testing for common arrivals of jumps for discretely...
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Testing for common arrivals of jumps for discretely observed
multidimensional processes
Jean Jacod ∗ and Viktor Todorov †
May 31, 2007
Abstract
We consider a bivariate process Xt = (X1t , X2
t ), which is observed on a finite time interval[0, T ], at discrete times 0, ∆n, 2∆n, · · ·. Assuming that its two components X1 and X2 have jumpson [0, T ], we derive tests to decide whether they have at least one jump occurring at the same time(“common jumps”) or not (“disjoint jumps”). There are two different tests, for the two possiblenull hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level,as the mesh ∆n goes to 0. We show on some simulations that these tests perform reasonably welleven in the finite sample case, and we also put them in use for some exchange rates data.
Keywords: Common jumps, tests, discrete sampling, high frequency data.
1 Introduction.
It seems more and more apparent, as high frequency data become available at a largescale, that many processes observed at discrete times like stock prices or exchange ratesdo have jumps. Now, finding models for discontinuous (continuous-time) processes whichare compatible with data is a hard task, especially if one wants tractable models; and thisis even more difficult if one wants to model several processes at once.
Among models for multidimensional processes with correlated components, the easiestones to tackle are those for which the various components do not jump together. Indeed,[16] assumes that jumps in individual stocks do not arrive together and can be diversifiedaway when stocks are aggregated in a portfolio. But are such models for asset pricescompatible with financial data ? The empirical studies of [2] and [8], using high-frequencydata, provide strong evidence for presence of jumps even on aggregate stock market level,which suggests that individual stocks contain systematic jump component. It is clearthat if we want to formally study the systematic and idiosyncratic jumps in individualasset prices we need formal tests for deciding whether the jumps in the different assets
∗Institut de Mathematiques de Jussieu, 175 rue du Chevaleret 75 013 Paris, France (CNRS – UMR7586, and Universite Pierre et Marie Curie–P6); e–mail: [email protected]
†Department of Economics, Duke University, Durham, NC 27708-0097; e-mail: [email protected]
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arrive together or not1. The main goal of this paper is to develop such tests in a (almost)completely general framework.
More specifically, we consider a d-dimensional process X = (X1, · · · , , Xd) whichevolves according to a model which we want to be as general as possible: we will takean Ito semimartingale, which essentially amounts to say that it is driven by a Wiener pro-cess and a Poisson random measures (this allows in particular for “infinite activity” of thejumps). This semimartingale is observed at regularly spaced times i∆n, where i = 0, 1, · · ·.We provide here some testing procedure, based on the observation of the Xi∆n ’s up tosome given terminal time T (that is, for i = 0, 1, · · · , [T/∆n]) to test the “null” hypothesisthat two components, say for example X1 and X2, have no common jumps (meaning thatthey never jump together) on the time interval [0, T ], and also the null hypothesis thatthey do have common jumps.
An important feature of this paper is that we want these tests to be as independent ofthe underlying model as it it is possible. A second – obvious – feature is that the problemis asymptotic, that is the time lag ∆n is “small” and in fact we study the asymptoticproperties of the tests as ∆n → 0, the horizon T being kept fixed. A third importantfeature is that we test for common jumps or no, for the path of t 7→ Xt on [0, T ]: somemodels allow for a positive probability of common jumps and simultaneously a positiveprobability for no common jump, and our tests try to give an answer for the observedpath, and of course not the model itself.
The tests exhibited here are based upon statistics involving suitable sums of functionsof the increments of the process X between successive observations. We will use theseincrements at two different scales, exactly as in [1], whose methods are somehow gener-alized here. The way the tests are conducted is however different and in a sense morecomplicated than in that paper. Tests for deciding whether a given path has jumps or noton the interval [0, T ] have been already developed (see [1], [4], [12], [14], [15], [17], [18]; [3]and [5] discuss some multivariate extensions of the test in [4]). Therefore, here we focuson the problem of testing whether there is at least one common jump time for the twocomponents, or none, supposing that there are jumps.
The paper is organized as follows. Sections 2 and 3 describe our setup and the teststatistics we use. We provide a Central Limit Theorem, or what plays the role of it for ourproposed statistics, in Section 4, and use them to construct the actual tests in Section 5.We report the results of some Monte Carlo simulations in Section 6, and in Section 7we put our tests in use on actual data, namely the exchange rates between two pairs ofcurrencies. Proofs are in Section 8.
2 Setting and assumptions.
Our problem in this paper is to determine whether any two components of a multi-dimensional process do jump at the same times. It thus amounts to solving this problem
1Recently, [7] analyze the relationship between the jumps in individual stocks and a diversified portfolioof these stocks and argue that for this we need formal tests about the possible common arrival of jumpsin individual series.
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separately for each pair of components. In other words, this is a truly 2-dimensional prob-lem, and it is no restriction to suppose that the underlying process X is 2-dimensional,with components denoted by X1 and X2.
As already mentioned, we do not want to make any specific model assumption onX, such as assuming some parametric family of models. We do need, however, a mildstructural assumption which is satisfied in all continuous time models used in finance, atleast as long as one wants to rule out arbitrage opportunities.
Our structural assumption is that X is an Ito semimartingale on some filtered space(Ω,F , (Ft)t≥0,P), which means that it can be written as
Xt = X0 +∫ t
0bsds +
∫ t
0σs dWs +
∫ t
0
∫κ δ(s, z)(µ− ν)(ds, dz)
+∫ t
0
∫κ′ δ(s, z)µ(ds, dz). (2.1)
where W and µ are a 2-dimensional Wiener process and a Poisson random measure on[0,∞)×E with (E, E) an auxiliary measurable space, on the space (Ω,F , (Ft)t≥0,P) andthe predictable compensator (or intensity measure) of µ is ν(ds, dz) = ds⊗λ(dz) for somegiven finite or σ-finite measure λ on (E, E). Above, b is a 2-dimensional adapted process,σ is a 2 × 2-dimensional adapted process, and δ is a 2-dimensional predictable functionon Ω×R+ ×E. Moreover κ is a continuous truncation function on R2, that is a functionfrom R2 into itself with compact support and κ(x) = x on a neighborhood of 0, andκ′(x) = x− κ(x).
Of course b, σ and δ should be such that the integrals in (2.1) make sense (see forexample [11] for a precise definition of the last two integrals). However we need a bitmore than just the minimal integrability assumptions, and also the non trivial fact thatthe volatility σt is also an Ito semimartingale, of the form
σt = σ0 +∫ t
0bs ds +
∫ t
0σs dWs +
∫ t
0σ′ dW ′
s +∫ t
0
∫
Eκ δ(s, x)(µ− ν)(ds, dx)
+∫ t
0
∫
Eκ′ δ(s, x)µ(ds, dx). (2.2)
In this formula, σt (a 2×2 matrix) is considered as an R4-valued process, and W ′ is another4-dimensional Wiener process independent of (W,µ), and b and δ, resp. σ, resp. σ′ areas above, with dimensions 4, resp. 4 × 2, resp. 4 × 4, and κ is a continuous truncationfunction on R4 and κ′(x) = x− κ(x).
In order to state the assumptions needed, we need some notation. We write
∆Xs = Xs −Xs−, τ = inf(t : ∆X1t ∆X2
t 6= 0) (2.3)
for the jumps of the X process and the infimum τ of the joint jumps times of the twocomponents. Set also Γ = (ω, t, x) : δ1(ω, t, x)δ2(ω, t, x) 6= 0 and, for i = 1, 2:
δ′it (ω) = ∫
(κi δ 1Γ)(ω, t, x) λ(dx) if the integral makes sense+∞ otherwise
(2.4)
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Assumption (H): (a) All paths t 7→ bt(ω) are locally bounded.
(b) All paths t 7→ bt(ω), t 7→ σt(ω), t 7→ σ′t(ω), t 7→ δ(ω, t, x) and t 7→ δ(ω, t, x) areleft-continuous with right limits.
(c) All paths t 7→ supx∈E|δ(ω,t,x)|
γ(x) and t 7→ supx∈E|δ(ω,t,x)|
γ(x) are locally bounded, whereγ is a (non-random) nonnegative function satisfying
∫E(γ(x)2 ∧ 1) λ(dx) < ∞.
(d) All paths t 7→ δ′it (ω) for i = 1, 2 are left-continuous with right limits on the interval[0, τ(ω)).
(e) We have∫ t0 |σs| ds > 0 a.s. for all t > 0. 2
The non-degeneracy condition (e) says that almost surely the continuous martingalepart of X is not identically 0 on any interval [0, t] with t > 0. It could be weakened,and in any case this condition is satisfied in all applications we have in mind. Apartfrom this non-degeneracy condition, Assumption (H) accommodates virtually all modelsfor stochastic volatility, including those with jumps, and allows for correlation betweenthe volatility and the asset price processes. For example if we consider a d-dimensionalequation
dYt = f(Yt−)dZt (2.5)
where Z is a multi-dimensional Levy process, and f is a C2 function with at most lineargrowth, then any of the components of Y satisfies Assumption (H) except perhaps (e).The same holds for more general equations driven by a Wiener process and a Poissonrandom measure.
Remark 2.1 Condition (d) above is implied by the others when∫
(γ(x) ∧ 1)λ(dx) < ∞,which essentially amounts to saying that (and implies that) the jumps of X are summable,i.e.
∑s≤t ‖∆Xs‖ < ∞ a.s. Otherwise it may appear as a strong assumption because we
can have δ′it (ω) = δ′′it (ω) = ∞ for “most” (ω, t). However, if At =∫ t0
∫1Γ(s, x) µ(ds, dx)
then Aτ ≤ 1 by construction, and by definition of the predictable compensator we also haveE
( ∫ τ0 ds
∫1Γ(s, x) λ(dx)
)= E(Aτ ) ≤ 1. Hence obviously
∫ τ0 |δ′is |ds) < ∞ a.s. Therefore
|δ′it (ω)| < ∞ for P(dω) ⊗ dt-almost all (ω, t) such that t ≤ τ(ω). Hence (d) is indeed arather weak technical assumption, similar to saying that bt is right-continuous with leftlimits instead of the “minimal” assumption saying that
∫ t0 ‖bs‖ ds < ∞ a.s. 2
In some of the results below we do not need that σt has the form (2.2) but simplythat it is right-continuous with left limits, or sometimes (d) or (e) are not needed. Forsimplicity we assume the full force of (H) throughout the whole paper.
3 The test statistics.
3.1 Preliminaries.
Recall that our process X is observed over a given time interval [0, T ], at times i∆n for alli = 0, 1, · · · , [T/∆n]. We cannot of course do any better than if the process were observed
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“continuously” over [0, T ], that is we can at the best decide in which of the following threesets Ω(j)
T (for “joint jumps”) or Ω(d)T (for “disjoint jumps”) or Ω(c)
T (for ”continuous”) belowthe particular “observed” outcome ω lies:
Ω(j)T = ω : on [0, T ] the process ∆X1
s ∆X2s is not identically 0
Ω(d)T = ω : on [0, T ] the processes ∆X1
s and ∆X2s are not identically 0,
but the process ∆X1s ∆X2
s isΩ(c)
T = ω : on [0, T ] at least one of X1 and X2 is continuous.
(3.1)
That is, under a “complete” observation of the path we cannot decide whether the actualmodel allows for joint jumps or not, but only that the observed path has this property. Ofcourse if we decide that the observed path has joint jumps then the model should allowfor this, but in the other case the model can still allow for joint jumps.
These three sets are disjoint and form a partition of Ω; however we may very well have
P(Ω(j)T ) > 0, P(Ω(d)
T ) > 0, P(Ω(c)T ) > 0, (3.2)
at least in the finite activity case for jumps, e.g. when λ is a finite measure. When bothcomponents have infinite activity we have P(Ω(c)
T ) = 0, but the first two probabilities in(3.2) may still be both positive.
A comprehensive testing procedure should encompass all three kinds of outcomes.However, using the procedure established in [1] (or other methods), we can in principledecide whether we are in Ω(c)
T or not. Here, we assume that this preliminary testing hasbeen performed. If the conclusion is that we are in Ω(c)
T then of course the procedure isended. Otherwise, we have to decide between Ω(j)
T and Ω(d)T : this is the aim of this paper.
In a first case, we set the null hypothesis to be “joint jumps”, that is that we are in Ω(j)T .
We will take a critical (rejection) region C(j)n at stage n, to be defined later, which should
depend only on the observations Xi∆n . Exactly as in [1], we do a kind of “conditional”test. Note that although X, hence Ω(j)
T as well, depend on the triple of coefficients (b, σ, δ)belonging to the set H of all coefficients satisfying Assumption (H), the observations Xi∆n
and thus C(j)n do not depend on (b, σ, δ) explicitly (the probability of C
(j)n does depend on
this triple, though).
Then, with obvious notation, we take the following as our definition of the asymptoticlevel, for a given triple of coefficients:
α(j) = sup(
lim supn
P(C(j)n | A) : A ∈ F , A ⊂ Ω(j)
T
). (3.3)
Here P(C(j)n | A) is the usual conditional probability with respect to the set A, with the
convention that it vanishes if P(A) = 0. If P(Ω(j)T ) = 0 then α(j) = 0, which is a natural
convention since in this case we want to reject the assumption whatever the outcome ωis. It would seem better to define the level as the essential supremum α′(j) (in ω) overΩ(j)
t of lim supn P(C(j)n | F); the two notions are closely related and α′(j) ≥ α(j), but we
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cannot exclude a strict inequality here, whereas we have no way (so far) to handle α′(j)t .
Note that α(j) features some kind of ”uniformity” over all subsets A ⊂ Ω(j)t .
As for the asymptotic power function, we define it as
β(j) = lim infn
P(C(j)n | F) restricted to Ω(d)
T , (3.4)
and we will see that we can handle this quantity (which, unlike α(j), is a random variable(defined on the whole of Ω, but its domain of interest is Ω(d)
T ).
In the second case we set the null hypothesis to be “disjoint jumps”, that is that weare in Ω(d)
T . We take a critical region C(d)n at stage n, again to be defined later, and the
asymptotic level and power for the triple of coefficients (b, σ, δ) in H are
α(d) = sup(
lim supn P(C(d)n | A) : A ∈ F , A ⊂ Ω(d)
T
),
β(d) = lim infn P(C(d)n | F) restricted to Ω(j)
T .
(3.5)
3.2 Construction of the critical regions.
We first need some notation. For any Borel function f on R2 we write
∆ni X = Xi∆n −X(i−1)∆n
, V (f, ∆n)t =[t/∆n]∑
i=1
f(∆ni X). (3.6)
Three functions will be of particular interest:
f(x) = (x1x2)2, g1(x) = (x1)4, g2(x) = (x2)4. (3.7)
The critical regions C(j)n and C
(d)n will be based upon the following two test statistics:
Φ(j)n =
V (f, k∆n)T
V (f, ∆n)T, Φ(d)
n =V (f, ∆n)T√
V (g1,∆n)T V (g2, ∆n)T
. (3.8)
Here, k is an integer not less than 2 (typically k = 2 or k = 3), which is fixed throughout.Note that Φ(j)
n depends on k, and both Φ(j)n and Φ(d)
n depend on T .
The asymptotic behavior of these two statistics is crucial, and in order to give adescription of it we need the notion of stable convergence in law, for which we refer forexample to [11]. We also need some (cumbersome) further notation to describe the limits.
Recall that (H) is assumed. We denote by (Sq)q≥1 a sequence of stopping times whichexhausts the “jumps” of the Poisson measure µ: this means that for each ω we haveSp(ω) 6= Sq(ω) if p 6= q, and that µ(ω, t × E) = 1 if and only if t = Sq(ω) for someq. There are many ways of constructing those stopping times, but it turns out that whatfollows does not depend on the specific description of them. Next, we consider an auxiliaryspace (Ω′,F ′,P′) which supports a number of variables and processes:
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• four sequences (Uq), (U ′q), (U q), (U ′
q) of 2-dimensional N(0, I2) variables;
• a sequence (κq) of uniform variables on [0, 1];
• a sequence (Lq) of uniform variables on the finite set 0, 1, · · · , k − 1, where k ≥ 2is some fixed integer;
and all these variables are mutually independent. Then we put
Ω = Ω× Ω′, F = F ⊗ F ′, P = P⊗ P′. (3.9)
We extend the variables Xt, bt, ... defined on Ω and Un, κn,... defined on Ω′ to the productΩ in the obvious way, without changing the notation. We write E for the expectation w.r.t.P. Finally, we let (Ft) be the smallest (right-continuous) filtration of F containing thefiltration (Ft) and such that Un, U ′
n, κn and Ln are FSn-measurable for all n. Obviously,µ is still a Poisson measure with compensator ν, and W and W ′ are still Wiener processeson (Ω, F , (Ft)t≥0, P). Finally we define the 2-dimensional variables
Rq = √κq σSq−Uq +
√1− κq σSqU
′q
R′q =
√Lq σSq−U q +
√k − 1− Lq σSqU
′q
R′′q = Rq + R′
q
(3.10)
Let us next define some auxiliary processes. Below, we write ct = σtσ?t (the diffusion
matrix of X). Set
Bt =∑
s≤t
(∆X1s )2 (∆X2
s )2, B′1t =
∑
s≤t
(∆X1s )4, B′2
t =∑
s≤t
(∆X2s )4, (3.11)
Ct =∫ t
0(c11
s c22s + 2(c12
s )2) ds, (3.12)
Dt =∑
q:Sq≤t
((∆X1
SqR2
q)2 + (∆X2
SqR1
q)2)
D′′t =
∑q:Sq≤t
((∆X1
SqR′′2
q )2 + (∆X2Sq
R′′1q )2
).
(3.13)
Gt = 2∑
q:Sq≤t
∆X1Sq
∆X2Sq
(∆X1
SqR′2
q + ∆X2Sq
R′1q
). (3.14)
The following theorem gives us the asymptotic behavior of our two test statistics, onthe union Ω(j)
T ∪ Ω(c)T . As said before, we supposedly know that we are not in Ω(c)
T , so thebehavior of the statistics on this set is of no importance for us. Recall that (H) is assumedthroughout.
Theorem 3.1 a) We have
Φ(d)n
P−→
BT /√
B′1T B′2
T > 0 on Ω(j)T
0 on Ω(d)T .
(3.15)
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b) We haveΦ(j)
nP−→ 1 on Ω(j)
T , (3.16)
and Φ(j)n converges stably in law, in restriction to the set Ω(d)
T , to a variable which is a.s.different from 1 and given by
Φ =D′′
T + kCT
DT + CT. (3.17)
The last claim means that E(h(Φ(j)n )Y 1
Ω(d)T
) → E(h(Φ)Y 1Ω
(d)T
) for all bounded F-measurable variables Y and all bounded continuous functions h on R. This theorem doesnot use the full force of (H), and in particular σt does not need to satisfy (2.2), althoughit should be right-continuous with left limits.
Of course, if any one of the two sets Ω(j)T or Ω(d)
T has a vanishing probability, thecorresponding statement above is empty.
As a consequence, we are lead to take critical regions of the form C(j)n = |Φ(j)
n −1| ≥ εnor C
(d)n = Φ(d)
n ≥ εn for suitable - and possibly random - sequences εn. However, todetermine the level of such tests we need to go a bit further and give a central limit theoremassociated with the convergences established in Theorem 3.1, at least in restriction to Ω(j)
T
for Φ(j)n and to Ω(d)
T for Φ(d)n .
4 Central limit theorems.
We have a genuine CLT for Φ(j)n , on Ω(j)
T . We do not really have it for Φ(d)n on Ω(d)
T , butit is replaced by the stable convergence in law towards a positive random variable, similarto the convergence in (3.17).
4.1 CLT for the non-standardized statistics.
We need first to complement the notation (3.11)-(3.14). Set
Ft =12
∑
s≤t
((∆X1
s )2(c22s− + c22
s ) + (∆X2s )2(c11
s− + c11s )
), (4.1)
F ′t = 2
∑
s≤t
(∆X1s ∆X2
s )2((∆X2
s )2(c11s−+ c11
s )+ (∆X1s )2(c22
s−+ c22s )+2∆X1
s ∆X2s (c12
s−+ c12s )
),
(4.2)
Theorem 4.1 a) In restriction to the set Ω(j)T the sequence 1√
∆n(Φ(j)
n −1) converges stablyin law to the variable Ψ = GT /BT which, conditionally on F , is centered with variance
E(Ψ2 | F) = (k − 1)F ′T /(BT )2, (4.3)
and is even Gaussian conditionally on F if the processes X and σ have no common jumps.
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b) In restriction to the set Ω(d)T the sequences 1
∆nΦ(d)
n converges stably in law to the
positive variable Φ′ = (DT + CT )/√
B′1T B′2
T which, conditionally on F , satisfies
E(Φ′ | F) = (FT + CT )/√
B′1T B′2
T . (4.4)
4.2 Some consistent estimators.
To evaluate the level of tests based on the statistic Φ(j)n or Φ(d)
n we need consistent esti-mators for the asymptotic mean or variance obtained in Theorem 4.1. That is, we needto estimate F ′
T and BT , resp. FT , CT , B′1T and B′2
T , on the set Ω(j)T , resp. Ω(d)
T .
For BT , B′1T and B′2
T a simple extension of [9], which is also used for the first part of(3.15) (see Subsection 8.2), gives us that
V (f, ∆n)TP−→ BT , V (g1, ∆n)T
P−→ B′1T , V (g2, ∆n)T
P−→ B′2T . (4.5)
For CT we can use multipower variations or truncated powers. This gives rise to twoalternative estimators:
A(∆n)T =π2
4∆n
[T/∆n]−3∑
i=1
(|∆n
i X1∆ni+1X
1∆ni+2X
2∆ni+3X
2|
+18|∆n
i (X1 + X2)∆ni+1(X
1 + X2))∆ni+2(X
1 + X2)∆ni+3(X
1 + X2)|
+18|∆n
i (X1 −X2)∆ni+1(X
1 −X2))∆ni+2(X
1 −X2)∆ni+3(X
1 −X2)|
−14|∆n
i (X1 + X2)∆ni+1(X
1 + X2))∆ni+2(X
1 −X2)∆ni+3(X
1 −X2)|)
(4.6)
A′(∆n)T =1
∆n
[T/∆n]∑
i=1
f(∆ni X)1‖∆n
i X‖≤α∆$n , (4.7)
where, for the second one, we choose α > 0 and $ ∈ (0, 12) arbitrarily.
For FT and F ′T things are more complicated, and we do as in [9]: take any sequence
kn of integers satisfyingkn →∞, kn∆n → 0, (4.8)
and then let In,−(i) = i−kn, i−kn+1, · · · , i−1 if i > kn and In,+ = i+1, i+2, · · · , i+kndefine two local windows in time of length kn∆n just before and just after time i∆n. Thenwe set for m, l = 1, 2 and 1 + kn ≤ i ≤ [t/∆n]− kn:
c(n,−)mli = 1
kn∆n
∑j∈In,−(i) ∆n
j Xm∆nj X l 1‖∆njX‖≤α∆$
n c(n, +)ml
i = 1kn∆n
∑j∈In,+(i) ∆n
j Xm∆nj X l 1‖∆n
j X‖≤α∆$n .
(4.9)
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Those are ”estimates” of the diffusion matrix ct on the left and on the right of time i∆n,respectively. Then the desired estimators are
F (∆n)t =1
2kn∆n
[t/∆n]−kn∑
i=1+kn
∑
j∈In,−(i)∪In,+(i)
((∆n
i X1)2(∆nj X2)2
+(∆ni X2)2(∆n
j X1)2)1‖∆n
i X‖>α∆$n , ‖∆n
j X‖≤α∆$n , (4.10)
F ′(∆n)t =2
kn∆n
[t/∆n]−kn∑
i=1+kn
∑
j∈In,−(i)∪In,+(i)
(∆ni X1)2(∆n
i X2)2
(∆n
i X1∆nj X2 + ∆n
i X2∆nj X1
)21‖∆n
j X‖≤α∆$n , ‖∆n
j X‖≤α∆$n . (4.11)
The following theorem establishes the consistency of these estimators:
Theorem 4.2 Let α > 0 and $ ∈ (0, 1/2). Then
A(∆n)TP−→ CT , (4.12)
∆nA′(∆n)TP−→ 0, A′(∆n)T
P−→ CT in restriction to Ω(d)T . (4.13)
F (∆n)TP−→ FT , F ′(∆n) P−→ F ′
T . (4.14)
The above is not quite enough for deriving the asymptotic size of a test (that is,the actual asymptotic level: they systematically give rise to tests with a size strictlysmaller, and often quite significantly smaller, than the prescribed level), except in case(a) of Theorem 4.1 when X and σ do not jump together. We need in fact a sort of”estimate” for the distribution of the variables GT and DT defined on the extended space,and conditionally on F . For this we first denote by σ(n,±)i an arbitrary (measurable)square-root of the matrix c(n,±)i of (4.9), and we define the 2-dimensional variables
R(n)i =√
κi σ(n,−)iUi +√
1− κi σ(n, +)iU′i
R′(n)i =√
Li σ(n,−)iU i +√
k − 1− Li σ(n,+)iU′i
(4.15)
(the variables (κi, Li, Ui, U′i , U i, Y
′i are the ones defined before (3.9)). Finally, on the
extended space we define the following processes:
D(∆n)t =[t/∆n]−kn∑
i=1+kn
((∆n
i X1 R(n)2i )2 + (∆n
i X2 R(n)1i )2)1‖∆n
i X‖>α∆$n , (4.16)
G(∆n)t = 2[t/∆n]−kn∑
i=1+kn
∆ni X1∆n
i X2(∆n
i X1 R′(n)2i +∆ni X2 R′(n)1i
)1‖∆n
i X‖>α∆$n . (4.17)
10
Theorem 4.3 Assume that we have a sequence Zn of positive variable going in probabilityto some variable Z > 0, on the space (Ω,F ,P). Then
P(|G(∆n)T | > Zn | F) P−→ P(|GT | > Z | F), (4.18)
P(D(∆n)T > Zn | F) P−→ P(DT > Z | F). (4.19)
4.3 CLT for the standardized statistics.
Combining Theorems 4.1 and 4.2, and in view of the properties of the stable convergencein law, we immediately get the following (at this stage this needs no proof):
Theorem 4.4 a) With
V (j)n =
√∆n (k − 1)F ′(∆n)T
V (f, ∆n)T, (4.20)
the variables (Φ(j)n − 1)/V
(j)n converge stably in law, in restriction to the set Ω(j)
T , to avariable which, conditionally on F , is centered with variance 1, and is even N (0, 1) if theprocesses X and σ have no common jumps.
b) With
V (d)n =
∆n(D(∆n)T + A(∆n)T )√V (g1, ∆n)T V (g2, ∆n)T
, V ′(d)n =
∆n(D(∆n)T + A′(∆n)T )√V (g1, ∆n)T V (g2, ∆n)T
, (4.21)
the variables Φ(d)n /V
(d)n and Φ(d)
n /V′(d)n converge stably in law, in restriction to the set Ω(d)
T ,to a positive variable which, conditionally on F , has expectation 1.
Another consequence of Theorems 4.1 and 4.3 is the following:
Theorem 4.5 Let Zn and Z be as in Theorem 4.3, and A ∈ F .
a) If A ⊂ Ω(j)T , we have
P(A ∩
|Φ(j)n − 1|V (f, ∆n)T√
(k − 1)∆n
> Zn
)→ P(A ∩ |GT | > Z). (4.22)
b) If A ⊂ Ω(d)T , and with either An = A(∆n)T or An = A′(∆n)T , we have
P(A ∩
Φ(d)n
√V (g1, ∆n)T V (g2, ∆n)T
∆n> Zn + An
)→ P(A ∩ DT > Z). (4.23)
5 Testing for common jumps.
We now use the preceding results to construct actual tests, either for the null hypothesisthat there are jumps but no common jumps for the two components of X, or for the nullhypothesis that there are common jumps.
11
5.1 When there are common jumps under the null hypothesis.
In a first case, we set the null hypothesis to be “common jumps”, that is we are in Ω(j)T .
For this we use the test statistics Φ(j)n and in view of (3.16) and (3.17) we associate the
critical region of the formC(j)
n = |Φ(j)n − 1| ≥ c(j)
n (5.1)
for some sequence c(j)n > 0, possibly even a random sequence, but in that case depending
only on the observations Xi∆n .
As usual, we fix a level α ∈ (0, 1), and wish to find c(j)n so that (5.1) asymptotically
achieves this level, that is for which α(j) ≤ α and of course α(j) = α if possible.
If we know that X and σ do not jump together, then (a) of Theorem 4.4 allows toachieve α(j) = α, and for this we need the α-absolute quantile of N (0, 1), that is thenumber zα such that P(|U | ≥ zα) = α for an N(0, 1) variable U . Otherwise we may relyon Bienayme-Tchebycheff inequality to construct a test for which α(j) ≤ α. Or, we canmake use of (a) of Theorem 4.5), in the following way:
Recall that at stage n we know the variables given by (4.9). Then we can use aMonte-Carlo procedure to simulate Nn copies of the variables R(n)i of (4.15) (that is,we simulate Nn copies of the variables (κi, Ui, U
′i)1≤i≤[T/∆n], and use the same observed
variables σ(n±)i always). Plugging these in (4.17), and again with the same observedincrements ∆n
i X, we thus obtain Nn copies (G(∆n; j)T : 1 ≤ j ≤ Nn). Then we take theorder statistics for the absolute values |Gn,1| ≥ |Gn,2| ≥ · · · ≥ |Gn,Nn | for this family, andwe set
Z(j)n (α) = |Gn,[(1−α)Nn]|, (5.2)
that is the α-absolute quantile of the empirical distribution of the family (G(∆n; j)T : 1 ≤j ≤ Nn). With this notation we construct three slightly different tests:
Theorem 5.1 a) Assume that the two processes X and σ do not jump together. If we set
c(j)n = zαV (j)
n , (5.3)
where V(j)n is given by (4.20), then the asymptotic level and power of the critical region
defined by (5.1) for testing the null hypothesis “common jumps” satisfy
α(j) ≤ α, β(j) = 1 on the set Ω(d)T , (5.4)
and if further P(Ω(j)T ) > 0 we have α(j) = α, and even
A ⊂ Ω(j)T , P(A) > 0 ⇒ P(C(j)
n | A) → α. (5.5)
b) If we setc(j)n = V (j)
n /√
α, (5.6)
where V(j)n is given by (4.20), then the asymptotic level and power of the critical region
defined by (5.1) for testing the null hypothesis “common jumps” satisfy (5.4).
12
c) Take a sequence Nn →∞, define Z(j)n (α) by (5.2), and set
c(j)n = Z(j)
n (α)
√(k − 1)∆n
V (f, ∆n)T. (5.7)
Then the asymptotic level and power of the critical region defined by (5.1) for testing thenull hypothesis “common jumps” satisfy (5.4). If further P(Ω(j)
T ) > 0 we have α(j) = α,and even (5.5).
Clearly (a) is preferable if it can be used, and otherwise (c) is preferable. The choice ofthe sequence Nn going to infinity is, asymptotically, arbitrary; but in practice n is givenand Nn should be big enough to have a good approximation of the ”true” α-quantile of theF-conditional distribution of G(∆n)T : this of course depends on α, and taking for exampleNn = 1000/α seems to be a reasonable choice (if α = 0.05, this means Nn = 20000; thislooks as a big number, but the simulation of our Nn = 20000 copies take only a few secondswhen the number of observations [T/∆n] is about 1000).
5.2 When there are no common jumps under the null hypothesis.
In a second case, we set the null hypothesis to be that “no common jumps”, that is weare in Ω(d)
T . We take the critical region to be
C(d)n = Φ(d)
n ≥ c(d)n (5.8)
for some sequence c(d)n > 0.
Here we have two ways for choosing c(d)n : we can use (b) of Theorem 4.4 and Bienayme-
Tchebycheff inequality, or we can use (b) of Theorem 4.5 and proceed as in the previoussubsection: we simulate Nn copies of D(∆n)T , giving rise to the order statistics Dn,1 ≥Dn,2 ≥ · · · ≥ Dn,Nn of this family, and we set
Z(d)n (α) = Dn,[α/Nn]. (5.9)
Theorem 5.2 a) If we setc(d)n = Vn/α, (5.10)
where Vn is either V(d)n or V
′(d)n , as given by (4.21), then the asymptotic level and power
of the critical region defined by (5.8) for testing the null hypothesis “no common jumps”satisfy
α(d) ≤ α, β(d) = 1 on the set Ω(j)T . (5.11)
b) Take a sequence Nn →∞, define Z(d)n (α) by (5.9), and put either An = A(∆n)T or
An = A′(∆n)T . Then if
c(d)n = (Z(j)
n (α) + An)∆n√
V (g1, ∆n)T V (g2,∆n)T
, (5.12)
13
the asymptotic level and power of the critical region defined by (5.8) for testing the nullhypothesis “no common jumps” satisfy (5.11). If further P(Ω(d)
T ) > 0 we have α(d) = α,and even
A ⊂ Ω(d)T , P(A) > 0 ⇒ P(C(d)
n | A) → α. (5.13)
Again, here (b) seems preferable to (a), and the simulation results given below stronglysuggest that one should use (b).
Remark 5.3 The test statistics above are insensitive to the scales used to measure X1 andX2: if we multiply each component Xi by a constant λi, the test statistics are unchanged.
However, this is not true of the standardized versions, because of the truncation α∆$n
which we use for the modulus ‖∆ni X‖ of the increments. This presupposes that both
components X1 and X2 have increments with roughly the same order of magnitude. Ifthis is not true, we should either first multiply the first component X1, say, by a suitableconstant, in such a way that the averages of |∆n
i X1| and of |∆ni X2| (or the averages after
deleting, say, the 10% biggest increments) are close one to the other; or, we can use twodifferent levels of truncation, that is replace the set ‖∆n
i X‖ ≤ α∆n by |∆ni X1| ≤
α1∆$n , |∆n
i X2| ≤ α2∆$n .
Remark 5.4 In fact, the choice of the truncation level α∆$n , in order to put our tests in
use, is a difficult one: asymptotically this choice does not matter, but in practice it doesmatter a lot. The idea is that α∆$
n should be bigger, but not too much, than ”most” ofthe increments when there is no jump, or no big jump: those increments being of order ofmagnitude ‖σt∆n
i W‖ with (i − 1)∆n ≤ t ≤ i∆n. A good choice, supported by empiricalevidence coming from simulation, seems to be $ = .48 or $ = .49, and α being of about3 or 4 times the ”average” value of ‖σt‖. The latter is unknown, but usually one has agood idea of its order of magnitude.
6 Simulation results.
In this section we check the performance of our tests on simulated data. In the simulationstudy we work with the following simple model for the prices
dX1t = X1
t σ1dW 1t + α1
∫
RX1
t−x1µ1(dt, dx1) + α3
∫
RX1
t−x3µ3(dt, dx3)
dX2t = X2
t σ2dW2t + α2
∫
RX2
t−x2µ2(dt, dx2) + α3
∫
RX2
t−x3µ3(dt, dx3), (6.1)
where cor(W 1, W 2) = ρ; the Poisson measures µ1, µ2 and µ3 are independent with com-pensators νi(dt, dxi) = λi
1(xi∈[−hi;−li]∪[li;hi])2(hi−li)
dtdxi for li < hi and i = 1, 2, 3. The initialvalues are X1
0 = X20 = 1.
In the Monte Carlo the observation length is one day, i.e. T = 1, consistent withthe literature on testing for jumps in individual series. We simulate from the above-given
14
process for a total of 5000 days. Since we are interested in the behavior of the tests on thesets Ω(j)
T and Ω(d)T , in the simulation we discard days on which there is no jump in either
of the two series. On each day we consider sampling at the following frequencies n = 100,n = 1600 and n = 25600, corresponding approximately to sampling every 5 minutes, 30seconds and 1 second for a trading day of 6.5 hours. In each simulation we compute theraw statistics Φ(j)
n and Φ(d)n as well as their standardized versions defined as:
T (j)n =
Φ(j)n − 1
V(j)n
, T (d)n =
Φ(d)n
V(d)n
,
where we use the notation (4.20) and (4.21). For the calculation of V(j)n and V
(d)n we use a
local window kn = 1/√
∆n and truncation level of αδ$n = 0.03× [δ0.49
n ]. For the calculationof CT we use the truncated power estimator A′(∆n)T defined in (4.7).
In Table 6 we report the parameter values for all cases considered. In all simulationscenarios σ2
1 = σ22 = 8 × 10−5 and therefore we do not report these values in the Table.
In all cases the variance of the jumps in the total price variation is kept constant at aproportion of 20%. Thus scenarios with higher on average number of jumps automaticallyimply that the jumps are of smaller in size. The different parameter settings differ in theaverage number of jumps (respectively their size), whether jumps arrive together or not,and in the correlation between the continuous components of the price.
Table 1: Parameter Setting for the Monte Carlo
Case Parameters
ρ α1 λ1 l1 h1 α2 λ2 l2 h2 α3 λ3 l3 h3
I-j 0 0.00 0.00 0.01 1 0.05 0.7484II-j 0 0.00 0.00 0.01 5 0.05 0.3187III-j 0 0.00 0.00 0.01 25 0.05 0.1238I-d0 0 0.01 1 0.05 0.7484 0.01 1 0.05 0.7484II-d0 0 0.01 5 0.05 0.3187 0.01 5 0.05 0.3187III-d0 0 0.01 25 0.05 0.1238 0.01 25 0.05 0.1238I-d1 1 0.01 1 0.05 0.7484 0.01 1 0.05 0.7484II-d1 1 0.01 5 0.05 0.3187 0.01 5 0.05 0.3187III-d1 1 0.01 25 0.05 0.1238 0.01 25 0.05 0.1238
On Figures 1 and 2 we plot the histograms of the raw statistics Φ(d)n and Φ(j)
n under thedifferent scenarios. On Figures 3 and 4 we plot the rejection curves of the standardizedtests T
(d)n and T
(j)n . For the calculation of the critical levels of the tests we use Theorem 5.2,
part(b) and Theorem 5.1, part(a) respectively. We summarize our findings from the MonteCarlo in the following:
15
0.9 0.95 10
100
200
300Φd , case I−j
0.8 0.9 10
50
100
150Φd , case II−j
0 0.5 10
20
40
60Φd , case III−j
0 0.05 0.10
100
200
300Φd , case I−d0
0 0.1 0.20
100
200
300Φd , case II−d0
0 0.5 10
50
100
150Φd , case III−d0
0 0.1 0.20
100
200
300Φd , case I−d1
0 0.2 0.40
50
100
150Φd , case II−d1
0 0.5 10
20
40
60Φd , case III−d1
Figure 1: Histograms of Φ(d)n in the different cases. The dashed line corresponds to sam-
pling frequency of n = 100, the dotted line to sampling frequency of n = 1600 and thesolid line to sampling frequency of n = 25600.
16
0 1 2 30
5
10
15Φj , case I−j
0 1 2 30
5
10Φj , case II−j
0 1 2 30
5
10
15Φj , case III−j
0 2 4 60
0.5
1Φj , case I−d0
0 2 4 60
0.5
1
1.5Φj , case II−d0
0 2 40
1
2
3Φj , case III−d0
0 2 4 60
0.5
1
1.5
2Φj , case I−d1
0 2 40
1
2
3Φj , case II−d1
0 2 40
2
4
6Φj , case III−d1
Figure 2: Histograms of Φ(j)n in the different cases. The dashed line corresponds to sam-
pling frequency of n = 100, the dotted line to sampling frequency of n = 1600 and thesolid line to sampling frequency of n = 25600.
17
0 0.5 10
0.5
1Td test, case I−j
0 0.5 10
0.5
1Td test, case II−j
0 0.5 10
0.5
1Td test, case III−j
0 0.5 10
0.5
1Td test, case I−d0
0 0.5 10
0.5
1Td test, case II−d0
0 0.5 10
0.5
1Td test, case III−d0
0 0.5 10
0.5
1Td test, case I−d1
0 0.5 10
0.5
1Td test, case II−d1
0 0.5 10
0.5
1Td test, case III−d1
Figure 3: Size and power of the test T(d)n (critical values computed using Theorem 5.2,
part(b)). The x-axis shows the nominal level of the corresponding test, while the y-axisshows the percentage of rejection in the Monte Carlo. The dashed line corresponds tosampling frequency of n = 100, the dotted line to sampling frequency of n = 1600 and thesolid line to sampling frequency of n = 25600.
18
0 0.5 10
0.5
1Tj test, case I−j
0 0.5 10
0.5
1Tj test, case II−j
0 0.5 10
0.5
1Tj test, case III−j
0 0.5 10
0.5
1Tj test, case I−d0
0 0.5 10
0.5
1Tj test, case II−d0
0 0.5 10
0.5
1Tj test, case III−d0
0 0.5 10
0.5
1Tj test, case I−d1
0 0.5 10
0.5
1Tj test, case II−d1
0 0.5 10
0.5
1Tj test, case III−d1
Figure 4: Size and power of the test T(j)n (critical values computed using Theorem 5.1,
part(a)). The x-axis shows the nominal level of the corresponding test, while the y-axisshows the percentage of rejection in the Monte Carlo. The dashed line corresponds tosampling frequency of n = 100, the dotted line to sampling frequency of n = 1600 and thesolid line to sampling frequency of n = 25600.
19
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case I−d0
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case II−d0
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case III−d0
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case I−d1
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case II−d1
0 0.2 0.4 0.6 0.8 10
0.5
1Td , case III−d1
Figure 5: Size of the test T(d)n using critical values computed via the simulation approach
in Theorem 5.2, part (b). The x-axis shows the nominal level of the corresponding test,while the y-axis shows the percentage of rejection in the Monte Carlo. The dashed linecorresponds to sampling frequency of n = 100, the dotted line to sampling frequency ofn = 1600 and the solid line to sampling frequency of n = 25600.
20
• Testing the null of common jumps. Under the null hypothesis Φ(j)n is concen-
trated around 1, with more dispersion from this value (and slight upward bias) forless frequent sampling and settings with higher number of (smaller) jumps. Underthe alternative hypothesis (disjoint) jumps, Φ(j)
n is concentrated around 2 as expectedfrom the result in Theorem 3.1, part (b). Comparing the case of zero correlationbetween the Brownian motions with that of perfect correlation we see that the Φ(j)
n
is more concentrated around 2 for the case of perfect correlation. Turning to thetesting of the null of common jumps, we see that T
(j)n has the right size in all scenar-
ios when the sampling frequency is n = 25600. On the other hand, for the case ofhigher number of smaller jumps, i.e. Case III-j, and sampling frequency n = 100, thetest over-rejects quite significantly. The reason is that for such sampling frequency,the relatively small jumps are hard to disentangle from the Brownian moves, whichin turn are independent. Finally, the second and third columns of Figure 4 revealthat T
(j)n has an excellent power against all considered alternatives.
• Testing the null of disjoint jumps. Under the null hypothesis, consistent withthe asymptotic results, Φ(d)
n is concentrated around zero. Upward bias appears whenthe sampling frequency is low (n = 100), the number of jumps is higher (withsmaller size) and the correlation between the Brownian motions in the prices isperfect. Under the alternative hypothesis Φ(d)
n takes values closer to 1. Turning tothe standardized test T
(d)n , we see that using the Bienayme-Tchebycheff inequality
in Theorem 5.2, part (b) leads to a significant underestimation of the size of thetest. This holds true for all considered simulation scenarios. On the other hand thefirst column in Figure 3 shows that the test has good power against the “commonjumps” alternatives considered in the Monte Carlo.
The results in Figure 3 suggest that it is recommendable to use the simulation approachin Theorem 5.2, part (c) to determine the critical values for testing the null of disjointjumps. We did this under the null hypothesis only and reported the results in Figure 5.As seen form the Figure we do not have size distortions anymore. Only in Case III-d1, thetest over-rejects. The reason for this is mainly in the underestimation of CT . Intuitively,when the jumps are very small, many of them enter in the estimation of CT and sincethey are independent they bias CT downwards.
7 Empirical application.
In the empirical part we use high-frequency data from the foreign exchange spot marketsfor two exchange rates DM/$ and U/$. The data covers the period December 1986 throughJune 1999 for a total of 3045 trading days. In each of the days we sample every 5 minutesin the 24-hours trading day and thus in each of the trading days we have 288 returnobservations. The DM/$ exchange rate data set has been used quite extensively in recentempirical studies for testing for presence of jumps (see e.g. [2]). Here we take this analysisone step further and test if the jumps in the two studied exchange rate series arrivetogether.
21
First, we test for presence of jumps in the individual exchange rate series and focusattention on those days in the sample for which both series exhibit jumps. To be consistentwith previous studies on the same data set (see [2]) we use the test based on the differencein logarithms of the realized variance and bi-power variation. The significance level of thetest we chose is 1%. For this significance level we found 288 days with jumps in the DM/$exchange rate series and 291 days with jumps in the U/$ exchange rate series. Out of thesedays there are 40 days in which the tests indicate that both series contain jumps. In Table 2we list these days, together with the raw statistics Φ(j)
n and Φ(d)n and their p-values (which,
following the conclusions from the Monte Carlo, are computed using Theorem 5.2, part(b)and Theorem 5.1, part(c) respectively). For the testing (i.e. calculation of the p-values)we used window of size kn = δ
−1/2n , truncation level of αδ$
n = 3.0√
BV (t)[δ0.49n ] = 16
applied to each individual series2 and the estimator A′(∆n)T of CT .
Based on commonly accepted levels of significance (i.e. 5% or 1%) we can separate thedays in Table 2 in the following four categories.
• Category 1. The first category is of days in which the two tests find that thereis common arrival of jumps. We notice that the number of days on which there iscojumping is significant and the p-values associated with testing the null of disjointjumps are very low (in most cases virtually 0).
• Category 2. The second category consists of days on which both tests indicateno common arrival of jumps. These days are much fewer. However, note that wealready found quite a significant number of days on which one of the series jumpswhile the other one does not. Importantly, the days in this category illustrate thepossibility that in spite of the fact that both series exhibit jumps during the day,they do so during different parts of the day. Note that if we were making our decisionsolely on the basis of individual tests for jumps, we would have misclassified the daysin this category as days with common arrival of jumps (i.e. Category 1). Thus daysin Category 2 underline the importance of the tests developed here.
• Category 3. The third category consists of days on which the null of both testscannot be rejected - these days are quite few. The possible explanations for suchan outcome are at least two3. First, it can be the case that on these days we haveboth common jumps and disjoint jumps with the magnitude of the common jumpsfar smaller as compared with the disjoint ones. The second possible explanation isthat we have common jumps with very weak dependence. In both possible scenariosthe value of Φ(j)
n is fairly close to 1, but for the above-mentioned reasons the valueof Φ(d)
n is close to zero. In general we will need more high-frequency observations forthe T
(d)n test to gain power against such scenarios. Alternatively we can perform the
tests on different parts of the day.
• Category 4. The last category is of days on which both tests reject their null2BV (t) is the bi-power variation of the corresponding individual series. The time-varying truncation
size reflects the fact that the variance of the continuous price component varies over time.3Apart from the Type II error in one of the two tests.
22
Table 2: Empirical Results for Common Jumps
Date Φd Φj p-value p-valueΦd Φj
09/11/1987 0.9938 1.0915 0.0000 0.419412/03/1987 0.6580 1.9831 0.0000 0.034212/10/1987 0.9933 1.1446 0.0000 0.271201/05/1988 0.5809 1.6876 0.0006 0.027601/15/1988 0.0040 1.6528 0.3663 0.529202/12/1988 0.9993 0.4100 0.0000 0.003805/17/1988 0.9658 1.0155 0.0000 0.856608/09/1988 0.5575 1.8825 0.0000 0.040409/14/1988 0.9984 0.7709 0.0000 0.230410/13/1988 0.9719 0.8011 0.0000 0.279210/26/1988 0.9731 1.3649 0.0000 0.154211/04/1988 0.9909 1.0527 0.0000 0.847605/17/1989 0.9860 0.6435 0.0000 0.176808/17/1989 0.9789 2.1938 0.0000 0.000209/27/1989 0.8255 1.1780 0.0000 0.660810/06/1989 0.9628 1.0647 0.0000 0.832010/17/1989 0.9732 1.4634 0.0000 0.106807/24/1991 0.8204 3.2959 0.0000 0.000208/02/1991 0.9753 1.2296 0.0000 0.384412/16/1991 0.2766 1.9990 0.0050 0.000201/10/1992 0.8595 0.6799 0.0000 0.343206/24/1992 0.9521 1.0435 0.0000 0.869208/24/1992 0.3306 2.0512 0.0018 0.002206/04/1993 0.9188 1.1350 0.0000 0.588009/16/1993 0.1866 1.2855 0.0222 0.640204/12/1994 0.2834 1.8069 0.0343 0.008806/17/1994 0.7766 2.6949 0.0000 0.000211/21/1994 0.1306 1.6013 0.3907 0.083403/17/1995 0.2787 2.7284 0.0267 0.000205/11/1995 0.6061 1.3020 0.0002 0.511811/13/1995 0.6948 2.2415 0.0000 0.003405/30/1996 0.5180 1.5381 0.0000 0.144006/27/1996 0.1544 0.7377 0.0010 0.476807/30/1997 0.1671 2.0925 0.7727 0.000403/30/1998 0.1203 2.4733 0.7621 0.000208/13/1998 0.1566 2.5072 0.2194 0.000610/05/1998 0.4035 1.4315 0.0164 0.167801/28/1999 0.1330 1.1790 0.0367 0.652403/01/1999 0.0498 1.9657 0.1661 0.021803/26/1999 0.2648 1.7011 0.0006 0.1178
23
hypothesis. In these days the value of the Φ(d)n statistics is above 0.5, i.e. it is
relatively high, but the value of the Φ(j)n test is fairly close to 2. One possible
explanation is the following4. We have common arrival of jumps in the series (ofthe same sign), but it happens that they arrive in consecutive 5-minute periods.This will cause the test statistic to have value much larger than 1 although we havecojumping. This is a finite-sample problem and can be solved by sampling morefrequently. Observation of the high-frequency data over these days reveals that thisexplanation seems to be the likely reason for the rejection by the T
(j)n test in these
days.
Finally, if we use less conservative significance level for determining presence of jumpsin the individual series of 5%, we find 113 days in which both series exhibit jumps. Further,if we use 1% significance level for testing for common and disjoint jumps we find that 57of these days are in Category 1, 14 in Category 2, 23 in Category 3 and 19 in Category 4.Thus, overall our empirical study shows that the exchange series exhibit both days withcommon arrival of jumps as well as days where jumps arrive at different times.
8 Proofs.
8.1 Preliminaries.
We begin by showing that the processes D, D′′ and G of (3.13) and (3.14) are actuallywell defined and finite-valued, and by stating some of their basic properties.
First, the process D is well-defined and increasing, but it might a priori take the value+∞. However, by taking the F-conditional expectation and using the properties of thevariables (κq, Uq, U
′q) we get
E(Dt | F) =∑
q:Sq≤t
E′((∆X1Sq
)2(R2q)
2 + (∆X2Sq
)2(R2q)
2)
=12
∑
q:Sq≤t
((∆X1
Sq)2(c22
Sq− + c22Sq
) + (∆X2Sq
)2(c11Sq− + c11
Sq))
= Ft (8.1)
(recall (4.1)). Then we deduce in particular hat D is finite-valued, and the same argumentshows that D′′ is also finite-valued. For G things are a bit more difficult, and we state theresult in the form of a lemma:
Lemma 8.1 Let φ and ψ be two functions on R2 with φ(x) = O(‖x‖) and ψ(x) = O(‖x‖)as x → 0.
a) The processC(φ, ψ)t :=
∑
s≤t
φ(∆Xs)ψ(∆Xs)(cs− + cs) (8.2)
4Of course this might be also due to Type I error in one of the tests.
24
takes its values in the set of 2 × 2 nonnegative symmetric matrices, and it is increasingfor the strong order in this set when ψ = φ.
b) The formulas (for i = 1, 2)
Zi(φ)t =∑
q: Sq≤t
φ(∆XSq) Riq, Z ′i(φ)t =
∑
q: Sq≤t
φ(∆XSq) R′iq (8.3)
define two R2-valued processes Z(φ) and Z ′(φ), and conditionally on F the 8-dimensionalprocess (Z(φ), Z ′(φ), Z(ψ), Z ′(ψ)) is a square-integrable martingale with independent in-crements, zero mean and variance-covariance given by
E(Zi(φ)tZj(ψ)t | F) = 1
2 C(φ, ψ)ijt
E(Z ′i(φ)tZ′j(ψ)t | F) = 0
E(Z ′i(φ)tZ′j(ψ)t | F) = k−1
2 C(φ, ψ)ijt .
(8.4)
Moreover, if X and c have no common jumps, the process (Z(φ), Z ′(φ), Z(ψ), Z ′(ψ)) is aGaussian martingale, conditionally on F .
Proof. This is proved exactly as Lemma 5.10 of [9]. The increasingness of C(φ, φ) for thestrong order comes from the fact that cs and cs− are nonnegative symmetric matrices. 2
Then obviously the process G is Z ′1(f ′1) + Z ′2(f ′2), where f ′1 and f ′2 and the two firstpartial derivatives of the function f , and further with the notation (4.2),
E((Gt)2 | F) = (k − 1)F ′t . (8.5)
We end this subsection by stating a strengthened version of Assumption (H):
Assumption (SH): We have Assumption (H), and ‖bt‖+ ‖σt‖+ ‖bt‖+ |σt‖+ ‖σ′t‖ ≤ Kand ‖δ(t, x)‖ ≤ γ(x) and ‖δ(t, x)| ≤ γ(x) and also γ(x) ≤ K for some constant K.
If we suppose that any of our limiting results holds under (SH), a localization procedureallows to get it under (H) only. This procedure is described in details in [9] and we omitit here. But in all the remainder of the paper we assume that (SH) holds.
8.2 Proof of Theorem 3.1-(a).
Observe that B′1T > 0 and B′2
T > 0 on the set Ω(j)T ∪ Ω(d)
T , whereas BT > 0 on Ω(j)T and
BT = 0 on Ω(d)T . Therefore (3.15) is a trivial consequence of (4.5), which in turns comes
from the following lemma:
Lemma 8.2 If h is a continuous function on R2 such that h(x) = o(‖x‖2) as x → 0, wehave for each t > 0:
V (h,∆n)tP−→ V (h)t :=
∑
s≤t
h(∆Xs). (8.6)
25
We even have the convergence in probability (for the Skorokhod topology) of the processesV (h,∆n) towards V (h).
Proof. Since V (h) has no fixed times of discontinuity, the last statement implies (8.6).When h vanishes around the origin the result is proved exactly as in step 2 of Theorem2.2 of [9]: the dimension of X plays no role here.
Next we turn to the general case. Let ψ(u) = (2 − u)+ ∧ 1 and ψε(x) = ψ(‖x‖/ε)for x ∈ R2. We have V (h(1 − ψε), ∆n) P−→ V (h(1 − ψε)) from what precedes, andV (h(1− ψε))t → V (h)t locally uniformly in t as ε → 0 by our assumptions on h, hence itis enough to prove that
limε→0
lim supn
P(supt≤T
|V n(hψε,∆n)t| > η) = 0 ∀η > 0. (8.7)
We have |hψε(x)| ≤ θ(x1) + θ(x2), where θ is a continuous function on R with θ(y) =o(y2) as y → 0. Obviously, it is enough to prove (8.7) with V n(hψε, ∆n)t substituted withV (i)(θψε,∆n)t =
∑[t/∆n]i=1 θ(∆n
i Xi)ψ(‖∆ni Xi|/ε), for i = 1, 2. That is, we only need to
prove (8.7) in the 1-dimensional case, and this again is done in Theorem 2.2 of [9]. 2
8.3 Proof of Theorem 4.1-(a).
Part (a) of Theorem 4.1 is an easy consequence of a general result, of independent interest.
Theorem 8.3 Let φ be a C2 function on R2 satisfying φ(0) = φ′i(0) = 0 and φ′′ik(x) =o(‖x‖) as x → 0, where φ′i and φ′′ik are the first and second order partial derivatives. The2-dimensional processes
1√∆n
(V (φ,∆n)t −
∑
s≤∆n[t/∆n]
φ(∆Xs), V (φ, k∆n)t −∑
s≤k∆n[t/k∆n]
φ(∆Xs))
(8.8)
converge stably in law, on the product D(R+,R) × D(R+,R) of the Skorokhod spaces, tothe process with components
(Z1(φ′1) + Z2(φ′2), Z1(φ′1) + Z2(φ′2) + Z ′1(φ′1) + Z ′2(φ′2)
). (8.9)
We have the (stable) convergence in law of the above processes, as elements of theproduct functional space D(R+,R)2, but usually not as elements of the space D(R+,R2)with the (2-dimensional) Skorokhod topology, because a jump of X entails a jump forboth components above, but “with a probability close to j/k” the times at which thesetwo component jump differ by an amount j∆n, for j = 1, · · · , k − 1.
Proof. The proof is essentially the same as in [9] for Theorem 2.12(i) of that paper. Letus fix ε > 0. We denote by S′q = S′q(ε) the successive jump times of the Poisson processµ([0, t]× x : γ(x) > ε). Put
X(ε)t = Xt −∑
q: S′q≤t
∆XS′q .
26
Let Ωn(t, ε) be the set of all ω such that each interval [0, t] ∩ (i∆n, (i + k)∆n] contains atmost one S′q, and S′1 > k∆n, and ‖∆n
i X(ε)‖ ≤ 2ε for all i ≤ t/∆n. We define the followingvariables on each set (ik + j)∆n < S′q ≤ (ik + j + 1)∆n (with 0 ≤ j < k) as • R−(n, q) = X(ε)(ik+j)∆n
−X(ε)ik∆n , R+(n, q) = X(ε)(i+1)k∆n−X(ε)(ik+j+1)∆n
• Rnq = ∆n
ik+j+1X(ε), R′nq = R−(n, q) + R+(n, q), R′′n
q = Rnq + R′n
q .
Exactly as in [1], we have (L−(s)−→ denoting the stable convergence in law):
(Rn
q /√
∆n, R′nq /
√∆n
)q≥1
L−(s)−→(Rq, R
′q
)q≥1
. (8.10)
For any process Y we write G(Y, φ,∆n)t =∑[t/∆n]
i=1 φ(∆ni Y )−∑
s≤∆n[t/∆n] φ(∆Ys). Onthe set Ωn(T, ε) we have for all t ≤ T and for k′ = 1 and k′ = k:
G(X, φ, k′∆n)t = G(X(ε), φ, k′∆n)t + Y (ε)(φ, k′∆n)t, (8.11)
where
Y (ε)(φ,∆n)t =∑
q: S′q≤∆n[t/∆n]
(φ(∆XS′q + Rn
q )− φ(∆XS′q)− φ(Rnq )
)
Y (ε)(φ, k∆n)t =∑
q: S′q≤k∆n[t/k∆n]
(φ(∆XS′q + R′′n
q )− φ(∆XS′q)− φ(R′′nq )
) (8.12)
Since φ is C2, we have
Y (ε)(φ,∆n)t =∑
q: S′q≤∆n[t/∆n]
(∑2i=1 φ′i(∆XS′q + Rn
q ) Rn,iq − φ(Rn
q ))
Y (ε)(φ, k∆n)t =∑
q: S′q≤k∆n[t/k∆n]
(∑2i=1 φ′i(∆XS′q + R′′n
q ) R′′n,iq − φ(R′′n
q ))
(8.13)
where Rnq and R′′n
q are between ∆XS′q and ∆XS′q +Rnq and between ∆XS′q and ∆XS′q +R′′n
q
respectively. Moreover ‖φ(x)‖ = o(‖x‖3), hence by (8.10)
1√∆n
(Y (ε)(φ, ∆n), Y (ε)(φ, k∆n)
) L−(s)−→( 2∑
i=1
Z(ε)i(φ′i),2∑
i=1
(Z(ε)i(φ′i) + Z ′(ε)i(φ′i)))
(for the product topology of D(R+,R)×D(R+,R)), where Z(ε)i(φ) and Z ′(ε)i(φ) are definedby (8.3), but with the sum taken only over the S′q(ε). As ε → 0, the right side above goeslocally uniformly in time to the right side of (8.9). Hence in view of (8.11) and sinceΩn(T, ε) → Ω as n →∞ for all ε > 0 and T > 0, it remains to prove that
limε→0
lim supn
P(
supt≤T
1√∆n
|G(X(ε), φ, k′∆n)t| > η)
= 0 (8.14)
for all η > 0 and T > 0 and for k′ = 1 and k′ = k.
Now, let ψ be a C∞ function on R+ with 1[0,1] ≤ ψ ≤ 1[0,2], and put φε(x) =φ(x)
∏di=1 ψ(|xi|/2kε). Then φε is a C2 function which coincide with φ when ‖x‖ ≤ 2kε
27
and vanishes for ‖x‖ > 4kε. Hence for each T , on a set of probability going to 1 as n →∞we have G(X(ε), φ, k′∆n)t = G(X(ε), φε, k
′∆n)t for all t ≤ T . Therefore it is enough toprove (8.14) with φε instead of φ. But this is exactly the last step in the proof of Theorem2-12-(i) of [9] (in which the dimension of X plays no role; this is where the hypothesisφ′′ij(x) = o(‖x‖) is used). Hence we are done. 2
Lemma 8.4 For any function φ on R2 such that φ(x) = O(‖x‖2) as x → 0, we have
1√∆n
∑
k∆n[T/k∆n]<s≤T
|φ(∆Xs)| P−→ 0. (8.15)
Proof. Denote by Un the left side of (8.15). Since under (SH) we have |φ(δ(s, z))| ≤ Kγ(z)for some constant K, we have
E(Un) ≤ K√∆n
E
(∫ T
k′∆n[T/k∆n]ds
∫γ(z)2λ(dz)
)≤ K ′k′
√∆n
for another constant K ′, and the result follows. 2
Proof of Theorem 4.1-(a). We can apply Theorem 8.3 with φ = f . Since X has nofixed time of discontinuity, and in view of the previous lemma, we deduce that
GnT :=
1√∆n
(V (f, k∆n)T − V (f, ∆n)T
) L−(s)−→ GT .
We also have Φ(j)n − 1 =
√∆n Gn
T /V (f, ∆n), which thus converges stably in law, inrestriction to the set Ω(j)
T = BT > 0, to GT /BT by (8.6). The end of the claim followsfrom Lemma 8.1 and from (8.5). 2
8.4 Proof of Theorems 3.1-(b) and 4.1-(b).
Clearly (3.16) follows from (8.6). For (3.17) and (b) of Theorem 4.1 we will use thefollowing theorem:
Theorem 8.5 The 2-dimensional processes( 1
∆nV (f, ∆n)t,
1∆n
V (f, k∆n)t
)t∈[0,T ]
(8.16)
converge stably in law, in restriction to the union Ω(d)T ∪Ω(c)
T and on the product D([0, T ],R)×D([0, T ],R), to the process (Dt + Ct, D
′′t + kCt)t∈[0,T ].
Proof. 1) As said before, we assume (SH). But another localization allows to do more:let τq = inf(t : ‖δ′t‖ ≥ q) (the process δ′ is defined in (2.4)). By (d) of (H) we have
28
limq τq ≥ τ , and of course τ > T on the set Ω(d)T , so it is enough to prove the result in
restriction to each set Ω(d)T ∩τq > T. Now, let X(q) be the process defined by (2.1), with
the coefficients
b(q)t = (bt − δ′t)1t≤τq, σ
(q)t = σt, δ(q)(t, z) = δ(t, z)1Γc(t, z).
These coefficients satisfy (SH). Moreover, we obviously have Xt = X(q)t for all t ≤ T on
the set (Ω(j)T )c ∩ τq > T, hence the process (8.16) and also Dt and D′
t for t ≤ T are thesame on that set, whether computed on the basis of X or on the basis of X(q). This meansthat for proving our result we can substitute X with X(q), which satisfies (SH) and whosetwo components have no common jumps by construction.
In other words, we can and will assume in the sequel that X satisfies (SH) and that X1
and X2 have no common jumps. We will then prove that in fact the stable convergencein law holds everywhere (not only on (Ω(j)
T )c), and for the time interval R+. We use thesame notation than in the proof of Theorem 8.3.
2) Pick ε > 0, and take S′q = S′q(ε). We still have (8.11) and (8.12) on Ωn(T, ε) andt ≤ T , for φ = f . Recalling that f and its first derivatives vanish at each ∆XS′q for S′q ≤ T
because X1 and X2 have no common jumps, instead of (8.13) we get
Y (ε)(f, ∆n)t =∑
q: S′q≤∆n[t/∆n]
(12
∑2i,j=1 f ′′ij(∆XS′q + Rn
q ) Rn,iq Rn,j
q − f(Rnq )
)
Y (ε)(f, k∆n)t =∑
q: S′q≤k∆n[t/k∆n]
(12
∑2i,j=1 f ′′ij(∆XS′q + R′′n
q ) R′′n,iq R′′n,j
q − f(R′′nq )
)
Recalling that f(x) ≤ ‖x‖4, we then readily deduce from (8.10) that
1∆n
(Y (ε)(f, ∆n), Y (ε)(f, k∆n)
) L−(s)−→(D(ε), D′′(ε)
)(8.17)
(for the product topology of D(R+,R)× D(R+,R)), where D(ε) and D′′(ε) are defined by(3.13), but with the sum taken only over the S′q(ε) (we have used the facts that f ′′11(x) = 2x2
2
and f ′′22(x) = 2x21; recall ∆X1
s ∆X2s ≡ 0). As ε → 0, we have D
(ε)t → Dt and D
′′(ε)t → D′′
t
locally uniformly in t. Hence in view of (8.11) and since Ωn(T, ε) → Ω as n → ∞, itremains to prove that, for all η > 0 and k′ = 1 and k′ = k,
limε→0
lim supn
P(
supt≤T
∣∣∣ 1∆n
G(X(ε), f, k′∆n)t − k′Ct
∣∣∣ > η)
= 0. (8.18)
3) Set Aε = z : γ(z) ≤ ε. We see that the process X(ε) can be written as
X(ε)t = X0 +∫ t
0b(ε)sds +
∫ t
0σsdWs +
∫ t
0
∫
Aε
δ(s, z)(µ− ν)(ds, dz),
where b(ε)t = bt −∫
(δ(t, z)1Aε(z) − κ δ(t, z))λ(dz) is bounded by K(1 + 1/ε) for someconstant K. Write
X ′(ε)t =∫ t
0b(ε)sds +
∫ t
0σsdWs, X ′′(ε)t = X(ε)t −X0 −X ′(ε).
29
Since X ′(ε) is continuous, it is well known (see Theorem 2.4-(i) of [9] for example) that1
∆nG(X ′(ε), f, k′∆n) converges in probability, locally uniformly in time, to k′C for each
fixed ε > 0. Hence (8.18) will be implied by
limε→0
lim supn
P(
supt≤T
∣∣∣ 1∆n
G(X(ε), f, k′∆n)t −G(X ′(ε), f, k′∆n)t
∣∣∣ > η)
= 0. (8.19)
4) We prove (8.19) for k′ = 1, the proof for k′ = k being similar. Since f(x) = x21x
22,
by expanding the squares we readily get for some constant A and all v > 0 and x, y ∈ R2:
|f(x + y)− f(x)| ≤ vf(x) +A
v
(x2
1y22 + x2
2y21 + y2
1y22 + x2
1y21 + x2
2y22
)(8.20)
We put Un(j, ε)t =∑[t/∆n]
i=1 ζni (j, ε) for j = 1, 2, 3, 4, 5, where
ζni (1, ε) = (∆n
i X ′′1(ε)X ′′2(ε))2,ζni (2, ε) = (∆n
i X ′1(ε)X ′′2(ε))2, ζni (3, ε) = (∆n
i X ′′1(ε)X ′2(ε))2,ζni (4, ε) = (∆n
i X ′1(ε)X ′′1(ε))2, ζni (5, ε) = (∆n
i X ′2(ε)X ′′2(ε))2.
We have |G(X(ε), f,∆n)t − G(X ′(ε), f, ∆n)t| ≤ vG(X ′(ε), f, ∆n)T + Av
∑5j=1 Un(j, ε)T if
t ≤ T . Use again the fact that 1∆n
G(X ′(ε), f, ∆n)TP−→ CT , which is smaller than A′T
for some constant A′, to get that
lim supn
P( v
∆nG(X ′(ε), f, ∆n)T >
η
2
)= 0
for all ε > 0, if v < η/2A′T . Therefore we see that (8.19) holds for all η > 0 as soon as
limε→0
lim supn
E( 1
∆nUn(j, ε)T
)= 0 (8.21)
for j = 1, 2, 3, 4, 5.
5) Let us recall that under our assumptions we have
E(‖X ′(ε)s+t −X ′(ε)s‖p | Fs) ≤ Kp
(tp/2 + tp
εp
),
E(‖X ′′(ε)s+t −X ′′(ε)s‖p | Fs) ≤ Kψ(ε)p/2 t1∧(p/2),
(8.22)
for all s, t ≥ 0 and p > 0 and with ψ(ε) =∫Aε
γ(z)2dz, which goes to 0 as ε → 0 (here Kand Kp are constants, which may change from line to line).
Since by hypothesis X ′′(ε)1 and X ′′(ε)2 have no common jumps, an application of Ito’sformula shows that
(X ′′(ε)1t )2 (X ′′(ε)2t )
2 = M(ε)t + H(ε)t
where M(ε) is a martingale and
H(ε)t =∫ t
0ds
∫
Aε
((X ′′(ε)1s)
2(δ2(s, z))2 + (X ′′(ε)2s)2(δ1(s, z))2
)λ(dz).
30
We deduce from (8.22) that
0 ≤ E((X ′′(ε)1t )
2 (X ′′(ε)2t )2)≤ Kψ(ε)2t2.
We can repeat the same argument with X ′′(ε)s+t −X ′′(ε)s instead of X ′′(ε)t: we obtainE(ζn
i (1, ε)) ≤ Kψ(ε)2∆2n. Then, since ψ(ε) → 0 as ε → 0, we deduce (8.21) for j = 1.
Another application of Ito’s formula yields
(X ′(ε)1t )2 (X ′′(ε)2t )
2 = M(ε)t + H(ε)t, H(ε)t =∫ t
0h(ε)sds
where M(ε) is a martingale and
h(ε)t = 2(X ′′(ε)2t )2 X ′(ε)1t b(ε)1t + (X ′′(ε)2t )
2 c11t + (X ′(ε)1t )
2
∫
Aε
(δ2(s, z))2 λ(dz).
Since ct is bounded and |b(ε)1t | ≤ K(1 + 1/ε), we easily deduce from (8.22) and Holder in-equality that E(|h(ε)t|) ≤ Kψ(ε)(t+t7/6/ε2). Hence E((X ′(ε)1t )2 (X ′′(ε)2t )2) ≤ Kψ(ε)t2(1+t1/6/ε2). Repeating the same argument for the increments, we obtain
E(ζni (j, ε)) ≤ Kψ(ε)∆2
n
(1 +
∆1/6n
ε2
)
for j = 2, and the same estimate hold for j = 3, 4, 5. Then obviously (8.21) holds for thesevalues of j. 2
Proof of Theorem 3.1-(b). First, the variable Φ defined by (3.17) on Ω(d)T and, say, 1
elsewhere, takes its values in (0,∞), and conditionally on F and in restriction to Ω(d)T it
has a density (recall (3.10) and (3.13), so Φ 6= 1 a.s. on Ω(d)T . Second, the convergence
Φ(j)n
L−(s)−→ Φ in restriction to Ω(d)T is obvious from Theorem 8.5. 2
Proof of Theorem 4.1-(b). In view of (8.6) applied to h = g1 and h = g2 and of
Theorem 8.5), it is obvious that Φ(d)n /∆n
L−(s)−→ Φ′ = (DT + CT )/√
B′1T B′2
T , in restriction
to Ω(d)T . Finally (4.4) follows from (8.1). 2
8.5 Proof of (4.12) and (4.13).
(4.12) is nothing else than a multidimensional version of (26) in [1], applied with q = 4and r = 1, and we leave the (simple) computations to the reader.
The first part of (4.13) is almost trivial. Indeed, if ψ is like in the proof of Theorem8.3 (C∞, with 1[0,1] ≤ ψ ≤ 1[0,2]), for any ρ > 0 we have
∆nA(∆n)T ≤ V (fρ, ∆n)T ,
31
where fρ(x) = f(x)ψ(‖x‖/ρ), and as soon as α∆$n < ρ. Now, (8.6) yields that
V (fρ, ∆n)TP−→
∑
s≤T
fρ(∆Xs),
and the right side above decreases to 0 as ρ → 0, hence the result.
The second part of (4.12) is a bit more involved and goes through several steps. Exactlyas in the proof of Theorem 8.5, we can assume (SH) and also that X1 and X2 do notjump together, and we will indeed prove the stable convergence on Ω instead of (Ω(j)
T )c.
Step 1) Here we prove that for any ρ > 0,
1∆n
V (fρ,∆n)L−(s)−→ D(fρ) + C, (8.23)
where for any C2 function g we have set
D(g)t :=12
∑
q: Sq≤t
(g′′11(∆XSq)(R2p)
2 + g′′22(∆XSq)(R1p)
2).
Exactly as for (8.17), one gets for each ε > 0:
1∆n
(Y (ε)(f, ∆n), Y (ε)(f − fρ, ∆n)
) L−(s)−→(D(ε), D(ε)(f − fρ)
),
where D(ε)(f − fρ) is defined as D(f − fρ), with the sum over the S′q(ε)’s only (notethat D(ε) = D(ε)(f)). Observe that f ′′ρ,12(x) = 0 if x1x2 = 0, so f ′′ρ,12(∆XSp) = 0)because X1 and X2 do not jump at the same times.If 2ε < ρ, on the set Ωn(T, ε) we have(f − fρ)(∆n
i X(ε)) = 0 for all i ≤ [T/∆n], so in view of (8.11) we have V (f − fρ, ∆n)t =Y (ε)(f − fρ, ∆n)t for all t ≤ T on this set. We also have D(ε)(f − fρ) = D(f − fρ) if ε < ρ.Hence
1∆n
(Y (ε)(f, ∆n), V (f − fρ,∆n)
) L−(s)−→(D(ε), D(f − fρ)
).
Making use of (8.11) and (8.18) (valid for all T ), we deduce
1∆n
(V (f,∆n), V (f − fρ, ∆n)
) L−(s)−→(C + D, D(f − fρ)
),
and (8.23) is obtained by difference, since D(f) = D.
Step 2) In this step we prove that for any A > 1 and t > 0,
1∆n
V (fA√
∆n,∆n)t
P−→∫ t
0ρσs(fA) ds, (8.24)
where ρσ(fA) = E(fA(σU)), where U is an N (0, I2) variable and σ a 2× 2 matrix.
We write X ′t and X ′′
t for the sum of the first three terms, resp. of the last two terms,in (2.1). By Theorem 2.4-(i) of [9] we have
1∆n
G(X ′, fA√
∆n,∆n)t
P−→∫ t
0ρσs(fA) ds,
32
hence it is enough to prove that
Gn :=1
∆n
∣∣∣G(X, fA√
∆n,∆n)t −G(X ′, fA
√∆n
, ∆n)t
∣∣∣ P−→ 0. (8.25)
Since (SH) holds, there is an increasing function F on R+ with F (ε) → 0 as ε → 0 suchthat the following holds for q ≥ 2, η, θ > 0 (use (45) of [1] and argue componentwise):
E(‖∆ni X ′′‖2
∧η2) ≤ K∆n
(η2 + ∆n
θ2+ F (θ)
). (8.26)
Since fε(x) ≤ Kε4 and |f ′ε,i(x)| ≤ Kε3 and |ψ′(u)| ≤ K, we clearly have |fε(x+y)−fε(x)| ≤Kε3(‖y‖ ∧ ε) for all x, y ∈ R2. Then if we use (8.26) we see that
E(Gn) ≤ 1∆n
[t/∆n]∑
i=1
E(|fA
√∆n
(∆ni X ′ + ∆n
i X ′′)− fA√
∆n(∆n
i X ′)|)
≤ KA3√
∆n
[t/∆n]∑
i=1
√E(‖∆n
i X ′′‖2∧
(A2∆n) ≤ KtA3
(A√
∆n
θ+
√F (θ)
)
for all θ > 0. Taking θ = θn = ∆1/4n , so F (θn) → 0, we deduce E(Gn) → 0, hence (8.25).
Step 3) Observe that for all A > 1 and ρ > 0 we have
1∆n
V (fA√
∆n, ∆n)T ≤ A(∆n)T ≤ 1
∆nV (fρ, ∆n)T ,
as soon as 2A√
∆n ≤ α∆$n ≤ ρ, that is for all n large enough. We also obviously have (for
each ω):
D(fρ)T →ρ→0 0,
∫ T
0ρσs(fA) ds →A→∞ CT .
At this stage, the convergence A(∆n)TL−(s)−→ CT readily follows from (8.23) and (8.24).
8.6 Proof of (4.14).
Step 1) In view of (4.1) and (4.2) and of (4.10) and (4.11), we have to prove that
Hnt :=
1kn∆n
[t/∆n]−kn∑
i=1+kn
hn(∆ni X)
∑
j∈In(i)
∆nj Xu∆n
j Xv1‖∆nj X‖≤α∆$
n
P−→ Ht =∑
s≤t
h(∆Xs)(cuvs− + cuv
s ) (8.27)
where In(i) = In,−(i) ∪ In,+(i) and u, v = 1, 2, and also hn(x) = h(x)1‖x‖>α∆$n with
h(x) = xu′1 xv′
2 with u′, v′ integers and u′ + v′ ≥ 2.
33
The proof is basically the same as for (27) of [1], and we first introduce (with the samenotation than in that paper) the 2-dimensional variables δn
j = σ(i−1)∆n∆n
i W . We use theshorthand notation En
i−1(Y ) for E(Y | F(i−1)∆n). In view of (SH) we have the following
estimates, for q ≥ 2 and η, θ > 0 and F the function occurring in (8.26) (one can arguecomponent by component, so (44) and (46) of [1] apply here as well):
Eni−1(‖δn
i ‖q) ≤ Kq∆q/2n , En
i−1(‖∆ni X − δn
i ‖q) ≤ Kq∆n
Eni−1(‖∆n
i X − δni ‖2
∧η2) ≤ K∆n
(η2+∆n
θ2 + F (θ))
.(8.28)
Step 2) For any given ρ ∈ (0, 1) we set ψρ(x) = 1 ∧ (2− |x|/ρ)+, and
H(ρ)′nt =1
kn∆n
[t/∆n]−kn∑
i=1+kn
ψρ(‖∆ni X‖)hn(∆n
i X)∑
j∈In(i)
∆nj Xu∆n
j Xv1‖∆nj X‖≤α∆$
n ,
H(ρ)n = Hn − H(ρ)′n,
H(ρ)′t =∑
q: Sq≤t
h(∆XSq)ψρ(‖∆XSq‖) (cuvSq− + cuv
Sq), H(ρ) = H −H(ρ)′.
Then it is obviously enough to prove the following three properties:
ρ → 0 ⇒ H(ρ)′tP−→ 0, (8.29)
limρ→0
lim supn
E(|H(ρ)′nt |) = 0, (8.30)
ρ ∈ (0, 1), n →∞ ⇒ H(ρ)nt
P−→ H(ρ)t. (8.31)
Step 3) The property (8.29) follows from Lebesgue Theorem. Next, if ρ ≤ 1/2 we have
|(ψρhn)(x + y)| ≤ K
(‖x‖4
ε2+ (‖y‖2 ∧ ρ2)
).
Using this with x = δni and y = ∆n
i X − δni , plus (8.28) with η = ρ and θ =
√ρ, we obtain
1∆n
Eni−1(ψρ(‖∆n
i X‖) |hn(∆ni X)|) ≤ Fn(ρ) := K(∆1−2$
n + ρ + F (√
ρ) + ∆nρ−1).
On the other hand Enj−1(|∆n
j X‖2) ≤ K∆n by (8.28), so by successive conditioning weobtain
E(∣∣∣hn(∆n
i X)∑
j∈In(i)
∆nj Xu∆n
j Xv1‖∆nj X‖≤α∆$
n ∣∣∣)≤ K∆2
nFn(ρ).
Then E(|H(ρ)′nt |) ≤ KtFn(ρ) and (8.30) follows.
Step 4) It remains to prove (8.31). Fix ρ ∈ (0, 1) and t > 0, and recall the jump timesS′q = S′q(ρ/2) and the set Ωn(T, ρ/2) of the proof of Theorem 8.3. On Ωn(t, ρ/2) there isno S′q in (0, kn∆n], nor in (t − (kn + 1)∆n, t] and there is at most one S′q in an interval
34
((i− 1)∆n, i∆n] with i∆n ≤ t, and if this is not the case we have ψρ(‖∆ni X‖) = 1. Hence
on Ωn(t, ρ/2),
H(ρ)nt =
∑
q: kn∆n<S′q≤t−(kn+1)∆n
ζnq
(c(n,−)uv
i(n,q) + c(n, +)uvi(n,q)
),
(recall (4.9)), where ζnq = hn(∆n
i(n,q)X)(1−ψρ(‖∆ni(n,q)X‖)) and i(n, q) = inf(i : i∆n ≥ S′q).
Observe also
H(ρ)t =∑
q: S′q≤t
h(∆XS′q)(1− ψρ(‖∆XS′q‖))(cuvS′q− + cuv
S′q).
The sum over q with S′q ≤ t are finite, and obviously ζnq → h(∆XS′q)(1 − ψρ(‖∆XS′q‖))
pointwise. Hence for (8.31) we need only to prove that
c(n,−)i(n,q)P−→ cS′q−, c(n,+)i(n,q)
P−→ cS′q . (8.32)
This is proved in [1] (see Lemma 1 and (71) in that paper), and we are finished.
8.7 Proof of Theorem 4.3.
Step 1) The proof is somewhat similar to the previous one, whose notation is generallyused (like Fn(ρ) or Ωn(t, ρ/2), and also S′q and i(n, q) when ρ is fixed). However we departof these notations by taking ψρ = 1[0,ρ] and by using the processes
Hnt =
[t/∆n]∑
i=1
hn(∆ni X)φ(R(n)i), Ht =
∑
q:Sq≤t
h(∆XSq)φ(Rq),
where hn and h are like in the previous proof, and φ is a monomial of degree at least 1,and R(n)i and Ri are either R(n)′i and R′
i (when proving (4.18)) or R(n, )i and Ri (whenproving (4.19)). Set also
H(ρ)′nt =[t/∆n]∑
i=1
ψρ(‖∆ni X‖)hn(∆n
i X)φ(R(n)i), H(ρ)n = Hn − H(ρ)′n,
H(ρ)′t =∑
q:Sq≤t
h(∆XSq)ψρ(‖∆XSq‖) φ(Rq), H(ρ) = H −H(ρ)′.
Then, with Zn and Z as in Theorem 4.3, it is clearly enough to prove that
Vn := P(|HnT | > Zn | F) P−→ V := P(|HT | > Z | F). (8.33)
Step 2) By taking first the conditional expectation w.r.t. F , we get E(|H(ρ)′nt ) ≤ KtFn(ρ),hence (like in (8.30)):
limρ→0
lim supn
E(|H(ρ)′nT ) = 0. (8.34)
35
Next, we consider the set B of all ρ > 0 such that outside a P-null set we have ‖∆Xs(ω)‖ /∈B for all s > 0. This set B has an at most countable complement. Suppose that ρ is fixedin B. On the set Ωn(T, ρ/2), we have
H(ρ)nT =
∑
q: S′q≤∆n[T/∆n]
hn(∆ni(n,q)X)(1− ψρ(‖∆n
i(n,q)X‖))φ(R(n)i(n,q)).
For each q we have hn(∆ni(n,q)X)(1 − ψρ(‖∆n
i(n,q)X‖)) → h(∆XS′q)(1 − ψρ(‖∆XS′q‖)) a.s.(because ρ ∈ B). Recalling that Ωn(T, ρ/2) → Ω as n →∞ and also that T −∆n ≤ S′q ≤T → ∅ a.s. for all q we deduce that H(ρ)n
T −H(ρ)nT
P−→ 0 as n →∞, where
H(ρ)nT =
∑
q: S′q≤T
h(∆XS′q)(1− ψρ(‖∆XS′q‖))φ(R(n)i(n,q)). (8.35)
Combining this with (8.34), we obtain for all η > 0:
HnT = H(ρ)n
T + H(ρ)′nT , where limρ→0, ρ∈B
lim supn
P(|H(ρ)′nT | > η) = 0. (8.36)
Moreover, in view of the definitions of H(ρ)n and H(ρ)′n, we have
conditionally on F , the variables H(ρ)nT and H(ρ)′nT are independent (8.37)
(this is where taking ψρ to be an indicator function is important, so that the two variablesabove involves the (κi, L,, Ui, U
′i , U i, U
′i) for disjoint sets of indices i).
In a similar, but much simpler, way we also get
limρ→0, ρ∈B P(|H(ρ)T | > η) = 0
conditionally on F , the variables H(ρ)T and H(ρ)′T are independent.
(8.38)
Step 3) The variable H(ρ)T has the form (8.35) as well, except that R(n)i(n,q) is substitutedwith R′
q, the variable equal to Rl on the set S′q = Sl. There is no connection betweenR(n)i(n,q) and R′
q. However, there is a strong connection between their F-conditional laws,hence between the F-conditional laws ζn
ρ (ω, dy) of |H(ρ)nT | and ζρ(ω, dy) of |H(ρ)T |.
More precisely, let σ and σ′ be two 2×2 matrices with squares c = σσ? and c′ = σ′σ′?.The variable φ
(√L1 σU1 +
√1− k − L1 σ′U ′
1
)or φ
(√κ1 σU1 +
√1− κ1 σ′U ′
1
)(according
to whether we prove (4.18) or (4.19)) has a law θc,c′ which depends only on (c, c′) and iscontinuous in (c, c′) (for the weak convergence) and is either the Dirac mass ε0 or has apositive density on R or R+ (depending on whether φ is odd or even). With this in mind,we see that ζn
ρ (ω, .) is in fact the law of∣∣∣
∑
q: S′q(ω)≤T
h(∆XS′q(ω))(1− ψρ(‖∆XS′q(ω)‖))Φq
∣∣∣,
where the Φq are independent variables, with the laws θc(n,−)i(n,q)(ω),c(n,+)i(n,q)(ω). Of courseζρ is the same, with (c(n,−)i(n,q), c(n,+)i(n,q)) substituted with (cS′q−, cSq). Then in view
36
of (8.32) and of the continuity of θc,c′ in (c, c′) and the fact that θc,c′ is either ε0 or has a
density, we readily deduce that for any F-measurable variables ZnP−→ Z > 0,
ζnρ ((Zn,∞)) P−→ ζρ((Z,∞)). (8.39)
Step 4) In view of the decomposition (8.36) and of (8.37), we have
Vn =∫
ζ ′nρ (dx) P(|H(ρ)nT + x| > Zn | F),
where ζ ′nρ denotes the F-conditional law of H(ρ)′nT . Then obviously, for any ε > 0,
ζnρ ((Zn + ε,∞))− ζ ′nρ ((−ε, ε)c) ≤ Vn ≤ ζn
ρ ((Zn − ε,∞)) + ζ ′nρ ((−ε, ε)c).
The same estimates hold for V , with ζ ′ρ being the conditional law of H(ρ)′T , and with Zinstead of Zn. It follows that
|Vn − V | ≤∣∣∣ζn
ρ ((Zn + ε,∞))− ζρ((Z + ε,∞))∣∣∣ +
∣∣∣ζnρ ((Zn − ε,∞))− ζρ((Z − ε,∞))
∣∣∣+2ζρ([Z − ε, Z + ε]) + ζ ′nρ ((−ε, ε)c) + ζ ′ρ((−ε, ε)c).
Moreover
E(ζρ([Z − ε, Z + ε])) = P(Z − ε ≤ |H(ρ)T | ≤ Z + ε)
≤ P(Z − 2ε ≤ |HT | ≤ Z + 2ε) + P(|H(ρ)′T | ≥ ε).
Then in view of (8.39) we obtain
lim supn
E(|Vn − V |) ≤ P(Z − 2ε ≤ |HT | ≤ Z + 2ε)
+3P(|H(ρ)′T | ≥ ε) + lim supn
P(|H(ρ)′nT | ≥ ε).
This holds for all ρ > 0 and ε > 0. But on the one hand the F-conditional law of |HT |is either the Dirac mass ε0 or it has a density, hence P(Z − 2ε ≤ |HT | ≤ Z + 2ε) → 0as ε > 0 because Z > 0 by hypothesis. On the other hand, we can apply (8.36) and thefirst part of (8.38), and we readily deduce that lim supn E(|Vn − V |) = 0, which of courseimplies (8.33).
8.8 Proof of Theorem 4.5.
Let us prove (4.23). Let us set Ψn = 1∆n
Φ(d)n
√V (g1, ∆n)T V (g2, ∆n)T . By Theorems 4.1
and 4.2 and by (4.5), the variables Ψn − An converge stably in law, in restriction to Ω(d)T ,
to DT , for the two possible choices of An, whereas on Ω(d)T the F-conditional law of DT
admits a density. Therefore,
P(A ∩ Ψn > Zn + An) → P(A ∩ DT > Z),which is (4.23). The proof of (4.22) is similar.
37
8.9 Proof of Theorem 5.1.
Set Un = (Φ(j)n − 1)/V
(j)n , and let A ∈ F .
We first prove (b), so we use (5.6). We have
P(C(j)n ∩A) = P(|Un| ≥ 1/
√α ∩A).
Theorem 4.4-(a) yields that Un converges stably in law, in restriction to Ω(j)T , to a variable
U with E(U2 | F) = 1, hence if A ⊂ Ω(j)T :
lim supn
P(C(j)n ∩A) ≤ P(|U | ≥ 1/
√α ∩A) ≤ αP(A),
where the last inequality follows from Bienayme-Tchebycheff applied to the conditional lawof U knowing F . This clearly yields α(j) ≤ α. On the other hand, proving β(j) = 1 amountsto proving that P(C(j)
n ∩Ω(d)T ) → P(Ω(d)
T ), or equivalently P(|Un| ≥ η∩Ω(d)T ) → P(Ω(d)
T ) forany fixed η > 0. Since Φ(j)
n − 1 converges stably in law to an almost surely non-vanishinglimit by Theorem 3.1-(b), on Ω(d)
T , the result will be implied by the property
V (j)n
P−→ 0 on the set Ω(d)T .
In view of (4.14) and (4.20), this amounts to proving that V (f, ∆n)T /√
∆nP−→∞ on Ω(d)
T .Since DT + CT > 0, this follows from Theorem 8.5, and this finishes the proof of (b).
Next we prove (a), so we use (5.3). We now have
P(C(j)n ∩A) = P(|Un| ≥ zα ∩A).
for each A ∈ F . If X and σ do not jump together, Theorem 4.4-(a) yields that Un
converges stably in law, in restriction to Ω(j)T , to a variable which is N (0, 1) conditionally
on F . Then if A ⊂ Ω(j)T we have
P(C(j)n ∩A) → P(|U | ≥ zα ∩A) = αP(A).
This yields (5.5), hence also the first part of (5.4), whereas β(j) = 1 is proved as for (b).
Finally we prove (c), so we use (5.7). We have
P(C(j)n ∩A) = P
(A ∩
|Φ(j)n − 1|V (f, ∆n)T√
(k − 1)∆n
> Z(j)n (α)
).
Hence in view of (4.22), the property (5.5) and thus the first part of (5.4) as well willfollow if we prove that
Z(j)n (α) P−→ Z(α) in restriction to Ω(j)
T , where Z(α) is positive andF-measurable and P(|GT | > Z(α) | F) = α in restriction to Ω(j)
T .
(8.40)
The law of |GT |, conditional on F and in restriction to Ω(j)T , admits a positive density
on R+, so Z(α) is uniquely defined and Z(α) > 0 a.s. on Ω(j)T . In restriction to a subset
38
Ωn which increases to Ω(j)T , the law of |G(∆n)T |, conditional on F also admits a positive
density on R+, so for all γ ∈ (0, 1) there is a (unique on Ωn) F-measurable variable Z ′n(γ)such that P(|G(∆n)T | > Z ′n(γ) | F) = γ. Moreover (4.18) implies that
γ ∈ (0, 1) =⇒ Z ′n(γ) P−→ Z(γ) on the set Ω(j)T . (8.41)
Now, recall from (5.2) that Z(j)n (α) is the [(1−α)Nn]th absolute order statistics for Nn
independents draws of G(∆n)T , conditionally on F , so by Biename-Tchebycheff inequalityapplied to the binomial distributions of size Nn and parameters α−ε and α+ε respectively(with ε < α ∧ (1− α)), we easily see that for all η > 0, and on the set Ω(j)
T again,
P(Z(j)
n (α) /∈ [Z ′n(α + ε), Z ′n(α− ε)] | F)≤ 1
ε2Nn.
Taking ε = 1/N1/4n , and using (8.41), we obtain for all η > 0, and on the set Ω(j)
T again,
P(Z(j)
n (α) /∈ [Z(α + N−1/4n )− η, Z(α−N−1/4
n ) + η] | F) P−→ 0,
and (8.40) follows.
It remains to prove that β(j) = 1, which amounts to proving P(C(j)n ∩Ω(d)
T ) → P(Ω(d)T ).
On the one hand, on Ω(d)T we have GT = 0, hence exactly as above we see that Z
(j)n (α) P−→ 0
on this set. On the other hand, combining Theorem 3.1-(b) and (4.5), we see that
|Φ(j)n − 1|V (f, ∆n)T√
(k − 1)∆n
P−→ +∞
on Ω(d)T . Then the result is obvious.
8.10 Proof of Theorem 5.2.
The proof is the same as for the previous theorem, with a few changes. In case (a), thatis of (5.10), we set Un = Φ(d)
n /V(d)n , so
P(C(d)n ∩A) = P(|Un| ≥ 1/α ∩A).
Theorem 4.4-(b) yields that Un converges stably in law, in restriction to Ω(d)T , to a variable
U > 0 with E(U | F) = 1, hence if A ⊂ Ω(d)T :
lim supn
P(C(d)n ∩A) ≤ P(|U | ≥ 1/α ∩A) ≤ αP(A),
(use again Bienayme-Tchebycheff), and thus α(d) ≤ α. The property β(d) = 1 amounts tohaving P(Un ≥ η ∩ Ω(j)
T ) → P(Ω(j)T ) for any fixed η > 0. By (4.5) and Theorem 3.1-(a)
we have Φ(d)n
P−→ BT /√
B′1T B′2
T > 0 on Ω(j)T , and also V
(d)n
P−→ 0 on this set by (4.5) again,
so UnP−→ +∞ on Ω(j)
T and the result readily follows.
Finally in case (b) of (5.12), the proof is exactly the same as for case (c) of Theorem5.1.
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