) =
n
n
∆i B + ∆ZT (n,i)1 1{τ (n,i)=1} if j = 11.
(7.9)
We will prove the following two lemmas, the second one clearly implying Proposition
7.2.
∑[t/∆ ]
u.c.p.
Lemma 7.4. In cases I, II and III we have i=1 n ζin (10) =⇒ 0.
Proof. We will prove that both arrays ξin = ∆ni M n and ξi′n = ∆ni An 1{τ (n,i)≥2} satisfy one
of the conditions of the Lemma 6.6.
In cases I and II we have E(ξin | F(i−1)∆n ) = 0 and E((ξin )2 ) ≤ K∆n vn2−β by (6.11), so
the array ξin satisfies (6.45). In case III, E(|ξin |) ≤ K∆n vn1−β by (6.11) again, so the array
ξin satisfies (6.44).
Next,
|ξi′n | ≤
∑
|∆ZT (n,i)j | 1{τ (n,i)≥2∨j} ,
(7.10)
j≥1
∑
j
j≥1 a ≤ Ka
K∆2n /vn2β−1 , when
hence (6.31) with χ = y = 1 (recall |∆Z| ≤ 1/2) and j = 0 and the property
when a ∈ (0, 1/2) yield that E(|ξi′n |) is smaller than K∆2n /vnβ , resp.
β < 1, resp. β > 1, for all n large enough. Hence the array ξi′n satisfies (6.44) in cases I
and III. In case II, (7.10) yields
(∑
(
)
2
E(|ξi′n |2 ) ≤
j≥1 E |∆ZT (n,i)j | 1{τ (n,i)≥2∨j}
(
))
∑
+2 k>j≥1 E |∆ZT (n,i)j ∆ZT (n,i)k | 1{τ (n,i)≥k} ,
CLT for Approximate Quadratic Variation
33
which by (6.31) with χ = 2 and (6.35) with χ = 1 is smaller than K∆2n (vn−β + ln2 ), so the
array ξi′n satisfies the second part of (6.45). Moreover (6.28) gives us
(
) |dT (n,i)j (1, vn )|
K
E ∆ZT (n,i) | Hvn
=
≤
j
(i−1)∆n
GT (n,i)j (vn )
GT (n,i)j (vn )
by (6.7) (we are in case II). Then, as in the proof of Lemma 6.2,
(
)k−1
(
)
E ∆ZT (n,i) 1{τ (n,i)≥k} | F(i−1)∆ ≤ K∆n C∆n
n
j
vn
when k ≥ j. It follows that |E(ξi′n | F(i−1)∆n )| ≤ K∆2n /vn , and the array ξi′n satisfies the
second part of (6.45). This completes the proof for case II.
∑[t/∆ ]
u.c.p.
2
Lemma 7.5. In all cases we have i=1 n (ζi′n (j) − σ(i−1)∆
ζ n (j)) =⇒ 0 for j = 1, · · · , 9.
n i
∑[t/∆ ]
u.c.p.
Furthermore, we have i=1 n ζin (j) =⇒ 0 when
• j = 1, 2, 4, 7, 9: in all cases;
• j = 3: in case I;
• j = 5: in cases I, II and III;
• j = 6: in cases I, II and III;
• j = 8: in cases II, III and IV.
Proof. We start with a few preliminary remarks. We set
2
ζi′′n (j) = ζi′n (j) − σ(i−1)∆
ζ n (j).
n i
For each of the arrays under consideration, we will prove that it satisfies one of the conditions of Lemma 6.6, and we observe that, because |σ| ≤ K, if an array ξin satisfies one of
∑[t/∆ ]
u.c.p.
2
those conditions, then so does the array σ(i−1)∆
ξ n . Therefore if i=1 n ζi′n (j) =⇒ 0 we
n i
∑[t/∆ ]
∑[t/∆ ]
u.c.p.
u.c.p.
also have i=1 n ζin (j) =⇒ 0 (by using σt ≡ 1) and i=1 n ζi′′n (j) =⇒ 0. Moreover we
∑[t/∆ ]
have B n = B ′n = 0 in case III. In case I, ∆1n (∆n /vnβ )2 → 0, hence i=1 n P(τ (n, i) ≥ 2) →
0 by (6.30) and the arrays ζi′n (j) satisfy (6.44) when j = 3, 4, 7, 9. All these considerations
show that we only need to prove to prove the following properties:

in case I: for j = 1, 2, 5, 6


[t/∆n ]

∑
in case II: for j = 1, 2, 4, 5, 6, 7, 8, 9
u.c.p.
(7.11)
ζi′n (j) =⇒ 0
in
case III: for j = 2, 4, 8, 9


i=1

in case IV: for j = 1, 2, 4, 7, 8, 9
∑
[t/∆n ]
i=1
ζi′′n (j) =⇒ 0
u.c.p.

in



in
 in


in
case
case
case
case
I: for j = 8
II: for j = 3
III: for j = 3
IV: for j = 3, 5, 6.
(7.12)
We will do this successively for all values of j. Each time, we conclude one of the conditions
of Lemma 6.6 by using the specific form of δn and vn , as given by (6.20), and also (6.21),
CLT for Approximate Quadratic Variation
34
without special mention. Note also that, since |σ| ≤ K, the estimates (6.11) and (6.12)
with v = vn hold for M ′n and A′n as well.
For (slightly) simpler notation, we write τ and ∆ZTj instead of τ (n, i) and ∆ZT (n,i)j ,
when no confusion arises. We also repeatedly use the following property: if Y is a [0, 1]valued variable (as |∆ZT | or |∆σT ∆ZT | for any stopping time T ) and H is an arbitrary
σ-field and χ, y > 0,
(
)
P(Y > y | H) ≤ y −χ E((Y ∧ y)χ | H),
E Y 1{Y ≤y} | H ≤ E((Y ∧ y)χ | H). (7.13)
• (7.11) for j = 1, cases I, II, IV: Using (7.7), the first part of (6.11) and the Cauchy3/2 1/2
Schwarz inequality, we get that E(|ζi′n (1)|) is smaller than Kδn ∆n vn in case II and
3/2
1/2
than Kδn ∆n ln vn in case IV, hence the array ζi′n (1) satisfies (6.44) in these two cases.
In case I we have ζi′n (1) = ξin + ξi′n , where
(
)
∫
′n
′n + σ
n (M ′n − M ′n
ξin = δn I(n,i) (Bs′n − B(i−1)∆
)
dM
b
)
ds
,
(i−1)∆n s
s
s
(i−1)∆n
n
∫
′n
n
′n
′n
ξi = δn I(n,i) bs (σs − σ(i−1)∆n )(Ms − M(i−1)∆n ) ds.
On the one hand, since bns in (7.7) is non random we have E(ξin | F(i−1)∆n ) = 0, whereas
(6.11) and (7.7) and the Cauchy-Schwarz inequality yield that E(|ξin |2 ) ≤ Kδn2 ∆3n vn4−3β ,
hence the array ξin satisfies (6.45). On the other hand, (6.14) and (6.11) and the Cauchy2−3β/2
Schwarz inequality again yield E(|ξi′n |) ≤ Kδn ∆2n vn
, hence the array ξi′n satisfies
(6.44). So (7.11) is proved in case I.
• (7.11) for j = 2, all cases: When β ≥ 1 we have E(ζi′n (2) | F(i−1)∆n ) = 0. Moreover,
with χ = 1 when β < 1 and χ = 2 otherwise, we deduce from a repeated use of (6.11) that
E(ζi′n (2)χ ) ≤ Kδnχ vn2χ−2β ∆2n = o(∆n ).
So the array ζi′n (2) satisfies (6.45) in cases I, II and IV, and (6.44) in case III.
• (7.12) for j = 3, cases II, III, IV: We have ζi′′n (3) =
∑3
n
q=1 ξi (q),
where
ξin (q) = δn Φq Φ′q ∆ZT1 ∆ZT2 1{τ =2} ,
Φ1 = σT1 − , Φ2 = σT1 − ∆σT1
2
2
′
Φ3 = σT1 − − σ(i−1)∆n , Φ1 = σT2 − − σT1 , Φ′2 = Φ′3 = 1.
)
(
Since Φq ∆ZT1 is FT1 -measurable and β ≤ 1, and since (uv) ∧ 1 ≤ u v ∧ u1 for any
u, v ≥ 0, (6.37) when q = 1 and (6.31) when q = 2, 3 yield
E(|ξin (q)| ∧ 1 | FT1 ) ≤ Kwn |δn Φq ∆ZT1 |β 1{τ ≥1} ,
 1+β/2
 ∆n
ln if q = 1
wn =
∆n ln
if q = 2, 3 and β = 1.

∆n
if q = 2, 3 and β < 1.
where
We have |Φq | ≤ K and |Φ3 | ≤ K|σT1 − − σ(i−1)∆n |. Then we deduce from (6.31), resp.
(
)
(6.33), resp. (6.37) that E |Φq ∆ZT1 |β 1{τ ≥1} is smaller than ∆n ln , resp. ∆n , resp.
CLT for Approximate Quadratic Variation
35
1+β/2
∆n
ln , when q = 1, resp. q = 2, resp. q = 3. Putting these estimates together
and with (6.20), we end up with

2+β/2 2 β

l n δn
if q = 1, 3
 K∆n
β
2
E(|ξin (q)| ∧ 1) ≤
(7.14)
K∆n ln δn
if q = 2 and β = 1


β
2
K∆n δn
if q = 2 and β < 1.
Hence the array ξin (q) satisfies (6.44) in all three cases when q = 1, 3, and in cases III and
IV when q = 2.
It remains to study ξin (2) in case II. We use (6.31) with χ = 2 and (6.32), together
with (7.13), to obtain

n
P(|ξ

in(2)| > 1 | FT1 )
E(ξ (2) 1{|ξn (2)|≤1} | FT )
≤ K∆n δn |Φ2 ∆ZT1 | 1{τ ≥1}
(7.15)
1
i
i

E(|ξin (2)|2 1{|ξin (2)|≤1} | FT1 )
The conditional expectation of the right side above, conditionally on F(i−1)∆n , is smaller
than K∆2n δn , by (6.33), hence the array ξin (2) satisfies (6.46).
• (7.11) for j = 4, cases II, III, IV: We have ζi′n (4) = ηin (1, 2) + ηin (1, 3) + ηin (2, 3) + ηi′n ,
where
ηin (j, k)
=
δn ∆XT′ j
∆XT′ k
1{τ =3} ,
ηi′n
= δn
∞ ∑
r−1
∑
∆XT′ k ∆XT′ r 1{τ ≥r∨4} .
r=2 k=1
We study ηi′n first. Since |σt | ≤ K and
we deduce from (6.36) with y = 1 that
(∑
xj
)β
≤
∑
xβk for any xk ≥ 0 because β ≤ 1,
)
∑
∑r−1 (
β
E(|η ′n | ∧ 1) ≤ E(|η ′n |β ) ≤ δnβ ∞
r=2
k=1 E |∆ZTk ∆ZTr | 1{τ ≥r∨4}
∑
∑r−1 ( C∆n )r∨4−2
≤ Kδnβ ln2 ∆2n ∞
≤ K∆4n δnβ ln2 /vn2β
β
r=2
k=1
vn
(recall C∆n /vnβ ≤ 12 ). Then the array ηi′n satisfies (6.44) in all cases. Another application
of (6.36), with y = 1/δn , yields that E(|η(j, k)| ∧ 1) for 1 ≤ j < k ≤ 3 is smaller than
K∆3n δn ln2 /vn when β = 1 and than K∆3n δnβ ln /vnβ when β < 1. This yields (6.44) for the
array ηin (j, k) in cases III and IV.
It remains to study ηin (j, k) in case II when 1 ≤ j < k ≤ 3. First, (6.36) with y = 1/δn
and χ = 2 yields, exactly as above, and since β = 1:
(
)
E (|η(j, k)| ∧ 1)2 | F(i−1)∆n ≤ Kδn ∆3n ln /vn ,
implying the last two properties in (6.46) for the array ηin (j, k). Next, we observe that
ηin (j, k) = δn σTj − σTk − ∆ZTj ∆ZTk 1{τ ≥3} . Since σTj − σTk − ∆ZTj 1{τ ≥3} is HTvnk − -measurable,
we deduce from (6.26) with y = |σTj − σTk − ∆ZTj | (which is again HTvnk − -measurable) that
( n
)
E ηi (j, k) 1{|η(j,k)|≤1} | Hvn ≤ Kδn |σT − σT − ∆ZT |
j
j
k
Tk −
1
1{τ ≥3} .
GTk (vn )
CLT for Approximate Quadratic Variation
36
At this stage, we can reproduce the proof of Lemma 6.2 to obtain, since |σt | ≤ 1:
( n
)
E ηi (j, k) 1{|η(j,k)|≤1} | F(i−1)∆ ≤ K∆3n δn ln /vn .
n
Hence the array ηin (j, k) satisfies all the properties in (6.46).
• (7.11) for j = 5, cases I and II: (7.7) and |σt | ≤ K yield that E(|ζi′n (5)|) is smaller than
Kδn ∆2n /vn2β−2 in both cases I and II, hence (6.44) holds.
• (7.12) for j = 5, case IV: We write Ψni = sups∈I(n,i) |σs − σ(i−1)∆n |. We deduce from
(7.7) and |σt | ≤ K that |ζi′′n (5)| ≤ Kδn ∆2n ln2 Ψni ; hence by (6.14) the array ζi′′n (5) satisfies
(6.44).
• (7.11) for j = 6, cases I and II: Here |ζi′n (6)| ≤ K∆n δn vn1−β |∆ZT1 | 1{τ ≥1} . Thus by
(6.31) we have E(|ζi′n (6)|) ≤ Kδn ∆2n vn2−2β , in case I, implying that the array ζi′n (6) satisfies
(6.44).
In order to study the case II, we observe that ζi′n (6) = ξin + ξi′n , where
2
ξin = ∆n δn ∆ni B n σ(i−1)∆
∆ZT1 1{τ =1} ,
n
|ξi′n | ≤ K∆n δn Ψni |∆ZT1 | 1{τ =1} .
(6.14) and (6.31) with χ = 2 and the Cauchy-Schwarz inequality yield E(|ξi′n |) ≤ Kδn ∆2n ,
so the array ξi′n satisfies (6.44). On the other hand, ∆ni B n is non-random, so (6.32) implies
|E(ξin | F(i−1)∆n )| ≤ Kδn ∆2n , whereas E((ξin )2 ) ≤ Kδn2 ∆3n , so the array ξin satisfies (6.45).
This completes the proof for case II.
• (7.12) for j = 6, case IV: Here |ζi′′n (6)| ≤ Kδn ∆n ln Ψni |∆ZT1 | 1{τ ≥1} , hence, by (6.14)
and (6.31) for χ = 2 and the Cauchy-Schwarz inequality, the expectation of this variable
is smaller than Kδn ∆2n ln , and ζi′′n (6) satisfies (6.44).
• (7.11) for j = 7, cases II, IV: Since |σt | ≤ K, we have with wn = 1 in case II and
wn = ln in case IV:
|ζi′n (7)| ≤ K∆n δn wn
∞
∑
|∆ZTj | 1{τ ≥j∨2} .
j=1
Therefore (6.31) gives us that E(|ζi′n (7)|) is smaller than Kδn ∆3n wn ln /vnβ . Hence in both
cases the array ζi′n (7) satisfies (6.44).
• (7.11) for j = 8, cases II, III, IV: We have
|ζi′n (8)| ≤ Kδn |∆ni M ′n | |∆ZT1 | 1{τ ≥1} .
(7.16)
The variable ∆ZT1 1{τ ≥1} is G0vn -measurable, hence by conditioning we deduce from (6.11)
(which holds for M ′n as well as for M n ) and (6.31) and the Cauchy-Schwarz inequality
when β = 1 that
{
3/2
1/2
Kδn ∆n ln vn
if β = 1
′n
E(|ζi (8)|) ≤
1−β
2
Kδn ∆n vn
if β < 1.
CLT for Approximate Quadratic Variation
37
Then the array ζi′n (8) satisfies (6.44) in cases III and IV.
n + η ′n , where
For dealing with case II, we write ζi′n (8) = ηi−
i+
M−′n,i
n = δ M ′n,i ∆X ′ 1
ηi±
where
n
±
T1 {τ =1} ,
′n,i
′n
′n
′n − M ′n
= MT1 ∧(i∆n ) − M(i−1)∆n , M+ = Mi∆
T1 ∧(i∆n ) .
n
n |
On the one hand, E(M+′n,i | GTvn1 ) = 0 and ∆XT′ 1 1{τ =1} is GTvn1 -measurable, hence E(ηi+
′n,i
F(i−1)∆n ) = 0. On the other hand, (6.26) and the fact that M− is FT1 − -measurable (M ′n
is adapted and does not jump at time T1 ) yield
( n
)
E ηi− | Hvn ≤ Kδn |M ′n,i |
−
T1 −
1
1{T1 ≤i∆n } .
GT1 (vn )
Then we use (6.11) for M ′n to deduce (recall β = 1)
( n
)
1/2
E ηi− | Hvn
≤ Kδn ∆1/2
n vn
(i−1)∆n
1
1{T1 ≤i∆n } .
GT1 (vn )
3/2 1/2
n |F
Finally, repeating the proof of Lemma 6.2, we deduce E(ηi−
(i−1)∆n ) ≤ Kδn ∆n vn
from the above. Putting these two partial results together yields |E(ζi′n (8) | F(i−1)∆n )| ≤
Kδn ∆n vn , hence the array ζi′n (8) satisfies the first part of (6.45). To check that it also
satisfies the second part, we use first (7.16) and next (6.11) and (6.31) to get
(
)
E ζi′n (8)2( | F(i−1)∆n
)
vn
≤ Kδn2 E |∆ZT1 |2 1{τ ≥1} E(|∆ni M ′n |2 | G(i−1)∆
) | F(i−1)∆n ≤ Kδn2 ∆2n vn .
n
3/2 1/2
Then the array ζi′n (8) satisfies the second part of (6.45).
• (7.12) for j = 8, case I: We use the notation M±′n,i of the previous step, and define M±n,i
∑
similarly, on the basis of M n . We can then write ζi′′n (8) = 5q=1 ξin (q), where (see (6.18)
for σ(vn )t , and recall that σ(vn )T1 = σ(vn )T1 − ):
ξin (1) = δn σT1 − ∆ZT1 (M+′n,i − σT1 M+n,i ) 1{τ =1}
ξin (2) = δn σT1 − ∆σT1 ∆ZT1 M+n,i 1{τ =1}
(
)
2
ξin (3) = δn σT21 − − σ(i−1)∆
∆ZT1 M+n,i 1{τ =1}
n
ξin (4) = δn (M−′n,i − σ(i−1)∆n M−n,i ) σT1 − ∆ZT1 1{τ =1}
(
)
ξin (5) = δn σ(i−1)∆n σ(vn )T1 − σ(vn )(i−1)∆n M−n,i ∆ZT1 1{τ =1} .
We will prove that
{
E(|ξin (q)|) ≤
2−3β/2
Kδn ∆2n vn
1+1/β 1−β/2 1/β
Kδn ∆n
vn
ln
if q = 1, 4, 5
if q = 3
(7.17)
and this implies that the arrays ξin (q) satisfy (6.44) when q = 1, 3, 4, 5. This will be
achieved by successive conditioning again. When q = 1 we apply (6.16) with χ = 2 and
the Cauchy-Schwarz inequality to M+′n,i − σT1 M+n,i after conditioning on FT1 , then we use
CLT for Approximate Quadratic Variation
38
(6.31). When q = 3 we apply (6.11) to M+n,i after conditioning on FT1 , then we apply
(6.37) with χ = β and Hölder’s inequality. For q = 4 we observe first that
∫
′n,i
T1 ∧(i∆n )
n,i
n
(M
(σ(vn )s − σ(vn )(i−1)∆n ) dMs .
− − σ(i−1)∆n M− )σT1 − ≤ (i−1)∆n
vn
Then for both q = 4 and q = 5 we first use (6.11) and (6.19) conditionally on G(i−1)∆
,
n
with respect to which ∆ZT1 1{τ =1} is measurable, and the Cauchy-Schwarz inequality, and
then we use (6.31).
It remains to study ξin (2), which is ξin (2) = ρ(1)ni + ρ(2)ni , where
ρ(k)ni = δn σT1 − ∆σT1 ∆ZT1 M (k)n,i
+ 1{τ =1}
n,i
n
n
M (k)+ = M (k)i∆n − M T1 ∧(i∆n )
∫t
∑
M (1)nt = s≤t ∆Zs 1{vn′ <|∆Zs |≤vn } − 0 (ds (1, vn′ ) − ds (1, vn )) ds
M (2)n = M n − M (1)n
and vn′ = ∆n , which satisfies vn′ < vn for all n large enough. The processes M (k)n are
vn
martingales relative to (Gtvn ), so E(M (k)n,i
+ | GT1 ) = 0 and, similar to (6.11), we have
1/β
vn
′1−β
E(|M (1)n,i
,
+ | | GT1 ) ≤ K∆n vn
vn
2
′2−β
E(|M (2)n,i
.
+ | | GT1 ) ≤ K∆n vn
Then, by successive conditioning and |∆σT | ≤ 2 and (6.31) we obtain
E(|ρ(1)ni |) ≤ Kδn ∆2n vn′1−β
E(ρ(2)ni | F(i−1)∆n ) = 0, E(|ρ(2)ni |2 ) ≤ Kδn2 ∆2n vn′2−β .
It follows that the arrays ρ(1)ni and ρ(2)ni satisfy (6.44) and (6.45), respectively, and the
proof for the case at hand is complete.
• (7.11) for j = 9, cases II, III, IV: We have
|ζi′n (9)| ≤ Kδn |∆ni M ′n |
∑
|∆ZTj | 1{τ ≥2∨j} .
j≥1
vn
G(i−1)∆
n
Then, upon conditioning first on
obtain
{
E(|ζi′n (9)|)
and the array
7.3
ζi′n (9)
≤
and using (6.11), and then using (6.31), we
√
5/2
Kδn ∆n ln / vn
if β = 1
Kδn ∆3n vn1−2β
if β < 1,
satisfies (6.44) in cases II, III and IV.
Proof of Proposition 7.3
1 - Preliminaries. Let Γ = (Z, U ′ ), where U ′ is a non-homogeneous Lévy process
′
U′
independent of Z and with spot characteristics (bU
t , 0, Ft ) defined by (3.5) in cases I and
III, and (3.6) in case II. Set also
∑
{
[t/∆n ]
Γnt
=
i=1
ξin ,
ξin,1
=
ζin (11),
ξin,2
=
2ζin (8) in case I
2ζin (3) in cases II and III.
(7.18)
CLT for Approximate Quadratic Variation
39
L
Then by Lemmas 7.4 and 7.5, it is enough to prove Γn =⇒ Γ, under (6.1).
To this end, in cases II and III, we use Lemma 6.8-(b), and we observe right away that
a part of (6.50), (6.51) and (6.52) is almost obvious: namely, we have:
(6.50) and (6.51) hold for j = 1, and (6.52) holds
for A ∈ A of the form A = A′ × R.
(7.19)
∑[t/∆ ]
L
Indeed, Z (n) → Z for the Skorokhod topology (for each ω), hence Γn,1
= i=1 n ζin (11) =⇒
t
Γ1 = Z by Lemma 7.4. Then (7.19) follows from Lemma 7.4 applied with d = 1 and
L
Γn,1 =⇒ Γ1 .
2 - Proof in case I. Here β > 1 and Btn =
∫t
n
0 bs ds
with bns = −ds (1, vn ) and bt = 0.
Step 1. We will apply the criterion
of Lemma 6.8. In) view of (3.5), the right side of
( ∫(a)
t
(6.49) for u = (u′ , u′′ ) ∈ R2 is exp 0 (ρ(u′ )s + ρ(u′′ )s ) ds , where
∫ (
∫
)
( iu′′ x
) ′
′
ρ(u′ )t =
eiu x − 1 − iu′ x Gt (dx), ρ(u′′ )t =
e
− 1 − iu′′ x FtU (dx)
′
∫
U′
{|x|>1} x Ft (dx)
and bt ≡ 0). With the notation
( ′ n
)
′′ n
Rjn (u′ , u′′ ) = E eiu ζj (11)+2iu ζj (8) − 1,
(for this, we use that bU
t =
and with help of Lemma 6.7, it is thus enough to prove that
∑
∫
[t/∆n ]
Rjn (u′ , u′′ )
u.c.p.
=⇒
t(
)
ρ(u′ )s + ρ(u′′ )s ds.
(7.20)
0
j=1
The process M n is a non-homogeneous Lévy process with spot characteristics (0, 0, Gt (dx) 1{|x|≤vn } ),
hence
∫
)
(
)
(∫
( ivx
)
iv∆n
Mn
j
E e
= exp
ds
e − 1 − ivx Gs (dx) .
{|x|≤vn }
I(n,i)
Moreover, ∆ZT (n,1)1 1{τ (n,1)=1} is independent of ∆nj M n , and ∆nj B n is non random. Then
with the notation
∫
∫
(
)
n
ds
e2ivδn xy − 1 − 2ivδn xy Gs (dy),
zj (x, v) =
I(n,j)
{|y|≤vn }
n
we deduce from (6.13) that, with the simplifying notation αjn = α(j−1)∆
,
n ,j∆n
∫
∫
(
)
)
( iu′ x+z n (x,u′′ )
′ n n
j
− 1 Gs (dx) − 1.
Rjn (u′ , u′′ ) = eiu ∆j B 1 + αjn
ds
e
{|x|>vn }
I(n,j)
Below, we fix u′ , u′′ in R, and set ρnt =
∫
{|x|>vn }
(
)
′
eiu x − 1 − iu′ x Gt (dx) and
}
)
zjn (x,u′′ )
e
−
1
G
(dx)
t
)
)( n ′′
∫{|x|>vn } ( ′
if t ∈ I(n, j).
ψtn = {|x|>vn } eiu x − 1 ezj (x,u ) − 1 Gt (dx)
ρnt =
∫
(
CLT for Approximate Quadratic Variation
40
Since bnt = −dt (1, vn ), we have
∫
( iu′ x+zn (x,u′′ )
)
e
− 1 Gt (dx) = ρnt + ρnt + ψtn − iu′ bnt ,
{|x|>vn }
hence
Rjn (u′ , u′′ ) = αj′n
αj′n
=
(
∫
I(n,j)
′ n n
αjn eiu ∆j B , anj
)
ρns + ρns + ψsn ds + anj ,
=e
n
iu′ ∆n
jB
(1 −
where
iu′ ∆nj B n αjn )
− 1.
Therefore (7.20) holds as soon as we have the following two properties, with the notation
t(n) = ∆n [t/∆n ]:
[t/∆n ]
∑
sup |αj′n − 1| → 0,
|anj | → 0.
(7.21)
j
j=1
∫ t(n) n u.c.p. ∫ t
ρ ds =⇒ ρ(u′ )s ds
∫0t(n) sn u.c.p. ∫0t
ρ ds =⇒ ρ(u′′ )s ds
∫0t(n) sn u.c.p. 0
ψs ds =⇒ 0.
0
(7.22)
Step 2. Here we prove (7.21). If wn = ∆n /vnβ , (6.24) and (7.7) yield |αjn − 1| ≤ Kwn
and |u′ ∆nj B n | ≤ Kvn wn . Since wn and vn go to 0, the first part of (7.21) follows. Next,
ix
e (1 − iαx) − 1 ≤ K|x| |1 − α| + Kx2 when α, x ∈ [−1, 1]. Applying this to x = u′ ∆n B n
j
2
and α = αjn , we see that for all n large enough we have |a′n
j | ≤ Kvn wn = K∆n
and the second part of (7.21) follows.
1+1/2β
/ln2β−1 ,
∫
Step 3. It remains to prove (7.22). Since |eix − 1 − ix| ≤ Kx2 and x2 Gt (dx) ≤ K,
we get |ρnt | ≤ K and, since vn → 0, we also have ρnt → ρt (u′ ) for all t by the dominated
convergence theorem. Since t(n) → t, the first part of (7.22) follows.
We also have
∫
∫
2
(|wx| ∧ (wx) )Gt (dx) = |w|
∫
{|x|>1/w}
|x|Gt (dx) + w
2
{|x|≤1/w}
x2 Gt (dx)
which is smaller than K|w|β (use (6.5) again). Recalling the definitions of zn (x, u′′ ) and
δn , and that u′′ is fixed, this implies
|zjn (x, u′′ )| ≤ K ∆n δnβ |x|β = K |x|β /ln .
(7.23)
( ′
)( n ′′
)
implies eiu x − 1 ezj (x,u ) − 1 ≤ K|x|1+β /ln , which yields |ψtn | ≤ K/ln because
∫This1+β
|x|
Gt (dx) ≤ K. Hence the last part of (7.22) holds.
′′n
For the second part of (7.22), we write ρnt = ρ′n
t + ρt , with
}
∫
n
′′
ρ′n
t = {|x|>vn } zj (x, u ) Gt (dx)
∫
if t ∈ I(n, j).
zjn (x,u′′ )
− 1 − zjn (x, u′′ )) Gt (dx)
ρ′′n
t = {|x|>vn } (e
n ′′
∫
Using (7.23) again, we get ezj (x,u ) −1−zjn (x, u′′ ) ≤ K|x|2β /ln2 and, since |x|2β Gt (dx) ≤
2
K we get |ρ′′n
t | ≤ K/ln . Hence it clearly remains to prove
∫ t
∫ t
u.c.p.
′n
ρs ds =⇒
ρ(u′′ )s ds.
(7.24)
0
0
CLT for Approximate Quadratic Variation
41
∫
Step 4. Here we prove (7.24). First, |ρnt | ≤ K by (7.23) and {|x|>vn } |x|β Gt (dx) ≤ Kln .
∫t
Thus, the functions 0 ρ′n
s ds are, locally, uniformly equi-continuous and, in order to prove
(7.24), it is enough to show the convergence for any fixed t.
∫ t(n)
∫ iu′′ x
Therefore in the sequel we fix t > 0. We have 0 ρ′n
− 1 − iu′′ x) Fn (dx),
s ds = (e
where the measure Fn (which depends also on t) is given by
[t/∆n ] ∫
Fn (A) =
∫
∑
j=1
ds dr
I(n,j)×I(n,j)
{|x|>vn }
∫
Gs (dx)
{|y|≤vn }
1A (2δn xy) Gr (dy).
∫t
∫
∫
′
′′
The measures H = Fn and H∫= 0 FsU ds satisfy
(|x| ∧ x2 ) H(dx) < ∞, hence (eiu x −
∫
1 − iu′′ x) H(dx) equals −iu′′ g(x) H(dx) + gu′′ (x) H(dx), where g(x) = x 1{|x|>1} and
′′
gu′′ (x) = eiu x −1−iu′′ x1{|x|≤1} are continuous outside {−1, 1} and smaller than a constant
∫t
′
times |x| ∧ x2 . Since the measure 0 FsU ds has no atom, the desired convergence result is
implied by the following properties, where w ranges through (0, ∞):

h = hw = 1(w,∞)


∫ t

′
h = h′w = 1(−∞,−w)
Fn (h) →
FsU (h) ds if
(7.25)
h(x) = h′ (x) = x2 1{|x|≤1}

0


h(x) = h′′ (x) = x1{|x|>1} .
Consider first h = hw . As soon as 2δn vn2 > w we have
(∫
( ( w )
)
∑[t/∆ ] ∫
+
Fn (hw ) = j=1 n I(n,j)×I(n,j) ds dr {x>vn } G+
r 2δn x − Gr (vn ) Gs (dx)
)
( ( w )
)
∫
− (v ) G (dx) .
+ {x<−vn } G−
−
G
n
s
r 2δn |x|
r
Since sup( 2vnwδn x : x > vn ) → 0, we deduce from (6.1) that
(
)
β β ∑[t/∆ ] ∫
(β, vn ) + θr− d−
Fn (hw ) = 2wβδn j=1 n I(n,j)×I(n,j) θr+ d+
s (β, vn ) ds dr)
(s
β
nt
+o ∆2β
+ δnw∆βn t sups≥0 ds (β, vn ) .
vn
Thus, (6.6) and
∆n /vn2β
→ 0 and log(1/vn ) ∼ ln /2β and δnβ = 1/∆n ln yield
[t/∆n ] ∫
( + +
)
2β−1 ∑
− −
Fn (hw ) =
θ
θ
+
θ
θ
ds dr + o(1).
r
s
r
s
∆n wβ
I(n,j)×I(n,j)
j=1
∫
Note that, if in (4.4) we take I(n,j)×I(n,j) f (s)f ′ (r) dr ds instead of the integral over the
∫t
triangle {(s, r) ∈ I(n, i)2 , r ≤ s}, the limit is 0 f (s)f ′ (s) ds. Thus what precedes yields
)
∫t
β−1 ∫ t (
′
Fn (hw ) → 2wβ 0 (θs+ )2 + (θs− )2 ds, which is 0 FsU (hw ) ds. In an analogous fashion we
∫t
β ∫t
′
find that Fn (h′w ) → w2 β 0 θs+ θs− ds = 0 FsU (h′w ) ds.
Next, consider h = h′ . As soon as 2δn vn2 > 1 we have
(
)
∫
∑[t/∆ ] ∫
Fn (h′ ) = 4δn2 j=1 n I(n,j)×I(n,j) ds dr {|x|>vn } d′r 2, 2δn1|x| x2 Gs (dx)
β δ β β ∑[t/∆ ] ∫
n
n
= 2 2−β
j=1
I(n,j)×I(n,j) θr ds (β, vn ) ds dr
)
(
+o δnβ ∆n t sups≥0 ds (β, vn )
β−1 β ∑[t/∆ ] ∫
n
= ∆2n (2−β)
j=1
I(n,j)×I(n,j) θr θs ds dr + o(1),
CLT for Approximate Quadratic Variation
42
where we have used (6.6) and log(1/vn ) ∼ ln /2β again. We deduce as above that Fn (h′ ) →
∫ t U′ ′
2β−1 β ∫ t
2
2−β
0 (θs ) ds = 0 Fs (h ) ds.
Finally, with h = h′′ and using similar argument, we get
Fn (h′′ ) = 2δn2
=
=
Thus Fn (h′′ ) →
7.3 in case I.
∑[t/∆n ] ∫
β
2β β δn
β−1
ds dr
I(n,j)×I(n,j)
(
∫
j=1
∑[t/∆n ] ∫
{|x|>vn }
(
)
dr 1, 2δn1|y| − dr (1, vn ) x Gs (dx)
′
I(n,j)×I(n,j) θr
ds (β, vn ) ds dr
∆n t
+o 2β−1 + δnβ ∆n t sups≥0
vn
2β−1 β ∑[t/∆n ] ∫
′ ′
j=1
I(n,j)×I(n,j) θr θs ds dr + o(1).
∆n (β−1)
2β−1 β
β−1
∫t
j=1
′ 2
0 (θs ) ds
=
(
∫t
0
)
ds (β, vn )
′
FsU (h′′ ) ds. This completes the proof of Proposition
3 - Proof in case II. In this case θt′ = 0 and the spot characteristics of Γ are (0, 0, FtΓ ),
where
∫
∫
θt2 ∞ 1
Γ
Ft (A) = 1A (x, 0) Gt (dx) +
1A (0, x) dx.
(7.26)
2 −∞ x2
{
}
∫∞
∫∞
We set C = z > 0 : 0 Gt ({z}) dt = 0 Gt ({−z}) dt = 0 , so (1, ∞) ⊂ C because
each Gt only charges (−1, 1)). We also set
A = {(w,
 ∞) : w ∈ C} ∪ {(−∞, −w) :
if
 {1A×B : A ∈ A, B ∈ B}
{1B×A : A ∈ A, B ∈ B}
if
Hi =

′
′
{1A×A : A, A ∈ A}
if
w ∈ C},
i=1
i=2
i = 3.
B = {[−w, w] : w ∈ C}
In the sequel, we use the following functions on R2 , for ε ∈ (0, 1]:
h′ (x, y) = y1{|y|≤1} ,
hw (x, y) = y 2 1{|y|≤w} .
We denote by Hin the law of ξin . With the notation (Vsn , Vtn ) of Lemma 6.5 we write
for (s, t) ∈ Dn and any function h on R and R2 and z ∈ R:
γsn (z, h) = Gs (vn ) E(h(z + Vsn , 0))
γ ns,t (z, h) = Gs (vn ) Gt (vn ) E(h(z, 2δn Vsn Vtn )).
(7.27)
By (6.13) and (7.18) and T (n, i)j ∈ Dn when j = 1, 2 and T (n, i)j < ∞, we then have for
any given i ≥ 1 and with the convention s0 = (i − 1)∆n and s3 = i∆n :
(
Hin (h) = αsn0 ,s3
)
∫
+ {s0 <s1 <s2 <s3 } αsn0 ,s1 αsn1 ,s2 Gs1 (vn ) Gs2 (vn )(1−αsn2 ,s3 ) ds1 ds2 h(∆ni B n , 0)
(7.28)
(∫
)
∫
s
+αsn0 ,s3 s03 γsn1 (∆ni B n , h) ds1 + {s0 <s1 <s2 <s3 } γ ns1 ,s2 (∆ni B n , h) ds1 ds2 .
CLT for Approximate Quadratic Variation
43
By (b) of Lemma 6.8 and (7.19), it is enough to prove the following properties:
∫ t
∑[t/∆n ]
n ′ u.c.p.
Hi (h ) =⇒
bΓ,2
(7.29)
s ds
i=1
0
∑[t/∆n ]
Hin (hw ) = 0
∫ t
∑[t/∆n ]
Hin (h) →
FsΓ (h) ds,
lim lim sup
w→0
(7.30)
i=1
n
i=1
∀ h ∈ H 1 ∪ H2 ∪ H3 .
(7.31)
0
We also set un = K∆n where K is the constant occurring in (7.7), case II, so |∆ni B n | ≤ un
for all i.
Proof of (7.29). With the notation (6.38) we have
∫
n ′
n
′′n
Hi (h ) = 2αs0 ,s3
f1/2
(s1 , s2 ) ds1 ds2 .
(7.32)
{s0 <s1 <s2 <s3 }
n ≤ 1 and (6.42) yield H n (h′ ) = o
Here we have δn log(1/vn ) ∼ 1/∆n . Then αs,t
u,n→∞ (∆n ),
2
i
Γ,2
implying (7.29) because bt = 0.
Proof of (7.30). We have, instead of (7.32):
∫
n
n
Hi (hw ) = 4αs0 ,s3
{s0 <s1 <s2 <s3 }
′n
fw/2
(s1 , s2 ) ds1 ds2 .
Then (6.40) yields for all n large enough:
Hin (h2,w ) ≤ Kw2−β ∆n + ou,n→∞ (∆n ),
and (7.30) follows.
Proof of (7.31). By (7.26), (7.27) and (7.28), when h ∈ H3 we have FtΓ (h) = 0 and
γsn (z, h) = 0 and also γ ns,t (∆ni B n , h) = 0, hence Hin (h) = 0 as well, as soon as un ≤ a,
where a is such that h(x, y) = 0 when |x| ≤ a; this occurs for all n large enough, hence
(7.31) holds when h ∈ H3 .
Next, let h = 1(w,∞)×[−w′ ,w′ ] with w ∈ C. We have γ ns,t (∆ni B n , h) = 0 when un < w, in
which case Hin (h) reduces to P(ζin (11) > w), and the property (7.31) follows from (7.19).
The same holds for h = 1( −∞,−w)×[−w′ ,w′ ] .
Finally, let h = 1[−w′ ,w′ ]×(w,∞) . As soon as un ≤ w′ we have
∫
( n
)
n
Hin (h) = αsn0 ,s3
fw/2,1,1 (s1 , s2 ) + fw/2,1,−1
(s1 , s2 ) ds1 ds2 .
(7.33)
{s0 <s1 <s2 <s3 }
Therefore, in view of (6.24) and (6.39) and β = 1 and δn log(1/vn ) ∼ 1/∆n again, and
recalling that θt+ = θt− = θt /2 in our case, we obtain
Hin (h) =
1
∆n w
∫
∫
i∆n
s
ds
(i−1)∆n
θs θt dt + ou,n→∞ (∆n ).
(i−1)∆n
CLT for Approximate Quadratic Variation
44
Then (4.4) implies (7.31), because FtΓ (h) = θt2 /2w. The same property holds for the
function h = 1[−w′ ,w′ ]×(−∞,−w) , and this ends the proof of (7.31).
4 - Proof in case III. Here, the spot characteristics of Γ are (bΓ , 0, F Γ ), where
∫
β (θt′ )2
bΓ,1
= bt = x Gt (dx),
bΓ,2
= 21−β (1−β)
t
t
∫
FtΓ (A) = (1A (x, 0) Gt (dx)
)
∫ ∞ (θt+ )2 +(θt− )2
∫ 0 2θt+ θt−
β
1
(0,
x)
dx
+
1
(0,
x)
dx
.
+ 21−β
A
A
1+β
1+β
0
−∞ |x|
x
Since ∆ni B n = 0,the law Hin of ξin is now given, when the function h on R2 satisfies
h(0, 0) = 0, by
∫
( ∫ s3
)
n
n
n
Hi (h) = αs0 ,s3
γs1 (0, h) ds1 +
γ ns1 ,s2 (0, h) ds1 ds2 .
{s0 <s1 <s2 <s3 }
s0
As in the previous subsection, it is enough to prove (7.29), (7.30) and (7.31).
We begin with (7.29). (7.32) still holds, and we apply (6.41) and δnβ log(1/vn ) ∼ 1/β∆n
and (6.24) to obtain
∫ s
∫ i∆n
β 2β
n ′
θs′ θt′ dt + ou,n→∞ (∆n )
Hi (h ) =
ds
∆n (1 − β) (i−1)∆n
(i−1)∆n
and (7.29) is implied by (4.4). (7.30) is proved exactly as in case II.
Finally, we prove (7.31). When h ∈ H3 and when h ∈ H1 , we argue as in case II again.
When h = 1[−w′ ,w′ ]×(w,∞) we still have (7.33) (for all n, here). Then if we use (6.39) and
δnβ log(1/vn ) ∼ 1/β∆n we deduce as before that
∫ i∆n
∫ s
2β
ds
(θs+ θt+ + θs− θt− ) dt + ou,n→∞ (∆n ).
Hin (h) =
∆n wβ (i−1)∆n
(i−1)∆n
Then (4.4) implies (7.31), because FtΓ (h) = ((θt+ )2 + (θt− )2 )/21−β wβ . If instead we use
n
n
n
the function h = 1[−w′ ,w′ ]×(−∞,−w) , one has to replace fw/2,1,1
+ fw/2,1,−1
by fw/2,−1,1
+
+
−
−
+
n
+
−
+
−
Γ
fw/2,−1,−1 in (7.33), hence θs θt + θs θt by θs θt + θs θt above, and now Ft (h) =
2θt+ θt− /21−β wβ , and we still get (7.31).
5
∫ t - ′2Proof in case IV. Here we need to prove (7.5). By Lemma 7.5, and letting Lt =
0 θs ds, it suffices to prove the following three convergences:
∑
[t/∆n ]
i=1
ζin (3)
1
=⇒ Lt ,
4
u.c.p.
∑
[t/∆n ]
i=1
ζin (5)
1
=⇒ Lt ,
2
u.c.p.
∑
[t/∆n ]
u.c.p.
ζin (6) =⇒ −Lt .
i=1
(6.6) implies bnt = −ln θt′ + ou(t),n→∞ (ln ), hence a simple computation shows
∫
∫ s
1
ζin (5) =
θs′ ds
θr′ dr + ou,n→∞ (∆n )
∆n I(n,i)
(i−1)∆n
(7.34)
CLT for Approximate Quadratic Variation
45
for each fixed ω. Then (4.4) implies the second convergence in (7.34).
Next, we prove the first convergence in (7.34). As for (7.28), and with the same s0 and
s3 , and notation (7.27), for any function h vanishing at 0 we have
∫
n
n
E(h(ζi (3))) = αs0 ,s3
γ ns1 ,s2 (0, g) ds1 ds2 ,
{s0 <s1 <s2 <s3 }
where g(x, y) = h(y/2). In particular, since δn (log(1/vn ))2 ∼ 1/∆n here, using (6.43) with
w = 1 and (6.24) yields
∫
E(ζin (3) 1{|ζin (3)|≤1} ) = αsn0 ,s3 {s0 <s1 <s2 <s3 } f1′′n (s1 , s2 ) ds1 ds2
∫
∫s
= 2∆1 n I(n,i) θs′ ds (i−1)∆n θr′ dr + ou,n→∞ (∆n ).
Then, (4.4) yields
∑
[t/∆n ]
u.c.p.
E(ζin (3) 1{|ζin (3)|≤1} ) =⇒
i=1
1
Lt .
4
Next, (6.36) applied with y = 1/δn implies E((|ζin (3)| ∧ 1)2 ) ≤ Kδn ∆2n ln ≤ K∆n /ln , hence
∑
[t/∆n ]
u.c.p.
E((|ζin (3)| ∧ 1)2 ) =⇒ 0.
i=1
In other words the sequence ξin = ζin (3) satisfies (with the dimension d = 1) the properties
(6.50) with the limit Lt /4, (6.51), and (6.52) with a vanishing limit. Then it follows from
Lemma 6.8 that the first convergence in (7.34) holds.
For the last convergence in (7.34), it is enough by Lemma 6.8 to show that
∑
∑
[t/∆n ]
[t/∆n ]
E(ζin (6))
u.c.p.
=⇒ −Lt ,
i=1
(7.35)
i=1
(6.13) and the definition of B n yield
∫
∫
n
E(ζi (6)) = δn
(bs − ds (1, vn )) ds
I(n,i)
E(ζin (6)2 ) → 0.
I(n,i)
n
n
α(i−1)∆
αs,i∆
d (1, vn ) ds.
n s
n ,s
Using again (6.24) and (6.6), we deduce
∫
)2
1 (
n
E(ζi (6)) = −
θs′ ds + ou,n→∞ (∆n ),
∆n I(n,i)
and using the càdlàg property of θt′ , we deduce the first property in (7.35). Finally, (6.31)
with χ = 2 and (7.7) yield E(ζin (6)2 ) ≤ Kδn2 ln2 ∆3n , implying the second property in (7.35).
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