ESTIMATION OF THE LEAD-LAG PARAMETER BETWEEN TWO STOCHASTIC
PROCESSES DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

arXiv:1705.10466v2 [] 12 Mar 2018

KOHEI CHIBA
Abstract. In this paper, we consider the problem of estimating the lead-lag parameter between two stochastic processes driven by fractional Brownian motions (fBMs) of the Hurst parameter greater than 1/2.
First we propose a lead-lag model between two stochastic processes involving fBMs, and then construct
a consistent estimator of the lead-lag parameter with possible convergence rate. Our estimator has the
following two features. Firstly, we can construct the lead-lag estimator without using the Hurst parameters
of the underlying fBMs. Secondly, our estimator can deal with some non-synchronous and irregular observations. We explicitly calculate possible convergence rate when the observation times are (1) synchronous
and equidistant, and (2) given by the Poisson sampling scheme. We also present numerical simulations of
our results using the R package YUIMA.

1. Introduction
Lead-lag effect is phenomenon that some asset prices follow the fluctuation of others with a small
time lag. Recently, milli-second level high frequency trading data are available, and the lead-lag effect
is observed at such a fine time scale. For a phenomenological perspective, see Huth and Abergel (2014)
and references therein. If we can effectively use the lead-lag effect, it ultimately leads to the prediction
of future behavior of stock prices. Hence it is important to analyse the lead-lag effect for developing a
trading strategy.
In order to analyse the lead-lag effect statistically, Hoffmann et al. (2013) introduced a regular semimartingale with the lead-lag parameter θ ∈ (−δ, δ) for δ > 0, and constructed a consistent estimator
for the lead-lag parameter θ. Here we briefly review their framework. Before describing the model, we
introduce some notations that are used throughout this paper. We follow the notations used in Hoffmann
et al. (2013).
Notation 1.1. Let T > 0 be a terminal time and δ > 0 the maximum temporal lead-lag allowed for the
model. We set Θ = (−δ, δ). The underlying probability space is denoted by (Ω, F, P), and we consider
a filtration F = (Ft )t∈[−δ,T +δ] satisfying the usual conditions on (Ω, F, P). We set τ−s (X)t = Xt+s
for a stochastic process X and a real number s ∈ R. For a subinterval [a, b] of [−δ, T + δ], we set
F[a,b] = (Ft )t∈[a,b] . Finally, for a lead-lag parameter θ ∈ (−δ, δ), the shifted filtration (Ft−θ )t∈[−δ+θ,T +δ+θ]
is denoted by Fθ .
They defined a regular semimartingale with the lead-lag parameter θ as follows.
Definition 1.2. The two-dimensional stochastic process {(Xt , Yt )}t∈[0,T +δ] is called a regular semimartingale with the lead-lag parameter θ ∈ [0, δ) if X and Y admit the decomposition X = X c + A
and Y = Y c + B respectively, where
(1) the process X c is a continuous F-local martingale, and the process Y c is a continuous Fθ[0,T +δ] -local
martingale,
(2) the quadratic variation processes hX c i and hY c i are absolutely continuous with respect to the
Lebesgue measure on [0, T + δ], and their Radon-Nikodym derivatives have a locally bounded
version, and
(3) the processes A and B are of finite variation on [0, T + δ].
2010 Mathematics Subject Classification. 62M09 and 60G22.
Key words and phrases. fractional Brownian motion; lead-lag effect; non-synchronous observations; contrast estimation.
1

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KOHEI CHIBA

Definition 1.3. If θ ∈ (−δ, 0] and the two-dimensional process (Y,X) is a regular semimartingale with
the lead-lag parameter −θ, then the process (X, Y ) is called a regular semimartingale with the lead-lag
parameter θ ∈ (−δ, 0].
Let us set TnX = {0 = sn0 < sn1 < · · · < snNX (n) = T + δ} and TnY = {0 = tn0 < tn1 < · · · < tnNY (n) = T + δ}
for some positive integers NX (n) and NY (n). The process X is assumed to be observed at the times in
TnX and the process Y at the times in TnY . Note that the observation times may be non-synchronous,
irregular and random. Since we keep in mind high-frequency data, we consider the asymptotics such that
the maximum mesh size of the observation times tend to 0 as n → ∞:
lim max{|sni − sni−1 |, |tnj − tnj−1 | | i = 1, . . . , NX (n), j = 1, . . . , NY (n)} = 0

n→∞

in probability. They constructed an estimator θ̂n for θ by maximizing the shifted Hayashi-Yoshida covariation contrast as follows. We define the shifted Hayashi-Yoshida covariation contrast Un (θ̃) by
X
Un (θ̃) = 1θ̃≥0
(Xsni − Xsni−1 )(Ytnj − Ytnj−1 )1(sn ,sn ]∩(tn −θ̃,tn −θ̃]6=∅
i−1

i,j:sn
i ≤T

+ 1θ̃<0

X

i

(Xsni − Xsni−1 )(Ytnj − Ytnj−1 )1(sn

j−1

j

n
n
n
i−1 ,si ]∩(tj−1 +θ̃,tj +θ̃]6=∅

i,j:tn
j ≤T

.

The lead-lag estimator is constructed by maximizing the contrast θ̃ 7→ |Un (θ̃)| over a finite subset G n of
Θ. More precisely, the estimator θ̂n is defined by a solution of the equation
θ̂n = max |Un (θ̃)|.
θ̃∈G n

They proved that the estimator θ̂n is consistent under appropriate assumptions on TnX , TnY and G n (see
Hoffmann et al. (2013) for detail).
In standard financial theory, it is assumed that there is no arbitrage in the market. Hence semimartingales, which satisfy this no-arbitrage assumption, are regarded as reasonable class for modeling stock
prices. However, non-semimartingales, especially ones involving fractional Brownian motion (fBM), also
attract attention as a model of stock prices recently. For example, many researchers investigate how arbitrage opportunities arise and how can we exclude them when we use non-semimartingales as a model of
stock prices (Bender et al. (2008, 2011); Cheridito (2003); Guasoni (2006); Jarrow et al. (2009) and Rogers
(1997) to name but a few). Using fBM in modeling stock prices reflects possible long range dependence
property empirically obeserved in some financial time series (for example Cutland et al. (1995); Greene
and Fielitz (1977); Hall et al. (2000); Henry (2002); Lo (1991); Teverovsky et al. (1999) and Willinger
et al. (1999)). An example of such models is the fractional Black-Scholes model. This model assumes that
the stock price dynamics are given by
dSt = µSt dt + σSt dBt , S0 > 0, t ≥ 0

(1.1)

where µ and σ are constants and the process B = (Bt )t≥0 is an fBM with Hurst parameter H ∈ (1/2, 1).
Rt
Here the stochastic integral σ 0 Ss dBs can be understood in various ways: if we use the Riemann-Stieltjes
integral, then the solution S (1) of the equation (1.1) is
(1)

St

= S0 exp(µt + σBt ),

and on the other hand, if we use the Wick-Itô-Skorokhod integral, then the solution S (2) of the equation
(1.1) becomes


σ2
(2)
St = S0 exp µt − t2H + σBt
2
(for reference, see Biagini et al. (2008); Mishura (2008) or Sottinen and Valkeila (2003)).
Now a natural question arises: if we consider estimation of the lead-lag parameter between nonsemimartingales, especially ones involving fBM, then can we construct a consistent estimator for the
lead-lag parameter θ?

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

3

Motivated by this quastion, we consider the problem of estimating the lead-lag parameter θ between
two stochastic processes driven by fBMs in this paper. The main aim of this paper is to construct a
consistent estimator of the lead-lag parameter θ with possible convergence rate.
Although our analysis is based on the framework by Hoffmann et al. (2013), there are different types
of frameworks for lead-lag estimation. For example, Robert and Rosenbaum (2010) studied the leadlag phenomenon using random matrix theory, and Koike (2016a) investigated the asymptotic structure
of the likelihood ratio process when the observed processes are correlated Brownian motions with the
microstructure noise.
The rest of this paper is organized as follows. In section 2, we give preliminary results about fractional
calculus and stochastic calculus involving fractional Brownian motion, which will be used in subsequent
sections of this paper. We also define a covariance structure between two fBMs in Section 2. In section
3, we describe our model. In Section 4, we construct an estimator of the lead-lag parameter θ. We prove
its consistency in Section 5. In Section 6, we give an example of our model and a numerical simulation.
2. Preliminaries
2.1. Tools from fractional calculus and stochastic calculus. In this section, we collect some results
about fBM that are used in this paper. Then we introduce a correlation between two fBMs. Let us define
a fBM at first.
Definition 2.1. Let T > 0 be a positive number. A centered Gaussian process {Bt }0≤t≤T is called
fractional Brownian motion (abbreviated fBM) of Hurst parameter H ∈ (0, 1) if its covariance function
R(s, t) = E{B(t)B(s)}, s, t ∈ [0, T ] satisfies
1
(2.1)
R(t, s) = (t2H + s2H − |t − s|2H )
2
for s, t ∈ [0, T ].
Remark 2.2. In the sequel, we always assume that the Hurst parameter H is greater than 1/2.
When we consider lead-lag relationship between time series, it is reasonable to assume that time
series are dependent on each other in appropriate sense. In order to introduce a dependent structure
between fBMs, we first define correlated BMs and then construct fBMs from them. Therefore we need a
representation of fBM via BM.
Proposition 2.3. Let W = (Wt )t∈[0,T ] be a standard BM. Consider the square integrable kernel
Z t
1/2−H
KH (t, s) = cH 1(0,t) (s)s
(u − s)H−3/2 uH−1/2 du, 0 ≤ t ≤ T
s

where cH =

q

H(2H−1)
B(2−2H,H−1/2) .

Then the process B = (Bt )t∈[0,T ] defined by
Z T
Bt =
KH (t, s) dWs

(2.2)

0

is an fBM of Hurst parameter H. Here the integral in (2.2) is interpreted as a Wiener integral.
Proof. Since the process B is clearly a centered Gaussian process, it suffices to show (2.1). Later we
calculate R(t, s) in a more general setting and (2.1) is obtained as a corollary, see Proposition 2.9 and
Corollary 2.10.

∗ : E → L2 ([0, T ])
Let E be the set of step functions on [0, T ]. Let us consider the linear operator KH
defined by
Z T
∂
∗
(KH ϕ)(s) =
ϕ(t) KH (t, s) dt.
∂t
s
Note that
∗
(KH
1[0,t] )(s) = KH (t, s)1[0,t] (s).

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KOHEI CHIBA

Let HH be the completion of E, with respect to the inner product
Z T Z T
du f (r)g(u)|r − u|2H−2
dr
hf, giHH = H(2H − 1)
0

0

∗
∗
= hKH
f, KH
giL2 ([0,T ]) .
∗ can be extended to an isometry between H and L2 ([0, T ]), and the extension is also
The operator KH
H
∗
denoted by KH . We set kf k2H = hf, f iH for f ∈ H. The space HH is isomorphic to a Gaussian Hilbert
space

HH := span {Bt ; t ∈ [0, T ]}

L2 (P)

.

We denote the isomorphism between HH and HH by B. For proofs of these results, see Nualart (2006).
In order to prove Theorem 4.6, we need the notion of multiple Wiener integral and its properties. Let
us collect some basic facts concerning multiple Wiener integral which will be used in subsequent sections
of this paper.
Definition 2.4. Let H be a real separable Hilbert space. A centered Gaussian family of random variables
(W (h))h∈H indexed by H is said to be an isonormal Gaussian process over H if the covariance function
E{W (g)W (h)} = hg, hiH for g, h ∈ H.
Example 2.5. Let H = L2 ([0, T ]; R2 ) with an inner product
h(f1 , g1 ), (f2 , g2 )iH = hf1 , g1 iL2 ([0,T ]) + hf2 , g2 iL2 ([0,T ]) .
Let W̃ = (W̃ 1 , W̃ 2 ) be a two-dimensional BM. Then
Z T
Z
W ((f, g)) :=
f (t) dW̃t1 +
0

T

g(t) dW̃t2

0

is an isonormal Gaussian process over H.
˜
˜
Let p be a positive integer and h ∈ H⊗p
, where H⊗p
is the pth symmetric tensor power of H. After
defining an isonormal Gaussian process over H, we can consider the pth multiple Wiener integral Ip (h).
Note that I1 (f ) coincides with W (f ) for f ∈ H. A detailed discussion about multiple Wiener integral can
be found in, for example, Chapter 2 of Nourdin and Peccati (2012). The following properties of multiple
Wiener integral are useful.
˜
Proposition 2.6 (Isometry property). Let p and q be positive integers with 1 ≤ p ≤ q. For f ∈ H⊗p
and
˜
g ∈ H⊗p
, we have
(
p!hf, giH⊗p if p = q,
E{Ip (f )Iq (g)} =
0
else.

Proof. For example, see Proposition 2.7.5 of Nourdin and Peccati (2012).
Theorem 2.7 (Product formula). Let p, q ≥ 1 be positive integers. For f ∈
  
p∧q
X
p
q
˜ r g),
Ip (f )Iq (g) =
r!
Ip+q−2r (f ⊗
r
r


˜
H⊗p

and g ∈

˜
H⊗p
,

we have

r=0

˜ r g denotes the symmetrization of the rth contraction of f and g (for a detail, see Appendix B
where f ⊗
of Nourdin and Peccati (2012)). In particular, if p = q = 1, then
˜
I1 (f )I1 (g) = hf, giH + I2 (f ⊗g),
˜ = (1/2)(f ⊗ g + g ⊗ f ).
where f ⊗g
Proof. For example, see Theorem 2.7.10 of Nourdin and Peccati (2012).



˜
Theorem 2.8 (Hypercontractivity). Let q > 0 be a positive number, p be a positive integer, and h ∈ H⊗p
.
Then there exists a constant C(p, q) ∈ (0, ∞) depending only on p and q such that

E{|Ip (h)|q }1/q ≤ C(p, q)E{|Ip (h)|2 }1/2 .

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

Proof. For example, see Theorem 2.7.2 of Nourdin and Peccati (2012).

5



2.2. Correlated fBMs. Let (Ω, F, P) be a stochastic basis supporting a two dimensional standard Brownian motion W = (W̃ 1 , W̃ 2 ). Here we assume that the σ-field F is complete and the filtration G satisfies
the usual conditions. For ρ ∈ [−1, 1], we set
(
Wt1 = W̃t1 ,
p
Wt2 = ρW̃t1 + 1 − ρ2 W̃t2 ,
for t ≥ 0. Then the processes W 1 and W 2 are Brownian motions satisfying
hW 1 , W 2 it = ρt, t ≥ 0.
We define correlated fractional Brownian motions on an interval [0, T ] by
(
RT
B̃t1 = 0 KH1 (t, s) dWs1 ,
RT
B̃t2 = 0 KH2 (t, s) dWs2 ,

(2.3)

for H1 , H2 ∈ (1/2, 1). Here the integrals in (2.3) are understood in the Wiener sense. In terms of an
isonormal Gaussian process, we can write (2.3) as
(
∗ 1
B̃t1 = W ((KH
, 0)),
1 [0,t]
p
(2.4)
∗ 1
2
∗
))
B̃t = W ((ρKH2 1[0,t] , 1 − ρ2 KH
2 [0,t]
(see Example 2.5). For given subinterval I ⊂ [0, T ], we set
(
∗ 1 , 0),
h1 (I) = (KH
1 I
p
2
∗
∗ 1 ).
h (I) = (ρKH2 1I , 1 − ρ2 KH
2 I

(2.5)

Note that since the kernel KHi (t, s) is a Volterra kernel, i.e., KHi (t, s) = 0 if s ∈ [t, T ], the process B̃ i is
G-adapted. We can take a modification of the process B̃ i which has Hi − -Hölder continuous trajectory
for each  ∈ (0, Hi ) and is denoted by B i . The covariance between B 1 and B 2 are given by (2.6) (see
Proposition 2.9 below).
Proposition 2.9. Let Bt1 = (Bt1 )t∈[0,T ] and B 2 = (Bt2 )t∈[0,T ] be fractional Brownian motions on [0, T ]
defined as above. Then we have
Z t Z s
1 2
E{Bt Bs } = ρcH1 cH2
du
dv β(u, v)|u − v|H1 +H2 −2 uH1 −H2 v H2 −H1 ,
(2.6)
0

0

where
(
B(H1 − 1/2, 2 − H1 − H2 )
β(u, v) =
B(H2 − 1/2, 2 − H1 − H2 )

if u ≤ v,
if v < u.

Here B(a, b) denotes the Beta function.
Proof. We have
E{Bt1 Bs2 } = ρ

T

Z

KH1 (t, r)KH2 (s, r) dr
Z t∧s
= ρcH1 cH2
dr r1−H1 −H2
0
Z t
Z s
H1 −3/2 H1 −1/2
×
du (u − r)
u
dv (v − r)H2 −2/3 v H2 −1/2
r
r
Z t
Z s
= ρcH1 cH2
du
dv uH1 −1/2 v H2 −1/2
0
0
Z u∧v
×
dr r1−H1 −H2 (u − r)H1 −3/2 (v − r)H2 −3/2 .
0

0

(2.7)

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KOHEI CHIBA

v−r
v
Suppose that u ≤ v. By the change of variables z = u−r
and x = uz
, we obtain
Z u
dr r1−H1 −H2 (u − r)H1 −3/2 (v − r)H2 −3/2
0
Z ∞
H1 +H2 −2
dz (zu − v)1−H1 −H2 z H2 −3/2
= (v − u)
v/u
H1 +H2 −2 1/2−H2 1/2−H1

= (v − u)

u

v

B(H1 − 1/2, 2 − H1 − H2 ).

(2.8)

The case where v < u can be obtained similarly:
Z v
dr r1−H1 −H2 (u − r)H1 −3/2 (v − r)H2 −3/2 = (u − v)H1 +H2 −2 u1/2−H2 v 1/2−H1
0

× B(H2 − 1/2, 2 − H1 − H2 ).
Plugging (2.7) and (2.8) into (2.9), we complete the proof.

(2.9)


Corollary 2.10. Assume that H1 = H2 = H and ρ = 1. Then we have
Z t Z s
1 1
2
dv |u − v|2H−2
du
E{Bt Bs } = cH B(H − 1/2, 2 − 2H)
0

0

1
= (t2H + s2H − |t − s|2H ).
2
We will use the representation (2.4) in the proof of Theorem 4.6. The next proposition shows that we
can always construct an isonormal Gaussian process W over L2 ([0, T ]; R2 ) satisfying (2.4) if we start with
fBMs satisfying (2.6).
Proposition 2.11. Let B 1 = (Bt1 )t∈[0,T ] and B 2 = (Bt2 )t∈[0,T ] be fBMs satisfying (2.6). Then there exists
an isonormal Gaussian process W over L2 ([0, T ]; R2 ) such that
 1
∗ 1
Bt = W ((KH
, 0)),
1 [0,t]
p
(2.10)
∗ 1
2
∗ 1
))
Bt = W ((ρKH
, 1 − ρ2 KH
2 [0,t]
2 [0,t]
holds.
∗ )−1 ϕ) for ϕ ∈
Proof. We define an isonormal Gaussian process over L2 ([0, T ]) by W l (ϕ) = B l ((KH
l
∗ 1
)
by
definition.
For
step
functions
f
,
f
,
it
holds that
L2 ([0, T ]) and l = 1, 2. Then Btl = W l (KH
1
2
[0,t]
l
∗
∗
E{B 1 (f1 )B 2 (f2 )} = ρhKH
f , KH
f i 2
.
1 1
2 2 L ([0,T ])

(2.11)

We can verify (2.11) holds for any f1 ∈ HH1 and f2 ∈ HH2 by approximating fl (with respect to the norm
k · kHHl ) by step functions. Therefore we have
E{W 1 (ϕ)W 2 (ψ)} = ρhϕ, ψiL2 ([0,T ])

(2.12)

for φ, ψ ∈ L2 ([0, T ]). We set
(
W̃ 1 (ϕ) = W 1 (ϕ)
p
W̃ 2 (ψ) = (W 2 (ψ) − ρW 1 (ψ))/ 1 − ρ2
and
W ((ϕ, ψ)) = W̃ 1 (ϕ) + W̃ 2 (ψ)

(2.13)

for φ, ψ ∈ L2 ([0, T ]). It is easy to check that the isonormal Gaussian process W defined by (2.13) satisfies
(2.10).

3. Model assumptions
Now we describe our model in this section. We give precise assumptions on the observed processes and
the observation times.

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

7

3.1. Lead-lag between stochastic processes involving fBM. Using the correlated fBMs introduced
in Section 2.2, we consider the following model for a lead-lag relationship between stochastic processes
involving fBM.
Assumption 3.1. Let (Ω, F, P) be an underlying probability space. If θ ∈ Θ≥0 := [0, δ), then we assume
that the two-dimensional process X = (Xt )t∈[0,T +δ] = ((Xt1 , Xt2 ))t∈[0,T +δ] is given by
(
1
Xt1 = X01 + σ1 Bt+θ
+ A1t ,
Xt2 = X02 + σ2 Bt2 + A2t ,
for t ∈ [0, T + δ], where
(A1) X01 and X02 are real-valued random variables,
(A2) the processes B 1 and B 2 are fBMs satisfying (2.6), and
(A3) the drift processes A1 and A2 are Lipschitz continuous P-almost surely.
Moreover, if θ ∈ Θ<0 := (−δ, 0), then we assume that the process X ? = (X 2 , X 1 ) has a representation
(
2
Xt2 = X02 + σ2 Bt−θ
+ A2t ,
Xt1 = X01 + σ1 Bt1 + A1t ,
for t ∈ [0, T + δ] where the conditions (A1)-(A3) are in force.
A lead-lag fractional Black-Scholes model, in which we are interested, can be described as follows.
Example 3.2. Let (Ω, F, P) be a probability space which supports fBMs B 1 and B 2 satisfying (2.6). Here
we assume that the σ-field F is complete.
Let us consider a system of stochastic differential equations
(
Rt
Rt
St1 = S01 + µ1 0 Ss1 ds + σ 1 0 Ss1 dBs1 ,
Rt
Rt
(3.1)
St2 = S02 + µ2 0 Ss2 ds + σ 2 0 Ss2 dBs2 ,
where t ∈ [T + 2δ], S0i > 0, µi ∈ R and σ i ∈ R \ {0} (i = 1, 2). Then, as we noted in Section 1, the solution
of system (3.1) can be written as
Sti = S0 exp(Ait + σ i Bti ),
i 2

where Ait = µi t if we use the Riemann-Stieltjes integral, and Ait = µi t − (σ2) t2H if we consider the
Wick-Itô-Skorokhod integral. Note that the function Ai is Lipschitz continuous in either case.
Let θ ∈ [0, δ). We define X = (X 1 , X 2 ) by
(
1
1
+ A1t+θ ,
Xt1 = log St+θ
= log S01 + σ 1 Bt+θ
Xt2 = log St2 = log S02 + σ 2 Bt2 + A2t ,
for t ∈ [0, T + δ]. Then the processes X 1 and X 2 satisfy Assumption 3.1.
We give another example of the process that satisfies Assumption 3.1 in Section 6.1.
3.2. Observation. Now we give the assumptions on the observation. We consider the problem of estimating the lead-lag parameter θ ∈ Θ from discretely observed X 1 and X 2 . Since we keep in mind that the
processes X 1 and X 2 are prices of stocks traded at high frequency, it is natural to consider asymptotics
where the number of observations tend to infinity as n → ∞. Therefore we assume that the process X 1
is observed at
1,n
1,n
T 1,n = {0 = t1,n
0 < t1 < . . . < tN1 (n) = T + δ},
and the process X 2 at
2,n
2,n
T 2,n = {0 = t2,n
0 < t1 < . . . < tN2 (n) = T + δ},

where n, N1 (n), and N2 (n) are positive integers. Note that they are in general unevenly spaced, nonsynchronous and may be random and depend on X. Now we introduce some notations that will be useful
for describing the assumptions on the observation times T 1,n and T 2,n .

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KOHEI CHIBA

Notation 3.3. We set
•
•
•
•
•
•
•
•
•
•
•

k,n
Iik,n = (tk,n
i−1 , 1i ], i = 1, . . . , Nk (n) for k = 1, 2,
|Iik,n | = tk,n
− tk,n
i
i−1 , i = 1, . . . , Nk (n) for k = 1, 2,
k,n
k,n
I
= {Ii | i = 1, . . . , Nk (n)} for k = 1, 2,
rn = max{|Ii1,n |, |Ij2,n | | i = 1, . . . , N1 (n), j = 1, . . . , N2 (n)} (the maximum mesh size of the
observation times at the step n),
I = inf{x | x ∈ I} for an interval I,
I = sup{x | x ∈ I} for an interval I,
Iθ = {x + θ | x ∈ I} for an interval I and θ ∈ R,
k,n
Iθk,n = {(I
Si )θ | i = 1, . . . , Nk (n)} for k = 1, 2 and θ ∈ R,
K(I) = k : Kk ∩I6=∅ Kk for given subintervals K := {Kk | k} and I (if there are no k such that
Kk ∩ I 6= ∅, then we set K(I) = ∅),
f (I) = f (I) − f (I) for a function f and an interval I, and
V≥0 = V ∩ [0, ∞) and V<0 = V ∩ (−∞, 0) for a subset V of R.

Notation 3.4. Let {ai }i∈I be a real sequence indexed
byPI ⊂ Z≥0 . If we omit the range of summation,
P
the index I is empty, then we set
then we mean the summation
is,
i ai =S i∈I ai . If T
S over I, that T
P
A
(resp.
A
)
as
A
(resp.
a
=
0.
We
also
denote
i
i
i
i
i Ai ) for sets {Ai }i∈I .
i
i∈I
i∈I
i∈I
Now we give precise assumptions on the observation times. We give some examples that satisfy Assumption 3.5 in Section 6.2.
Assumption 3.5. Let (Ω, F, P) be an underlying probability space. Suppose that there are random
variables {Nl (n) ∈ N | l = 1, 2; n ∈ N} and {tl,n
i | i = 0, 1 . . . , Nl (n); l = 1, 2; n ∈ N} such that, for l = 1, 2
and n ∈ N,
l,n
l,n
0 = tl,n
0 < t1 < . . . < tNl (n) = T + δ,

T
hold P-almost surely. We also assume that l=1,2 {i | Iil,n ≤ T } =
6 ∅ for all n ∈ N. We set T l,n = {tl,n
i |
i = 1, . . . , Nl (n)}.
The observation times T 1,n and T 2,n satisfy following conditions.
(B1) The σ-algebra σ(∪n∈N T 1,n ) generated by the observation times T 1,n , n ∈ N is independent of the
σ-algebra σ(∪n∈N T 2,n ) generated by T2,n , n ∈ N. Moreover the σ-algebra σ(T ) := σ(∪n∈N (T 1,n ∪
T 2,n )) is independent of the fBMs B 1 and B 2 .
(B2) There exist positive constants c∗ > 0 and  > 0 such that
 P



|Ii1,n |H1 +H2
1,n
1,n


i:Ii ≤T,Ii ≥
≥ c∗ → 1
(3.2)
P qP
q


1,n 2H1 P
2,n 2H2


|I
|
|I
|
i i
j j
and
 P

2,n H1 +H2


 j:Ij2,n ≤T,Ij2,n ≥ |Ii |

P qP
≥ c∗ → 1
qP


1,n 2H1
2,n 2H2


i |Ii |
j |Ij |

(3.3)

hold as n → ∞.
Moreover, we assume that there exists a deterministic sequence {vn }n∈N such that vn ∈ (0, δ) for all
n ∈ N, vn → 0 as n → ∞ and the following properties (B3) and (B4) hold.
(B3) For any µ > 0, there exists γ(µ) > µ such that
rn2H1 −1+µ
P

i:Ii1,n ≤T

|Ii1,n |2H1

= Op (vnγ(µ) )

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

and

rn2H2 −1+µ
P

j:Ij2,n ≤T

|Ij2,n |2H2

9

= Op (vnγ(µ) )

hold.
(B4) It holds that
lim v 2−(H1 ∨H2 ) (E{#I 1,n }
n→∞ n
Remark 3.6. The assumption

T

| Iil,n ≤ T } =
6 ∅ for all n ∈ N is in fact not a constraint. Since

 P
l,n H1 +H2


|I
|

o
[ 
i:Iil,n ≤T,Iil,n ≥ i
qP
=0
>T ⊂
qP


1,n 2H1
2,n 2H2

l=1,2 
i |Ii |
j |Ij |

l=1,2 {i

[ \ n l,n
Ii
l=1,2 i

+ E{#I 2,n }) = 0.

T
S
T
holds, we have P{ l=1,2 i {Iil,n ≤ T }} → 1 as n → ∞ by (B2). Hence we can assume l=1,2 {i | Iil,n ≤
T} =
6 ∅ outside an asymptotically negligible set.
Assumption 3.5 is for consistency of the lead-lag estimator defined later in Definition 4.4. These
conditions may seem strange in comparison with the assumptions in Hoffmann et al. (2013). In Section
4, we compare Assumption 3.5 with the assumptions in Hoffmann et al. (2013).
4. Construction of an esimator for the lead-lag parameter
Let us now turn to construct an estimator for the lead-lag parameter θ ∈ Θ. First we explain the idea
for our estimator. Then we define our estimator in Definition 4.4 and state the main theorem of this
paper (Theorem 4.6). Finally we compare our consistency result with that of Hoffmann et al. (2013). In
particular, we consider differences of the assumptions on the sampling scheme and the grid between our
case and the semimartingale case.
4.1. The idea for the lead-lag estimator. To explain the idea for our estimator, let us outline the
proof of consistency of the lead-lag estimator in semimartingale case. For now, we assume that the
observed process X = (X 1 , X 2 ) is a regular semimartingale with lead-lag parameter θ ∈ [0, δ). We also
assume that the drift terms A1 and A2 are zero for simplicity. Let θ̃n ∈ [0, δ) ∩ G n .
Suppose first that |θ̃n − θ| converges to 0 rapidly as n → ∞. Then the Hayashi-Yoshida covariance
contrast Un (θ̃n ) is “nearly” the Hayashi-Yoshida estimator:
X
Un (θ̃n ) =
X 1 (Ii1,n )X 2 (Ij2,n )1I 1,n ∩(I 2,n ) 6=∅
1,n

i,j:I i

=

i

j

−θ̃n

≤T

X

X 1 (Ii1,n )τ−θ̃n (X 2 )((Ij2,n )−θ̃n )1I 1,n ∩(I 2,n )
i

1,n
i,j:I i ≤T

j

−θ̃n 6=∅

.

Therefore, the limit
Un (θ̃n ) →p hX, τ−θ (Y )iT

(4.1)

holds as n → ∞. This is Proposition 4 of Hoffmann et al. (2013). For the properties of the Hayashi-Yoshida
estimator, we refer to Hayashi and Kusuoka (2008); Hayashi and Yoshida (2005, 2008) for example.
On the other hand, we can prove
|Un (θ̃)| →p 0

sup
θ̃∈G n ∩{θ̃∈Θ||θ̃−θ|≥2v

(4.2)

n}

as n → ∞. This is a consequence of Proposition 3 of Hoffmann et al. (2013). If |θ̃ − θ| ≥ 2vn , then the
contrast function Un (θ̃) can be rewritten as the terminal value of a martingale outside of an asymptotically
negligible set. To handle the probability of the supremum, the Bürkholder-Davis-Gundy inequality is used
in Hoffmann et al. (2013).

10

KOHEI CHIBA

By (4.1) and (4.2), the contrast function Un (θ̃) has a peak at θ̃ = θ asymptotically on the set
{hX, τ−θ (Y )iT 6= 0}. Hence we can conclude that the maximum contrast estimator is consistent.
Now we go back to the original problem. Let us assume that the observed process X satisfies Assumption
3.1. We essentially follow the strategy above. However, we can not use the Hayashi-Yoshida covariance
contrast function directly. This is because the Hayashi-Yoshida estimator may converge to zero if the
observed process is not a semimartingale. For example, let us set A1 = A2 = 0, σ1 = σ2 = 0, H1 = H2 = H
and ρ = 1 in Assumption 3.1 and T l,n = {(1 + δ)i/n | i = 1, . . . , n} for l = 1, 2 in Assumption 3.5. We
also assume θ = 0 for simplicity. Then we obtain
Un (0) =

bn/(1+δ)c

1

n

X

n2H−1
2H−1

1
1
)2
− B(i−1)/n
(Bi/n

i=1

→p 0
as n → ∞ by ergodicity and self-similarity of fractional Gaussian noise. To ensure that the contrast
function is away from zero when |θ̃n − θ| converges to 0 rapidly as n → ∞, we consider the “correlation
version” of the contrast Un (θ̃). We consider the shifted Hayashi-Yoshida correlation contrast function
Uncor : Θ → R defined by
P
X 1 (Ii1,n )X 2 (Ij2,n )1I 1,n ∩(I 2,n ) 6=∅
i,j:Ii1,n ≤T
i
j
−θ̃
cor
Un (θ̃) = 1Θ≥0 (θ̃) r
qP
P
2 2,n 2
X 1 (Ii1,n )2
1,n
j X (Ij )
≤T

i:Ii

X 1 (Ii1,n )X 2 (Ij2,n )1(I 1,n ) ∩I 2,n 6=∅
i
j
θ̃
rP
qP
.
1 (I 1,n )2
2 (I 2,n )2
X
X
2,n
i
i
j

P
+ 1Θ<0 (θ̃)

i,j:Ij2,n ≤T

j:Ij

≤T

Then we can ensure that the contrast Uncor (θ̃n ) is away from zero when |θ̃n − θ| converges to 0 rapidly as
n → ∞.
We have another problem. In the semimartingale case, the proof of (4.2) relies on the martingale
property of the observed process. However, we cannot expect it in our case. We overcome this problem
by exploiting the Gaussian property of the observed process: the Wiener chaos decomposition and the
hypercontractivity.
4.2. Definition of the lead-lag estimator. Following the discussion above, we now define the lead-lag
estimator in our case.
Definition 4.1. We define the shifted Hayashi-Yoshida correlation contrast function Uncor : Θ → R by
P
X 1 (Ii1,n )X 2 (Ij2,n )1I 1,n ∩(I 2,n ) 6=∅
1,n
≤T
i,j:I
i
j
−θ̃
Uncor (θ̃) = 1Θ≥0 (θ̃) r i
qP
P
2 2,n 2
X 1 (Ii1,n )2
1,n
j X (Ij )
≤T

i:Ii

X 1 (Ii1,n )X 2 (Ij2,n )1(I 1,n ) ∩I 2,n 6=∅
i
j
θ̃
rP
qP
.
1,n
2,n
1
2
2
2
X (Ij )
2,n
i X (Ii )

P
+ 1Θ<0 (θ̃)

i,j:Ij2,n ≤T

j:Ij

if

P

i:Iil,n ≤T

X l (Iil,n )2 6= 0 for l = 1, 2. When

P

i:Iil,n ≤T

(4.3)

≤T

X l (Iil,n )2 = 0 for some l, we set Uncor (θ̃) = 0.

Hereafter, we denote the contrast function Uncor (θ̃) in (4.3) by Un (θ̃) for notational simplicity.
Our estimator θ̂n is obtained by maximizing the contrast function |Un (θ̃)| over a finite grid G n in the
parameter space Θ. The assumptions on the grid G n are as follows.
Assumption 4.2. Let G n be a finite subset of the parameter space Θ such that
(C1) 0 ∈ G n ,

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

11

(C2) #G n = O(vn−β ) for some β > 0, and
(C3) for some deterministic sequence ρn > 0 and positive constant ς ∈ (0, 1 − (H1 ∨ H2 )), we have
[
Θ⊂
[θ̃ − ρn , θ̃ + ρn ]
θ̃∈G n

and
lim ρ1−ς (E{#I 1,n }
n→∞ n

+ E{#I 2,n }) = 0.

Here the sequence (vn )n∈N is from Assumption 3.5.
Example 4.3. An example of the grid G n satisfying Assumption 4.2 is
G n = {vn2−(H1 ∨H2 )+ m | m ∈ Z} ∩ Θ
for any  > 0 by (B4).
Now we are ready to define our lead-lag estimator.
Definition 4.4. Let Un (θ̂n ) be the shifted Hayashi-Yoshida correlation contrast function defined in Definition 4.1. We define the lead-lag estimator θ̂n as a solution of
|Un (θ̂n )| = max |Un (θ̃)|.
θ̃∈G n

Remark 4.5. We can assume that θ̂n is chosen to be measurable. For example, let θ̂n∗ denote the largest
element in the set arg maxθ̃∈G n |Un (θ̃)|. Then we have
(
)
{θ̂n∗ ≤ r} =

max
θ̃∈G n ∩(r,δ)

|Un (θ̃)| <

max
θ̃∈G n ∩[0,r]

|Un (θ̃)|

for any r ∈ [0, δ), and hence θ̂n∗ is measurable.
Let us state our main theorem.
Theorem 4.6. Suppose that
(a) the processes X 1 and X 2 are defined as in Assumption 3.1 with ρ 6= 0,
(b) the observation times T 1,n and T 2,n satisfy Assumption 3.5, and
(c) the grid G n satisfies Assumption 4.2.
Then the estimatior θ̂n satisfies
vn−1 (θ̂n − θ) →p 0

(4.4)

as n → ∞. Here the symbol →p denotes convergence in probability. Note that the sequence (vn )n∈N is
from Assumption 3.5.
Remark 4.7.
(1) It is interesting to consider removing the assumption H ∈ (1/2, 1). By constructing
correlated fBMs from correlated BMs as in Section 2.2, we could consider a lead-lag model between two
processes involving fBMs with any Hurst parameter. However, we use the representation (2.6) of the
covariance when we prove the consistency of the lead-lag estimator.
(2) As we shall see in Section 6, the convergence rate vn essentially equals n−1 when the observation
times are synchronous and equispaced with n points or given by the Poisson sampling scheme with frequency proportional to n. However, as mentioned in Hoffmann et al. (2013), the convergence rate vn could
be improved in our case. In fact, Koike (2016a) analyzed the lead-lag model based on Gaussian likelihood
and suggested the possibility such that the convergence rate becomes n−3/2 when the observed processes
are correlated Brownian motions with the microstructure noise and the observation is synchronous and
evenly spaced with n points.

12

KOHEI CHIBA

(3) For financial high-frequency data, the true price process are considered to be contaminated by
market microstructure noise, although we do not take into account the presence of it in this paper. In the
semimartingale case, it is suggested in Hoffmann et al. (2013) that using a contrast function based on the
pre-averaged Hayashi-Yoshida estimator gives a consistent lead-lag estimator even if microstructure noise
is present. However, it is unclear whether the same approach works or not in our non-semimartingale
case. For the properties of the pre-averaged Hayashi-Yoshida estimator, see Koike (2014, 2016b) and
references therein.
Before proceeding the proof of Theorem 4.6, let us compare our consistency result with the one of
Hoffmann et al. (2013).
The condition (B1) in Assumption 3.5 is a counterpart of the condition [B2] of Hoffmann et al. (2013).
Although they allowed the observation times to be dependent on the process, we content ourselves with
the assumption that the observation times are independent of the process.
The condition (B2) has no counterpart in Hoffmann et al. (2013). This is because (B2) is naturally
satisfied in semimartingale case. Let us set H1 = H2 = 1/2 in (B2). Then (B2) becomes as follows.
(B20 ) There exist positive constants c∗ > 0 and  > 0 such that
)
(
X
l,n
P
|Ii | ≥ c∗ →p 1
i:Iil,n ≤T,Iil,n ≥

holds as n → ∞ for l = 1, 2.
Let us assume rn = op (1). Since we consider high-frequency asymptotics, this is a minimum requirement.
Then (B20 ) is derived from rn = op (1) since the inequality
(
P

)
X

|Iil,n | ≥ T /4

≥ P{rn < T /8}

i:Iil,n ≤T,Iil,n ≥T /2

holds.
The condition (B3) is a counterpart of the condition [B1] of Hoffmann et al. (2013), that is, rn = op (vn ).
If H1 = H2 = 1/2, then the condition (B3) becomes as follows.
γ(µ)

(B30 ) For any µ > 0 there exists γ(µ) > µ such that rnµ = Op (vn

) holds as n → ∞.

(B30 )

The condition
clearly implies rn = op (vn ). Hence (B3) is a bit stronger than the condition [B1] of
Hoffmann et al. (2013).
The condition (B4) also has no counterpart in the semimartingale case. Unfortunately, the author has
no explanation for the condition (B4). We use the condition (B4) only when we show that the contrast
Un (θ̃n ) is away from zero if |θ̃n − θ| converges to 0 rapidly as n → ∞. In semimartingale case, the
consistency of the Hayashi-Yoshida estimator is exploited here.
Finally, Assumption 4.2 is a counterpart of [B3] in Hoffmann et al. (2013).

5. Proof of Theorem 4.6
In this section, we give the proof of Theorem 4.6. Without loss of generality, we can assume that
θ ∈ Θ≥0 . By Assumption 3.1, the case where θ ∈ Θ<0 is equivalent to the case of −θ ∈ Θ≥0 with indices
1 and 2 interchanged. Therefore the proof when θ ∈ Θ≥0 can be applied in the case of θ ∈ Θ<0 . First we
prove Theorem 4.6 assuming that A1 = A2 = 0, and then we remove this assumption.
Notation 5.1. Let {an }n∈N and {bn }n∈N be sequences of real numbers. If there exists a positive constant
c > 0 independent of n such that an ≤ cbn for all n, then we write an . bn .

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

13

5.1. Proof of Theorem 4.6 when A1 = A2 = 0. Under the additional assumption A1 = A2 = 0, the
contrast function Un (θ̃) becomes
P
B 1 ((Ii1,n )θ )B 2 (Ij2,n )1I 1,n ∩(I 2,n ) 6=∅
i,j:Ii1,n ≤T
i
j
−θ̃
qP
Un (θ̃) = 1Θ≥0 (θ̃)
qP
2,n
2
2
B 1 ((Iin )θ )2
j B (Ij )
i:Ii1,n ≤T
P
B 1 ((Ii1,n )θ )B 2 (Ij2,n )1(I 1,n ) ∩I 2,n 6=∅
i,j:Ij2,n ≤T
i
j
θ̃
rP
+ 1Θ<0 (θ̃) pP
(5.1)
1 ((I n ) )2
2 (I 2,n )2
B
B
2,n
i
i θ
j
j:Ij

≤T

if the denominators appearing in (5.1) are not equal zero. Note that they are not equal zero P-almost
surely. We start with the following lemma.
Lemma 5.2. Let J denote [0, T ] or [0, T + δ]. Then we have, for all a ∈ [0, δ),
P
B l ((Iil,n )a )2
i:Iil,n ∈J
→p 1
P
l,n 2Hl
|I
|
l,n
i

(5.2)

i:Ii ∈J

as n → ∞ for l = 1, 2.
Proof. First we define an auxiliary set An (K) by



\  rn2Hl −1+µ vn−γ(µ)
An (K) =
≤K .
P


|Iil,n |2Hl
l,n
l=1,2
i:Ii ≤T

Since

P

i:Iil,n ≤T

|Iil,n |2Hl ≤ T rn2Hl −1 holds, we have
rn ≤ (T K)1/µ vnγ(µ)/µ ,

on the set An (K). In particular, it holds that (rn /vn )1An (K) → 0 as n → ∞ (recall that γ(µ) > µ).
Thanks to (B3), it holds that supn P(An (K)c ) <  for each  > 0 if K = K() is sufficiently large. Let
P
P
R1l,n denote (
B l ((Iil,n )a )2 )/(
|Iil,n |2Hl ) for simplicity. For any positive number 1 > 0, we
l,n
l,n
have

i:Ii ∈J

i:Ii ∈J

n
P

o
o
n
R1l,n − 1 ≥ 1 ≤ P R1l,n − 1 ≥ 1 , An (K) + P{An (K)c }.

Hence we obtain
n
P

o
n
o
R1l,n − 1 ≥ 1 ≤ P R1l,n − 1 ≥ 1 , An (K(2 )) + 2


2
l,n
−2
≤ 1 E R1 − 1 1An (K(2 )) + 2 .

By using [B1] and conditioning, we can calculate the expectation in (5.3) as


2
E R1l,n − 1 1An (K(2 ))
  P


l ((I l,n ) )2 − |I l,n |2Hl ) 2


(B
 



a
i
i
i:Iil,n ∈J

=E E 
σ(T
)
1
P
An (K(2 ))


|I l,n |2Hl


 

i:Iil,n ∈J i
(
)


 2
P


l,n
l,n 2Hl


l
2


1
E
B
((I
)
)
−
|I
|
σ(T
)
a


l,n
A
(K(
))
n
i
i
2


i:Ii ∈J
=E
.
P


(
|Iil,n |2Hl )2


l,n


i:Ii ∈J





(5.3)

14

KOHEI CHIBA

Since B l ((Iil,n )a )2 − |(Iil,n )|2Hl = B l ((Iil,n )a )2 − |(Iil,n )a |2Hl coincides with the multiple Wiener integral
I2 (hl ((Iil,n )a )⊗2 ) (recall that the definition of hl (I) is given in (2.5)) by Theorem 2.7, we have



 
2











X
 X
l,n
l,n 2Hl 
l,n
l
2
l
⊗2 
2
E 
B ((Ii )a ) − |Ii |
h
((I
)
)
σ(T
)
σ(T
)
=
E
I


a
2
i








l,n
l,n
i:Ii ∈J

i:Ii ∈J

=

X

=

X

hh ((Iil,n )a ), hl ((Ijl,n )a )i2L2 ([0,T +2δ];R2 )
l

i,j

h1(I l,n )a , 1(I l,n )a iHHl
i

i,j

j

Since |h1(I l,n )a , 1(I l,n )a iHHl | ≤ rn2Hl because of Cauchy-Schwarz inequality, we have
i

j



2






X
 X
l,n
l,n 2Hl 
l
2
h1(I l,n )a , 1(I l,n )a iHHl
σ(T
)
≤ rn2Hl
B ((Ii )a ) − |Ii |
E 

i
j




i,j
l,n
i:Ii ∈J

. rn2Hl .
Note that h1(I l,n )a , 1(I l,n )a iHHl is non-negative. Plugging this into (5.3), we obtain
i

n
P

j

R1l,n − 1 ≥ 1

o



P
E
. −2
1


2

rnHl
i:Iil,n ∈J

|Iil,n |2Hl

 1A (K( ))
n
2




+ 2





2

 r2Hl −1+µ v −γ(µ)
n
n
1−Hl −µ γ(µ) 

r
v
1
+ 2
= −2
E
P
An (K(2 ))
n
n
1


|I l,n |2Hl
l,n
i:Ii ∈J

i

2/µ 2(1−Hl )
vn
+ 2 ,
. −2
1 (T K(2 ))

if we choose sufficiently small µ > 0 such that 1 − H − µ > 0 holds. Since 1 and 2 are arbitrary, we
obtain (5.2) by letting n → ∞.

Let us set
B 1 ((Ii1,n )θ )B 2 (Ij2,n )1I 1,n ∩(I 2,n )

X

R2n (θ̃) = 1Θ≥0 (θ̃)

i

−θ̃ 6=∅

j

i,j:Ii1,n ≤T

+ 1Θ<0 (θ̃)

X

B 1 ((Ii1,n )θ )B 2 (Ij2,n )1(I 1,n )
i

2,n
θ̃ ∩(Ij )6=∅

,

i,j:Ij2,n ≤T

and
v X
sX
u
1,n 2H1
u
D (θ̃) = 1Θ≥0 (θ̃)t
|Ii |
|Ij2,n |2H2
n

i:Ii1,n ≤T

j

sX
v X
1,n 2H1 u
+ 1Θ<0 (θ̃)
|Ii | u
t
i

|Ij2,n |2H2 .

j:Ij2,n ≤T

Lemma 5.2 reduces calculation of Un (θ̃) to that of R2n (θ̃)/Dn (θ̃). Next we show that R2n (θ̃)/Dn (θ̃) goes
to zero as n → ∞ if θ̃ is distant from θ.

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

15

Proposition 5.3. We have, for any  > 0,
sup
G n ∩{θ̃∈Θ||θ̃−θ|≥vn }

R2n (θ̃)
→p 0
Dn (θ̃)

(5.4)

as n → ∞.
Proof. The set G n ∩ {θ̃ ∈ Θ | |θ̃ − θ| ≥ vn } can be decomposed as
n
n
G n ∩ {θ̃ ∈ Θ | |θ̃ − θ| ≥ vn } = (G≥0
∩ {θ̃ ∈ Θ | θ̃ ≥ θ + vn }) ∪ (G≥0
∩ {θ̃ ∈ Θ | θ̃ ≤ θ − vn })
n
∪ (G<0
∩ {θ̃ ∈ Θ | θ̃ ≤ θ − vn })

=: G1n ∪ G2n ∪ G3n .
Clearly (5.4) is equivalent to
sup
Gin

R2n (θ̃)
→p 0
Dn (θ̃)

(5.5)

for i = 1, 2, 3. We only prove (5.5) when i = 1. The other cases are similar.
Let us assume H1 ≤ H2 for the moment. Since θ̃ ∈ G1n , we have
X
B 1 ((Ii1,n )θ )B 2 ((Ij2,n ))1(I 1,n )θ ∩(I 2,n ) 6=∅
R2n (θ̃) =
i

j

θ−θ̃

i,j:Ii1,n ≤T

X

=

2,n
B 1 ((Ii1,n )θ )B 2 (Iθ−
((Ii1,n )θ )θ̃−θ )
θ̃

i:Ii1,n ≤T

X

=

2,n
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ii1,n )θ )θ̃−θ )iL2 ([0,T +2δ];R2 )
θ̃

i:Ii1,n ≤T

+

X 

2,n
B 1 ((Ii1,n )θ )B 2 (Iθ−
((Ii1,n )θ )θ̃−θ )
θ̃

i:Ii1,n ≤T
2,n
− hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ii1,n )θ )θ̃−θ )iL2 ([0,T +2δ];R2 )
θ̃



=: R3n (θ̃) + R4n (θ̃).
Hence it suffices to show
R3n (θ̃)
→p 0
n
D (θ̃)

(5.6)

R4n (θ̃)
→p 0
Dn (θ̃)

(5.7)

R3n (θ̃)
1An (K) → 0
Dn (θ̃)

(5.8)

sup
θ̃∈G1n

and
sup
θ̃∈G1n

as n → ∞.
The limit (5.6). It suffices to show
sup
θ̃∈G1n

as n → ∞ for each K > 0. By (2.6) in Proposition 2.9, it holds that
Z
X Z
|R3n (θ̃)| = |ρ|cH1 cH2
du
i:Ii1,n ≤T

(Ii1,n )θ

I 2,n ((Ii1,n )θ )θ̃−θ
θ−θ̃

× |u − v|H1 +H2 −2 uH1 −H2 v H2 −H1 .

dv β(u, v)

16

KOHEI CHIBA

Note that for each K > 0 and  > 0 we can choose n0 = n0 (K, ) such that n ≥ n0 implies

(T K)1/µ vn(γ(µ)/µ)−1 ≤ .
3
Hence if n ≥ n0 then rn 1An (K) ≤ (vn /3) holds. We can always assume n ≥ n0 in this proof. Since θ̃ ∈ G1n
and rn ≤ (vn )/3 on An (K), we have |u − v| ≥ (vn )/3. Therefore
Z
X Z
du uH1 −H2
dv v H2 −H1
(5.9)
|R3n (θ̃)| . (vn )H1 +H2 −2
i:Ii1,n ≤T

(Ii1,n )θ

I 2,n ((Ii1,n )θ )θ̃−θ
θ−θ̃

on An (K). Since H1 ≤ H2 , we have
Z
I 2,n ((Ii1,n )θ )θ̃−θ

dv v H2 −H1 . rn .

(5.10)

θ−θ̃

Since the integral

R

(Ii1,n )θ

du uH1 −H2 is summable (note that H1 − H2 ∈ (−1/2, 1/2)), we have
|R3n (θ̃)| . (vn )H1 +H2 −2 rn .

(5.11)

R3n (θ̃)
1An (K) . vnγ(µ)−µ 1An (K) .
Dn (θ̃)

(5.12)

By (5.11) and (B3), we have

Since the right-hand-side of (5.12) does not depend on θ̃, we obtain (5.8).
˜ 2 ([0, T + 2δ]; R2 ) by
The limit (5.7). We define h̃ni (θ̃) ∈ L2 ([0, T + 2δ]; R2 )⊗L
˜ 2 (I 2,n ((Ii1,n )θ )θ̃−θ (ω)).
h̃ni (θ̃)(ω) = h1 ((Ii1,n )θ (ω))⊗h
θ−θ̃
Let 1 and 2 be positive numbers. By the same conditioning technique as in the proof of Lemma 5.2, we
have
)
(

2p
P




n


(
)
h̃
(
θ̃))
σ(T
)
1
E
I
(
2
1,n


A
(K(
))
i
n
2


i:Ii ≤T
n
X
R4 (θ̃)
−2p
E
P sup
≥ 1 ≤ 2 + 1
(5.13)
n


|Dn (θ̃)|2p


θ̃∈G1n D (θ̃)
n


θ̃∈G1




for any p ≥ 1. Using Theorem 2.8, we obtain

 
 
 2p







 X


 X
h̃ni (θ̃) σ(T ) ≤ Cp E I2 
h̃ni (θ̃)
E I2 






1,n
1,n
i:Ii

≤T

i:Ii

≤T

2

σ(T )

p



(5.14)




where Cp is a positive constant depending only on p ≥ 1. Let R5n (θ̃) denote the expectation in the
right-hand-side of (5.14). A simple calculation using Proposition 2.6 yields
X
R5n (θ̃) =
hh1 ((Ii1,n )θ ), h1 ((Ik1,n )θ )iL2 ([0,T +2δ];R2 )
i,k:Ii1,n ≤T, Ik2,n ≤T
2,n
2,n
× hh2 (Iθ−
((Ii1,n )θ )θ̃−θ , h2 (Iθ−
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 )
θ̃
θ̃
X
2,n
+
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 )
θ̃
i,k:Ii1,n ≤T, Ik2,n ≤T
2,n
× hh1 ((Ik1,n )θ ), h2 (Iθ−
((Ii1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 )
θ̃
n
n
=: R5,1
(θ̃) + R5,2
(θ̃).

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

17

(1) The first term can be estimated as follows. Since
2,n
2,n
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) . rn2H2 ,
hh2 (Iθ−
((Ii1,n )θ )θ̃−θ , h2 (Iθ−
θ̃
θ̃

we have
n
|R5,1
(θ̃)| . rn2H2

(5.15)

on An (K(2 )). Note that I 7→ h1 (I) is linear so that
X
hh1 ((Ii1,n )θ ), h1 ((Ik1,n )θ )iL2 ([0,T +2δ];R2 ) < ∞.
i,k:Ii1,n ≤T, Ik1,n ≤T

(2) The second term can be estimated as follows. We first recall that
2,n
|hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) |
θ̃
Z
Z
du
dv β(u, v)
= |ρ|cH1 cH2
(Ii1,n )θ

I 2,n ((Ik1,n )θ )θ̃−θ
θ−θ̃

× |u − v|H1 +H2 −2 uH1 −H2 v H2 −H1 .
By the same reasoning as (5.9), we have
2,n
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) .
θ̃

 v H1 +H2 −2
n

3

rn |Ii1,n |

if i ≤ k, and
2,n
hh1 ((Ik1,n )θ ), h2 (Iθ−
((Ii1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) .
θ̃

 v H1 +H2 −2
n

3

rn |Ik1,n |

if i > k, on An (K(2 )). Therefore, it holds that
n
|R5,2
(θ̃)| . rn vnH1 +H2 −2

X

2,n
((Ii1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) |
|Ii1,n ||hh1 ((Ik1,n )θ ), h2 (Iθ−
θ̃

i≤k

!
+
.

X

2,n
((Ik1,n )θ )θ̃−θ iL2 ([0,T +2δ];R2 ) |
|Ik1,n ||hh1 ((Ii1,n )θ ), h2 (Iθ−
θ̃

k>i
1+H2 H1 +H2 −2
rn
vn

(5.16)

on An (K(2 )). Plugging (5.15) and (5.16) into (5.13), we obtain
(
)
(
)
X
1An (K(2 )) |rn2H2 + rn1+H2 vnH1 +H2 −2 |p
R4n (θ̃)
−2p
E
P sup
≥ 1 . 2 + 1
n
|Dn (θ̃)|2p
θ̃∈G1n D (θ̃)
n
θ̃∈G1

. 2 +

1−2p (#G n )vnp(1−H1 )
p(1−H )

1
by (B3). Thanks to (C2), we have limn→∞ (#G n )vn
= 0 if we choose sufficiently large p ≥ 1. This
gives (5.7).
Thanks to (5.6) and (5.7), we obtain (5.5) when H1 ≤ H2 . Now let us consider the case where H1 > H2 .
In the case where H1 > H2 , we use an alternative expression of R2n (θ̃):
X
R2n (θ̃) =
B 1 ((Ii1,n )θ )B 2 (Ij2,n )1(I 1,n ) ∩I 2,n 6=∅
i

θ̃

j

i,j:Ii1,n ≤T

=

X

B 2 (Ij2,n )B 1 (Iθ̃1,n,≤T (Ij2,n )θ−θ̃ ),

j

where the symbol Iθ̃1,n,≤T denotes the family of shifted intervals
Iθ̃1,n,≤T = {(Ii1,n )θ̃ | Ii1,n ≤ T }.

(5.17)

18

KOHEI CHIBA

Using the expression (5.17), we can prove (5.5) when H1 > H2 in the same manner as we did in the case
where H1 ≤ H2 .

Let us show that R2n (θ̃n )/Dn (θ̃n ) is away from zero if |θ̃n − θ| decreases rapidly as n → ∞.
Proposition 5.4. Suppose that θ̃n satisfies |θ − θ̃n | ≤ ρn for all n ∈ N. Then there exists c0∗ > 0 such
that
(
)
R2n (θ̃n )
0
P
≥ c∗ → 1
(5.18)
Dn (θ̃n )
as n → ∞.
Proof. We formally extend B 1 and B 2 by setting Bt1 = Bt2 = 0 for t < 0. Let us first suppose that
θ̃n ∈ Θ≥0 . This is the case for sufficiently large n if θ ∈ (0, δ). Then we decompose R2n (θ̃n ) into three
terms:


X
1,n
2,n
1,n
2 2,n
)
−
B
(I
((I
)
)
∩
[0,
T
+
2δ])
B 1 ((Ii1,n )θ ) B 2 (Iθ−
((I
)
)
R2n (θ̃n ) =
θ
θ θ̃n −θ
i
i
θ−θ̃
θ̃
n

n

i:Ii1,n ≤T

+

X 

2,n
((Ii1,n )θ ) ∩ [0, T + 2δ])
B 1 ((Ii1,n )θ )B 2 (Iθ−
θ̃
n

i:Ii1,n ≤T


1,n
2,n
− hh1 ((Ii1,n )θ ), h2 (Iθ−
((I
)
)
∩
[0,
T
+
2δ])i
2
2
θ
L ([0,T +2δ];R )
i
θ̃n
X
2,n
+
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ii1,n )θ ) ∩ [0, T + 2δ])iL2 ([0,T +2δ];R2 )
θ̃
n

i:Ii1,n ≤T

=: R6n (θ̃n ) + R7n (θ̃n ) + R8n (θ̃n ).

(5.19)

2,n
2,n
((Ii1,n )θ ) ∩ [0, T + 2δ]. Let us show
((Ii1,n )θ )+ = Iθ−
To simplify the notation, we set Iθ−
θ̃
θ̃
n

n

R6n (θ̃n )
→p 0,
Dn (θ̃n )

(5.20)

R7n (θ̃n )
→p 0
Dn (θ̃n )

(5.21)

and that there exists c∗ > 0 such that
(
P

R8n (θ̃n )
≥ c0∗
n
D (θ̃n )

)
→1

(5.22)

as n → ∞.
The limit (5.20). Thanks to (B3), it suffices to show
R6n (θ̃n )
1An (K) →p 0
Dn (θ̃n )

(5.23)

as n → ∞. By the conditioning argument as in the proof of Lemma 5.2, we have
(
)
(
|R6n (θ˜n )|1An (K)
1An (K) X
E
≤E
E{|B 1 ((Ii1,n )θ )|
n|
n
|D
|D (θ̃n )|
i
)
2,n
2,n
× |B 2 (Iθ−
((Ii1,n )θ )θ̃n −θ ) − B 2 (Iθ−
((Ii1,n )θ )+ )| | σ(T )} .
θ̃
θ̃
n

n

(5.24)

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

19

Let p > 1 and q > 1 be conjugate indices: 1/p + 1/q = 1. Using (conditional) Hölder’s inequality and
Theorem 2.8, we have
X

2,n
2,n
E{|B 1 ((Ii1,n )θ )||B 2 (Iθ−
((Ii1,n )θ )θ̃n −θ ) − B 2 (Iθ−
((Ii1,n )θ )+ )| | σ(T )}
θ̃
θ̃
n

i

≤
.

X

n

1/q
1/p X
 2 2,n
1,n
1,n
1,n
p
2 2,n
+ q
E{|B((Ii )θ )| | σ(T )}
E |B (Iθ−θ̃ ((Ii )θ )θ̃n −θ ) − B (Iθ−θ̃ ((Ii )θ ) )| | σ(T )

i
H1 −1/p
rn
((#I 1,n )|θ

− θ̃n |qH2 )1/q .

(5.25)

Plugging (5.25) into (5.24) with p = 1 −
(
E

n

n

i

|R6n (θ˜n )|1An (K)
|Dn (θ̃n )|

)

(
≤E


H2 −1
1−ς


H2 −1
,
1−ς

and q =

we obtain

H ς

2 −µ
H1 +H2 −1+µ+ 1−ς

(#I 1,n )|θ − θ̃n |1−ς

rn

 H2

)

1−ς

|Dn (θ̃n )|

1An (K) .

(5.26)

2ς
Now we can choose sufficiently small µ > 0 satisfying H
1−ς − µ > 0. Applying (B3), (C3) and Jensen’s
inequality to (5.26), we complete the proof of (5.20).
The limit (5.21). This can be proved in the same way as (5.7). Note that in this case the term
n (θ̃) is
corresponding to R5,2

2,n
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ik1,n )θ )+ )iL2 ([0,T +2δ];R2 )
θ̃

X

n
R7,2
(θ̃n ) :=

n

i,k:Ii1,n ≤T, Ik2,n ≤T
2,n
((Ii1,n )θ )+ )iL2 ([0,T +2δ];R2 ) .
× hh1 ((Ik1,n )θ ), h2 (Iθ−
θ̃
n

This is estimated as follows:
n
|R7,2
(θ̃n )| . rnH1 +H2

2,n
hh1 ((Ii1,n )θ ), h2 (Iθ−
((Ik1,n )θ )+ )iL2 ([0,T +2δ];R2 )
θ̃

X

n

i,k:Ii1,n ≤T, Ik2,n ≤T

. rnH1 +2H2 (#I 1,n )
= rn2H1 −1+µ rn2H2 −1+µ rn2−H1 −2µ (#I 1,n ).
This, combined with (B4), completes the proof of (5.21).
The limit (5.22). Finally we analyze the term R8n (θ̃n ). Recall that
|R8n (θ̃n )| = |ρ|cH1 cH2

X Z
i:Iin ≤T

Z

(Ii1,n )θ

du

I 2,n ((Ii1,n )θ )+

dv β(u, v)

θ−θ̃n

× |u − v|H1 +H2 −2 uH1 −H2 v H2 −H1 .
2,n
Since (Ii1,n )θ ⊂ Iθ−
((Ii1,n )θ )+ for all i with Ii1,n ≤ T , it holds that
θ̃
n

|R8n (θ̃n )| ≥ |ρ|cH1 cH2

X Z
i:Ii1,n ≤T

(Ii1,n )θ

Z
du

(Ii1,n )θ

× |u − v|H1 +H2 −2 uH1 −H2 v H2 −H1 .

dv β(u, v)
(5.27)

20

KOHEI CHIBA

To obtain a lower bound for (5.27), we note the following fact: for any  > 0, it holds that
Z
X Z
du
dv β(u, v)|u − v|H1 +H2 −2 uH1 −H2 v H2 −H1
(Ii1,n )θ

i:Ii1,n ≤T

(Ii1,n )θ

i:Ii1,n ≤T,Ii1,n ≥

du

i:Ii1,n ≤T,Ii1,n ≥

(Ii1,n )θ

Z

Z

X

&

Z

Z

X

≥

(Ii1,n )θ

(Ii1,n )θ

du

(Ii1,n )θ

dv β(u, v)|u − v|H1 +H2 −2 uH1 −H2 v H2 −H1

dv |u − v|H1 +H2 −2 .

(5.28)

Combining (5.27) nad (5.28), we obtain
|Ii1,n |H1 +H2

X

|R8n (θ̃n )| &

(5.29)

i:Ii1,n ≤T,Ii1,n ≥

for any  > 0. Therefore we obtain
P
|I 1,n |H1 +H2
i:Ii1,n ≤T,Ii1,n ≥ i
R8n (θ̃n )
.
& qP
q
1,n 2H1 P
2,n 2H2
Dn (θ̃n )
|I |
|I |
i

j

i

j

We complete the proof of (5.22) by (B2) (especially (3.2)).
As we noted above, if the true parameter value θ is positive, then θ̃n is in Θ≥0 for sufficiently large
n since |θ̃n − θ| ≤ ρn . Therefore we can conclude (5.18) by (5.20)-(5.22) if θ > 0. However θ̃n may be
negative for any n when θ = 0. In order to obtain (5.18) when θ = 0, it suffices to show (5.20)-(5.22) hold
for θ = 0 and θ̃n ∈ [−ρn , 0). This is completely analogous to the case where θ̃n ∈ Θ≥0 , so that we only
add a few remarks and omit the proof.
If θ = 0 and θ̃n ∈ [−ρn , 0), then (5.19) becomes
X
R2n (θ̃n ) =
B 1 (Ii1,n )B 2 (Ij2,n )1(I 1,n ) ∩I 2,n 6=∅
i

j

θ̃n

i,j:Ij2,n ≤T

X

=




B 1 (Iθ̃1,n (Ij2,n )−θ̃n ) − B 1 (Iθ̃1,n (Ij2,n )+ ) B 2 (Ij2,n )
n

n

j:Ij2,n ≤T

+

X



B 1 (Iθ̃1,n (Ij2,n )+ )B 2 (Ij2,n ) − hh1 (Iθ̃1,n (Ij2,n )+ ), h2 (Ij2,n )iL2 ([0,T +2δ];R2 )
n



n

j:Ij2,n ≤T

+

X

hh1 (Iθ̃1,n (Ij2,n )+ ), h2 (Ij2,n )iL2 ([0,T +2δ];R2 ) .

(5.30)

n

j:Ij2,n ≤T

Here the symbol Iθ̃1,n (Ij2,n )+ denotes Iθ̃1,n (Ij2,n ) ∩ [0, T + 2δ] as before. Let R̂8n (θ̃n ) denote the last term in
n
n
(5.30). Then the inequality corresponding to (5.29) is
X
|Ii2,n |H1 +H2 .
|R̂8n (θ̃n )| &
j:Ij2,n ≤T,Ij2,n ≥

In particular, we use (3.3) of (B2) in this case.



The following corollary is immediate from Propositions 5.3 and 5.4 and Lemma 5.2.
Corollary 5.5.

(1) We have
|Un (θ̃)| →p 0

sup
θ̃∈G n ∩{θ̃∈Θ||θ̃−θ|≥v

n}

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

21

as n → ∞ for any  > 0.
(2) Suppose that θ̃n ∈ G n satisfies |θ − θ̃n | ≤ ρn for all n ∈ N. Then there exists c∗ > 0 such that
n
o
P |Un (θ̃n )| ≥ c∗ → 1
as n → ∞.
Now we are ready to prove Theorem 4.6 under the additional assumptions A1 = A2 = 0.
Proof of Theorem 4.6 when A1 = A2 = 0. Thanks to (C3), there is a sequence (θ̃n )n∈N ⊂ G n satisfying
|θ̃n − θ| ≤ ρn . By the definition of θ̂n , we have |Un (θ̃n )| ≤ |Un (θ̂n )|. On the other hand, one obtains
|Un (θ̂n )| ≤ supθ̃∈G n ∩{θ̃∈Θ||θ̃−θ|≥vn } |Un (θ̃)| on the set {vn−1 |θˆn − θ| ≥ }. Therefore, we obtain
)
(
−1 ˆ
|Un (θ̃)| .
(5.31)
P{v |θn − θ| ≥ } ≤ P |Un (θ̃n )| ≤
sup
n

θ̃∈G n ∩{θ̃∈Θ||θ̃−θ|≥vn }

The right-hand-side of (5.31) tends to 0 as n → ∞ by Corollary 5.5.



5.2. Proof of Theorem 4.6. Let Un0 (θ̃) denote the right-hand-side of (5.1). Now we consider the general
case: we no longer assume A1 = A2 = 0. We reduce the general case to the previous one. We establish
a relation between Un (θ̃) and Un0 (θ̃) in Proposition 5.8 below. We begin with a lemma that relates the
denominators of Un (θ̃) and those of Un0 (θ̃).
Lemma 5.6. Let J denote [0, T + δ] or [0, T ]. We set a1 = θ and a2 = 0. Then we have
P
l,n
l l,n 2
2 l
2
i:Iin ∈J (X (Ii ) − σl B ((Ii )al ) )
→p 0
P
l,n 2Hl
|I
|
n
i:I ∈J i

(5.32)

i

as n → ∞ for l = 1, 2.
Proof. By the Hölder continuity of B l , we have, for any  > 0,
|X l (Iil,n )2 − σl2 B l ((Iil,n )al )2 | = |Al (Iil,n )2 + 2σl Al (Iil,n )B l ((Iil,n )al )|
. |Iil,n |2Hl + |Iil,n |1+Hl −
. rn |Iil,n | + rnHl − |Iil,n |.
Since Hl − = (2Hl −1)+1−Hl −, we obtain (5.32) by (B3) if we choose  > 0 such that 1−Hl − > 0. 
We set
X

Ř2n (θ̃) = 1Θ≥0 (θ̃)

X 1 (Ii1,n )X 2 (Ij2,n )1I 1,n ∩(I 2,n )
i

−θ̃ 6=∅

j

i,j:Ii1,n ≤T

+ 1Θ<0 (θ̃)

X

X 1 (Ii1,n )X 2 (Ij2,n )1(I 1,n )
i

2,n
θ̃ ∩(Ij )6=∅

.

i,j:Ij2,n ≤T

The next lemma relates the numerators of Un (θ̃) and those of Un0 (θ̃).
Lemma 5.7. We have
sup
θ̃∈G n

Ř2n (θ̃) − σ1 σ2 R2n (θ̃)
→p 0
n
D (θ̃)

as n → ∞.
Proof. Note that
X 1 (Ii1,n )X 2 (Ij2,n ) = A1 (Ii1,n )A2 (Ij2,n ) + σ2 A1 (Ii1,n )B 2 (Ij2,n )
+ σ1 B 1 ((Ii1,n )θ )A2 (Ij2,n ) + σ1 σ2 B 1 ((Ii1,n )θ )B 2 (Ij2,n ).

(5.33)

22

KOHEI CHIBA

Obviously (5.33) is equivalent to
sup
θ̃∈G n ∩Θ≥0

Ř2n (θ̃) − σ1 σ2 R2n (θ̃)
→p 0
Dn (θ̃)

(5.34)

Ř2n (θ̃) − σ1 σ2 R2n (θ̃)
→p 0.
Dn (θ̃)

(5.35)

and
sup
θ̃∈G n ∩Θ<0

We only prove (5.34). The proof of (5.35) is completely analogous. Since θ̃ ∈ Θ≥0 , we have
X
2,n 1,n
|Ř2n (θ̃) − R2n (θ̃)| ≤
|A1 (Ii1,n )A2 (I−
(I )θ̃ )|
θ̃ i
i:Ii1,n ≤T

+

2,n 1,n
|σ2 A1 (Ii1,n )B 2 (I−
(I )θ̃ )|
θ̃ i

X
i:Ii1,n ≤T

+

X

|σ1 B 1 (Iθ1,n,≤T ((Ij2,n )θ−θ̃ ))A2 (Ij2,n )|

j

. rn + rnH1 + r2H2
= rnH1 +H2 −1+µ (rn2−H1 −H2 −µ + rn1−H2 −µ + rn1−H1 −µ ).
We can choose µ > 0 such that 1 − (H1 ∨ H2 ) − µ > 0 so that we conclude (5.34) by (B3).

P
T
X l (Iil,n )2 6= 0 . Note that P{Sn } → 1 as n → ∞ by Lemmas 5.2 and
Let us set Sn = l=1,2
l,n
i:Ii ≤T

5.6. The next proposition shows that we can reduce calculation of Un (θ̃) to that of Un0 (θ̃).
Proposition 5.8. It holds that


1 + op (1) 0
U (θ̃) 1Sn .
Un (θ̃) = ōp (1)(θ̃) +
1 + op (1) n

(5.36)

Here ōp (1)(θ̃) denotes a quantity such that
sup ōp (1)(θ̃) →p 0.
θ̃∈G n

In particular,
(1) we have
|Un (θ̃)| →p 0

sup
θ̃∈G n ∩{θ̃∈Θ||θ̃−θ|≥v

n}

as n → ∞ for any  > 0, and
(2) if θ̃n ∈ G n satisfies |θ̃n − θ| ≤ vn1+α , then there exists c∗ > 0 such that
n
o
P |Un (θ̃n )| ≥ c∗ → 1
as n → ∞.
Proof. We have (5.36) by Lemmas 5.2, 5.6 and 5.7.



Now we are ready to prove Theorem 4.6.
Proof of Theorem 4.6. Thanks to Proposition 5.8, (4.4) follows by the same line of argument as in the
proof of Theorem 4.6 when A1 = A2 = 0.

6. Examples
In this section, we give some examples that satisfy the assumptions of Theorem 4.6.
6.1. Processes.

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

23

6.1.1. The solution of a stochastic differential equation driven by fBM. Let (Ω, F, G = (Gt )t∈[0,T +2δ] , P)
be a stochastic basis which supports fBMs B 1 and B 2 satisfying (2.6). Here we assume that the σ-field
F is complete and the filtration G satisfies the usual conditions. Let us consider a stochastic differential
equation
Z t
b(s, Xsl ) ds + σ l Btl
(6.1)
Xtl = X0l +
0

for t ∈ [0, T + 2δ] and l = 1, 2. We assume that X0 is a real-valued random variable, σ l ∈ R \ {0} and the
function b satisfies the following conditions (D1) and (D2).
Assumption 6.1. The function b : [0, T + 2δ] × R → R satisfies that
(D1) for each positive integer N there exists a positive constant LN > 0 such that
|b(t, x) − b(t, y)| ≤ LN |x − y|
for all x ∈ [−N.N ], y ∈ [−N, N ] and t ∈ [0, T + 2δ], and
(D2) there exists a positive constant L0 > 0 such that
|b(t, x)| ≤ L0 (1 + |x|)
for all x ∈ R and t ∈ [0, T + 2δ].
Then there exists a unique solution X l to the equation (6.1) which has 1/2-Hölder continuous sample
paths almost surely. This is a special case of Theorem 2.1 of Nualart and Răsçanu (2002).
Let us check the processes τ−θ (X 1 ) and X 2 satisfy Assumption 3.1. Since X l satisfies the equation
(6.1), it suffices to verify (A3). We have
|b(s, Xsl )| ≤ L0 (|Xsl | + 1) ≤ L0 (|Xsl − X0l | + |X0l | + 1) ≤ L0 (C(T + 2δ)1/2 + |X0l | + 1)
for some Hölder constant C > 0. Hence it holds that
|Alt − Als | ≤ L0 (C(T + 2δ)1/2 + |X0l | + 1)|t − s|
for 0 ≤ s ≤ t ≤ T + 2δ.
6.2. Sampling scheme.
6.2.1. Synchronous and evenly spaced observations. Let us check that the synchronous and evenly spaced
observations satisfy Assumption 3.5. Let il,n
∗ (a) denote
l,n
il,n
∗ (a) = inf{i | Ii ≥ a}

for a > 0. Note that, using il,n
∗ (a), we obtain the following relation:
P
Pil,n
∗ (T )−1
|Iil,n |H1 +H2
l,n
l,n
|Iil,n |H1 +H2
i:Ii ≤T,Ii ≥
i=il,n
∗ ()+1
qP
= qP
.
q
q
1,n 2H1 P
2,n 2H2
1,n 2H1 P
2,n 2H2
|I
|
|I
|
|I
|
|I
|
i i
j j
i i
j j
Proposition 6.2. Let us set
T

1,n

=T

2,n


=


i
(T + δ) | i = 0, . . . , n .
n



1−(H1 ∨H2 )
Then T 1,n and T 2,n satisfy Assumption 3.5 with vn = (1/n)1−ξ for any ξ ∈ 0, 2−(H
.
1 ∨H2 )

24

KOHEI CHIBA

Proof. First (B1) is clear. Note that il,n
∗ () =

l

n
T +δ

m

in this case. Let us check (B2):

Pil,n
∗ (T )−1
|Iil,n |H1 +H2
|Iil,n |H1 +H2
i=il,n
()+1
qP
= qP ∗
q
q
1,n 2H1 P
2,n 2H2
1,n 2H1 P
2,n 2H2
|I
|
|I
|
|I
|
i i
j j
i i
j |Ij |




Tn
n
1
−1−
=
n
T +δ
T +δ
T −
→
T +δ
as n → ∞. Next we verify (B3):
 µ  
−1
 
−1
µ
1
1
Tn
1
Tn
rn2Hl −1+µ
1−ξ
=
= vn
.
P
n
n T +δ
n T +δ
|Iil,n |2Hl
l,n
P

i:Iil,n ≤T,Iil,n ≥

i:Ii ≤T

We can take γ(µ) = µ/(1 − ξ). Finally we verify (B4):
vn2−(H1 ∨H2 ) E{#I 1,n }

=

vn2−(H1 ∨H2 ) E{#I 2,n }

 (1−ξ)(2−(H1 ∨H2 ))−1
1
=
.
n



1−(H1 ∨H2 )
, then (1 − ξ)(2 − (H1 ∨ H2 )) − 1 > 0 so that (B4) holds.
If ξ ∈ 0, 2−(H
1 ∨H2 )



6.2.2. Poisson Sampling. We check that the Poisson sampling scheme satisfies Assumption 3.5. Let
p1 > 0 and p2 > 0 be positive numbers, and N 1,n and N 2,n be mutually independent Poisson processes
with intensity p1 n and p2 n, respectively. We also assume N 1,n and N 2,n are independent of the underlying
fractional Brownian motions B 1 and B 2 . We set Til,n = inf{t ≥ 0 : Ntl,n = i} ∧ (T + δ), T̃il,n = inf{t ≥
l,n
l,n
l,n
0 : Ntl,n = i}, Iil,n = (Ti−1
, Til,n ], I˜il,n = (T̃i−1
, T̃il,n ] and λl,n (t) = dpl nte. Note that il,n
∗ (t) = Nt− + 1 in
this case.
Proposition 6.3. We set
n
o
T 1,n = Ti1,n | i = 0, . . . , i1,n
(T
+
δ)
∗
and

n
o
T 2,n = Tj2,n | j = 0, . . . , i2,n
∗ (T + δ) .



1 ∨H2 )
Then T 1,n and T 2,n satisfy Assumption 3.5 with vn = (1/n)1−η for any η ∈ 0, 1−(H
2−(H1 ∨H2 ) .
In order to prove Proposition 6.3, we use the following lemmas.
Lemma 6.4. Let
!
r̃n =

max

1≤i≤λ1,n (T +δ)∨i1,n
∗ (T +δ)

|I˜i1,n |

!
∨

max

1≤j≤λ2,n (T +δ)∨i2,n
∗ (T +δ)

|I˜j2,n |

.

Then for any q ≥ 1, we have
E{r̃nq } = o(nα ),
for any 0 < α < q.
Proof. See Lemma 8 of Hayashi and Yoshida (2008).



Lemma 6.5. Let  ∈ (1/2, 1). Then, for each T > 0, it holds that

P{|λl,n (T ) − il,n
∗ (T )| ≤ n } → 1,

as n → ∞ for l = 1, 2.
Proof. See Lemma 9 of Hayashi and Yoshida (2008).



LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

25

Before starting the proof of Proposition 6.3, we introduce some notation: for a ∈ (0, T + δ] and
H ∈ (1/2, 1), we set
Pil,n
∗ (a)−1
|Iil,n |H1 +H2
l,n
R9 (a) = qP i=1
,
q
1,n 2H1 P
2,n 2H2
|I
|
|I
|
i i
j j
l,n,±
R10
(a)

Pλl,n (a)±d2n2/3 e ˜l,n H1 +H2
|I |
i=1
q
=
,
q i
Pλl,n (T +δ)∓d2n2/3 e ˜1,n 2H Pλl,n (T +δ)∓d2n2/3 e ˜2,n 2H
1
2
|
I
|
|
I
|
i=1
i
j=1
j

l,n,±
R11
(H, a)

λl,n (a)±d2n2/3 e 

X

=

i=1

Γ(2H + 1)
|I˜il,n |2H −
(npl )2H


,

and
l,n,±
R12
(H, a) =

Γ(2H + 1)(λl,n (a) ± d2n2/3 e)
.
(npl )2H

Note that we have the following relation:





l,n,± H1 + H2
l,n,± H1 + H2
l,n,±
, a + R12
,a
R10 (a) = R11
2
2
−1/2

1,n,∓
1,n,∓
(H1 , T + δ)
(H1 , T + δ) + R12
× R11

−1/2
2,n,∓
2,n,∓
× R11
(H2 , T + δ) + R12
(H2 , T + δ)
.

(6.2)

Proof of Proposition 6.3. By assumption, (B1) holds. Let us show that Poisson sampling scheme satisfies
condition (B2).
Claim. Let a ∈ (0, T + δ]. Then it holds that
a
R9n (a) →p
cl
(6.3)
T +δ
as n → ∞. Here cl is a constant depending on l, H1 , H2 , p1 and p2 :
cl =

H −1/2 H2 −1/2
p2
Γ(H1 + H2 + 1)
.
H1 +H2 −1
pl
Γ(2H1 + 1)Γ(2H2 + 1)

p1 1

In particular, (B2) holds.
Let us show (6.3). First note that a straightforward calculation yields E{|I˜il,n |2H } = Γ(2H + 1)/(npl )2H
and Var{|I˜il,n |2H } = (Γ(4H + 1) − Γ(2H + 1)2 )/(npl )4H for H ∈ (1/2, 1). By Lemma 6.5, we can assume
l,n
l,n

l,n

l,n

|λl,n (a) − il,n
∗ (a)| ≤ n and |λ (T + δ) − i∗ (T + δ)| ≤ n for  ∈ (1/2, 1). Since |λ (a) − i∗ (a)| ≤ n , we
l,n
l,n
l,n
l,n
l,n

l,n

have λ (a) − d2n e < i∗ (a) and hence Ii = I˜i if i ≤ λ (a) − d2n e. Note that clearly |Ii | ≤ |I˜il,n |
holds for all i. Hence we have
l,n,−
l,n,+
R10
(a) ≤ R9l,n (a) ≤ R10
(a).

In order to prove (6.3), it suffices to show
a
cl
T +δ
as n → ∞. Thanks to (6.2), we can further reduce (6.4) to the following limits:
l,n,±
R10
(a) →p

l,n,±
n2H−1 R12
(H, a) →

aΓ(2H + 1)
(T + δ)pl2H−1

(6.4)

(6.5)

and
l,n,±
R11
(H, a) = op (n1−2H )

(6.6)

26

KOHEI CHIBA

as n → ∞ for each H ∈ (1/2, 1). First (6.5) is straightforward. Next, Kolmogorov’s inequality shows that
o λl,n (a) ± d2n2/3 e
n
l,n,±
≥ n−ξ ≤
(Γ(4H + 1) − Γ(2H + 1)2 )
(6.7)
P n2H−1 R11
n2(1−ξ) p4H
l
holds for H ∈ (1/2, 1), and the right-hand-side of (6.7) goes to 0 as n → ∞ if ξ < 1/2. Hence (6.6) holds.
As a consequence of (6.3), we can conclude that (B2) holds: we have
P
|I l,n |H1 +H2
i:Iil,n ≤T,Iil,n ≥ i
T −
qP
→p
cl
qP
T +δ
1,n 2H1
2,n 2H2
|I
|
|I
|
i i
j j
as n → ∞.
Next we show that (B3) holds. Note that for each η > 0, we can choose ζ > 0 satisfying η > ζ/(1 + ζ).
Claim. If ζ > 0 satisfies η > ζ/(1 + ζ), then, for each µ > 0 and γ(µ) := (1 + ζ)µ > µ, it holds that
rn2Hl −1+µ
P

i:Iil,n ≤T

|Iil,n |2Hl

= op (vnγ(µ) ).

In particular, (B3) holds.
Indeed, let  > 0 be any positive number. Then we have


)
(
 r2Hl −1+µ v −γ(µ)

r̃n2Hl −1+µ n(1−η)(1+ζ)µ
n
n
l,n
l,n
2/3
P P
≥ , |λ (T ) − i∗ (T )| ≤ n
≥  ≤ P P l,n
λ (T )−d2n2/3 e ˜l,n 2H


l
|Iil,n |2Hl
|
I
|
l,n
i=1
i
i:Ii ≤T
o
n
2/3
+ P |λl,n (T ) − il,n
(T
)|
>
n
∗
(
)
r̃n2Hl −1+µ n2Hl −1+(1−η)(1+ζ)µ
≤P
≥  + o(1).
l,n,±
l,n,±
n2Hl −1 R11
(Hl , T ) + n2Hl −1 R12
(Hl , T )
Hence it holds that


)
(
 r2Hl −1+µ v −γ(µ)

T
Γ(2H
+
1)
n
n
l
P P
+ o(1)
≥  ≤ P r̃n2Hl −1+µ n2Hl −1+(1−η)(1+ζ)µ ≥
l,n 2Hl
l −1


2(T + δ)p2H
|I
|
l,n
i
l
i:Ii ≤T

≤

T Γ(2Hl + 1)
l −1
2(T + δ)p2H
l

!

−1
2Hl −1+µ

E{r̃n }n

2Hl −1+(1+ζ)(1−η)µ
2Hl −1+µ

+ o(1).

(6.8)

Since (1 + ζ)(1 − η)µ < µ, Lemma 6.4 implies that the right-hand-side of (6.8) goes to zero as n → ∞.
l,n
Note
 that E{#I
 } = pl n(T + δ). We can verify (B4) easily: since (1 − η)(2 − (H1 ∨ H2 )) > 1 when
1−(H1 ∨H2 )
η ∈ 0, 2−(H
, the condition (B4) holds.

1 ∨H2 )
6.3. Simulation study. In this subsection, we present a numerical simulation. We use the R package
YUIMA to generate the data, plot the paths, and calculate the estimated values.
We consider the following setting. We set dXt1 = dBt1 and dXt2 = dBt2 where B 1 and B 2 satisfy (2.6)
with H1 = 0.6, H2 = 0.7. We vary the correlation parameter ρ as ρ = 0.25, 0.50 and 0.75. The time
horizon T equals 1. We assume that the processes X 1 and X 2 are observed by independent Poisson
sampling schemes whose intensities are both n. We also vary the intensity n as n = 100, 300 and 500.
We set the true lead-lag parameter θ = 0.02, so that the process X 1 has a lead of θ = 0.02 over the
process X 2 . To calculate the estimator, we use the grids G1 = {−1 + 10−3 k | k ∈ Z≥0 } ∩ [−1, 1] and
G2 = {−1 + (3 · 10−3 )k | k ∈ Z≥0 } ∩ [−1, 1]. Hence the grid G1 contains the true parameter θ = 0.02, and
the grid G2 does not.
First we fix the grid. Then we repeat 300 simulations and compute the value of the estimator θ̂n each
time, letting ρ and n vary as we explained above. That is, we consider the following nine cases after we

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

27

fix the grid: (ρ, n) = (0.25, 100), (0.25, 300), (0.25, 500), (0.50, 100), (0.50, 300), (0.50, 500), (0.75, 100),
(0.75, 300) and (0.75, 500).
Figure 1 (resp. 2) shows a plot of estimated values θ̂n with the grid G1 (resp. G2 ). The red dashed
lines show the true parameter θ = 0.02.
Figure 3 shows a simulated observation data. Figure 4 shows a plot of the contrast function Un (θ̃)
using the simulated data shown in Figure 3. We use the grid G = G1 . Again the red dashed line shows
the true parameter θ = 0.02. In this simulation, the estimated parameter value is θ̂n = 0.02.
Acknowledgement
The author would like to express deepest gratitude to Professor Nakahiro Yoshida for introducing him
to this problem and for many valuable suggestions. He is also very grateful to two anonymous referees
for their really helpful comments and suggestions. He could never have improved the results and the
presentation of this paper without their helpful comments.
This work was supported by JST CREST and the Program for Leading Graduate Schools, MEXT,
Japan.
Conflict of interest
The author has no conflicts of interest to declare.
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E-mail address: kchiba@ms.u-tokyo.ac.jp

300

0

0

100

200

300

Experiment (rho=0.75 and n=100)

200

300

0

100

200

300

0.018

0.020

0.022

Experiment (rho=0.50 and n=500)

Estimated value

−0.2
−0.6
0

100

Experiment (rho=0.50 and n=300)

Estimated value

0.2 0.4

Experiment (rho=0.50 and n=100)

300

−0.15

−0.4
300

0.017 0.019 0.021 0.023

200

200

Experiment (rho=0.25 and n=500)

Estimated value

0.2

Estimated value
100

100

0.05

200

Experiment (rho=0.25 and n=300)

0.5
0.0
−0.5

Estimated value

0

Estimated value

100

−0.05

0

Experiment (rho=0.25 and n=100)

0.2
−0.6

−0.6
300

0.0

200

−0.2

100

−0.2

Estimated value

0.6
0.2
−0.2

Estimated value

0.5
0.0
−0.5

Estimated value

0

29

0.6

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

0

100

200

300

Experiment (rho=0.75 and n=300)

0

100

200

300

Experiment (rho=0.75 and n=500)

Figure 1. A plot of estimated values θ̂n (colored online). We fix a grid G1 = {−1+10−3 k |
k ∈ Z≥0 } ∩ [−1, 1], repeat 300 simulations and plot the computed values of the estimator
θ̂n . The red dashed lines show the true parameter θ = 0.02.

−0.6
300

0

−0.15
100

200

300

200

300

Experiment (rho=0.75 and n=100)

0

100

200

300

Experiment (rho=0.75 and n=300)

100

200

300

Experiment (rho=0.50 and n=500)

Estimated value

0.016
100

0

Experiment (rho=0.50 and n=300)

Estimated value

0.2
−0.2
−0.6

Estimated value

0

300

0.0190 0.0200 0.0210 0.0220

0

Experiment (rho=0.50 and n=100)

200

Experiment (rho=0.25 and n=500)

Estimated value

0.4
0.2

300

0.024

200

0.020

100

100

0.05

200

Experiment (rho=0.25 and n=300)

Estimated value

0.4
0.0
−0.4

100

−0.05

0

−0.8
0

−0.2

Estimated value

0.2
−0.6

300

0.0

200

−0.2

100

Experiment (rho=0.25 and n=100)

Estimated value

−0.2

Estimated value

0.5
0.0
−0.5

Estimated value

0

0.2

0.6

KOHEI CHIBA

0.6

30

0

100

200

300

Experiment (rho=0.75 and n=500)

Figure 2. A plot of estimated values θ̂n (colored online). We fix a grid G2 = {−1 + (3 ·
10−3 )k | k ∈ Z≥0 } ∩ [−1, 1], repeat 300 simulations and plot the computed values of the
estimator θ̂n . The red dashed lines show the true parameter θ = 0.02. Note that the grid
G2 does not contain the true parameter θ = 0.02.

31

0.2
−0.2

0.0

X2

0.2

0.4

−0.2

0.0

X1

0.4

0.6

LEAD-LAG ESTIMATION BETWEEN TWO PROCESSES DRIVEN BY FBM

0.0

0.2

0.4

0.6

0.8

1.0

Time

Figure 3. A simulated observation data. The underlying processes are dXt1 = dBt1 and
dXt2 = dBt2 where B 1 and B 2 satisfy (2.6) with H1 = 0.6, H2 = 0.7 and ρ = 0.50. The time
horizon is T = 1, and we use independent Poisson sampling schemes both with intensity
n = 300. The lead-lag parameter is θ = 0.02, so that the process X 1 has a lead of θ = 0.02
over the process X 2 .

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KOHEI CHIBA

−0.2

0.0

U(θ)

0.2

0.4

1 vs 2 (positive theta means 1 leads 2 )

−1.0

−0.5

0.0

0.5

1.0

θ

Figure 4. A plot of the contrast function Un (θ̃) (colored online). We use the simulated
data shown in Figure 3. In order to maximize the contrast, we use the grid G1 = {−1 +
10−3 k | k ∈ Z≥0 } ∩ [−1, 1]. The red dashed vertical line shows θ̃ = 0.02. In this simulation,
the estimated value is θ̂n = 0.02.

