EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED
FRACTIONAL BROWNIAN MOTIONS
ENKELEJD HASHORVA AND LANPENG JI
Abstract: Let {Xi (t), t ≥ 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with
constant cross-correlation. In this paper we derive the exact asymptotics of the joint survival function
(
)
sup X1 (s) > u, sup X2 (t) > u
P
s∈[0,1]
t∈[0,1]
as u → ∞. A novel finding of this contribution is the exponential approximation of the joint conditional first passage
times of X1 , X2 . As a by-product we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality
for 2-dimensional Gaussian random fields.
Key Words: Extremes; first passage times; Borell-TIS inequality; Piterbarg inequality; fractional Brownian motion;
Gaussian random fields.
AMS Classification: Primary 60G15; secondary 60G70
1. Introduction and main results
Let {Xi (t), t ≥ 0}, i = 1, 2 be two standard fractional Brownian motion’s (fBm’s) with Hurst indexes αi /2 ∈ (0, 1), i =
1, 2, i.e., Xi is a centered Gaussian process with a.s. continuous sample paths and covariance function
Cov(Xi (t), Xi (s)) =
1 αi
(|t| + |s|αi − | t − s |αi ),
2
s, t ≥ 0,
i = 1, 2.
√
Hereafter (X1 , X2 ) are assumed to be jointly Gaussian with cross-correlation function r(s, t) = E {X1 (s)X2 (t)} / sα1 tα2 ∈
(−1, 1). Calculation of the following joint survival function
(
(1)
Pr (u) := P
)
sup X1 (s) > u, sup X2 (t) > u ,
s∈[0,1]
u > 0.
t∈[0,1]
is important for various applications in statistics, mathematical finance and insurance mathematics. The special simple
model of two correlated Brownian motions (i.e., α1 = α2 = 1) with r(s, t) = r a constant has been well studied in the
literature; see e.g., [26] and [30]. Therein an explicit expression for (1) was given through the modified Bessel function
and in the form of series; recently [39] obtained some computable bounds for (1). We refer to [27] for related results.
Explicit calculation of (1) is only possible for correlated Brownian motions. Since typically in applications calculation
of the joint survival probability is needed for large thresholds u, one can rely on the asymptotic theory to find adequate
approximations of this survival probability. In [38] logarithmic asymptotics of (1) as u → ∞ for general correlated
Gaussian processes X1 , X2 was obtained; see also [13] for a general treatment of the multidimensional case. So far in the
literature there are only two contributions that derive exact asymptotics of (1) for certain Gaussian processes, namely
Cheng and Xiao [7] obtained an exact asymptotic expansion of (1) for two correlated smooth Gaussian processes
X1 , X2 . In the aforementioned paper the result was obtained by studying the geometric properties of the processes.
The second contribution is from Anshin [3] where the exact asymptotics for two correlated non-smooth Gaussian
processes X1 , X2 is derived by relying on a modified double-sum method (see [19, 32, 34, 35, 36] for details on the
double-sum method). The assumptions in [3] are such that our model of two standard fBm’s with r(s, t) = r ∈ (−1, 1)
is not included. Indeed, the conditions C1-C3 therein are all invalid for our model. Due to wide applications of fBm’s
and their exit probabilities, we consider in this paper the exact asymptotics of Pr (u) given as in (1) with X1 , X2 being
1
2
ENKELEJD HASHORVA AND LANPENG JI
the two standard fBm’s above with a constant cross-correlation function r ∈ (−1, 1). Another merit of choosing fBm’s
rather than other (general) Gaussian processes is that it allows for somewhat explicit formulae. In order to proceed
with our analysis using a modified double-sum method, we shall extend the celebrated Borell-TIS inequality and the
Piterbarg inequality for 2-dimensional Gaussian random fields in Theorem 2.1 and Theorem 2.2, respectively. These
results are of independent interest given their importance in the theory of Gaussian processes and random fields; see
[29] for new developments in this direction.
Before presenting our main results we recall the definition of the well-known Pickands constant
Hα := lim
(2)
T →∞
1 0
H [ΛT ], α ∈ (0, 2],
T α
ΛT := [0, T ],
where
b
Hα
[Λ]
(3)
√
α
, Λ ⊂ R, b ∈ R.
:= E exp sup 2Bα (t) − |t| − bt
t∈Λ
Here {Bα (t), t ∈ R} is a standard fBm defined on R with Hurst index α/2 ∈ (0, 1]. By the symmetry about 0 of the
fBm, for any T ∈ (0, ∞) we have (with ΛT given as in (2))
0
0
Hα
[[−T, 0]] = Hα
[ΛT ], H1−b [[−T, 0]] = H1b [ΛT ],
(4)
b ∈ R.
We refer to [2, 5, 10, 8, 9, 12, 17, 28, 36] for the basic properties of the Pickands and related constants.
Our first principle result is presented below.
Theorem 1.1. Let {Xi (t), t ≥ 0}, i = 1, 2 be two standard fBm’s with Hurst indexes αi /2 ∈ (0, 1), i = 1, 2, respectively.
If (X1 , X2 ) are jointly Gaussian with a constant cross correlation function r ∈ (−1, 1), then as u → ∞
3
u2
(1 + r) 2
√
(5)
(1 + o(1)),
Υ1 (u)Υ2 (u)u−2 exp −
Pr (u) =
1+r
2π 1 − r
where
Υi (u) =

1− α1
1− α2

i (1 + r)
i

 2
2
1
αi −2
,
αi Hαi u
2+r
1+r ,
if αi ∈ (0, 1),
if αi = 1,


 1,
i = 1, 2.
if αi ∈ (1, 2),
Remarks: a) The case r = 0 can be confirmed by the fact that (cf. [15] or [36]), for a standard fBm Bα
(
)
2
1
u
(6)
(1 + o(1)),
as u → ∞,
P sup Bα (t) > u = √ Fα (u)u−1 exp −
2
2π
t∈[0,1]
where Fα (u) is equal to 21−1/α α−1 Hα u2/α−2 if α ∈ (0, 1), 2 if α = 1, and 1 if α ∈ (1, 2).
b) It follows from Theorem 1.1 and (6) that M1 := sups∈[0,1] X1 (s) and M2 := supt∈[0,1] X2 (t) are asymptotically
independent, i.e.,
lim
(7)
u→∞
P {M1 > u, M2 > u}
= 0.
P {M2 > u}
We refer to [16] for the concept of the asymptotic independence of two random variables. The result in (7) improves
that in Corollary 1.1 in [39] where an upper bound (1/2) for the left-hand side of (7) for two correlated Brownian
motions was obtained.
c) In Theorem 1.1 we considered the joint extremes of two standard fBm’s on the time interval [0, 1]. Next, we briefly
discuss the case where the time interval is [0, S], with S some positive constant. It follows by the self-similarity of the
fBm’s that, for S α1 −α2 ≥ 1 we have
(
P
sup X1 (s) > u,
t∈[0,S]
)
sup X2 (t) > u
t∈[0,S]
o
n
α2
α2
= P M1 > c(S − 2 u), M2 > S − 2 u ,
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
with c = S
α2 −α1
2
3
∈ (0, 1]. By slight modifications of the proofs of Theorem 1.1 and Lemma A we conclude that similar
results can be obtained for the case c > r. However, the case c ≤ r can not be dealt with similarly since we do not
observe a similar result as Lemma A which is crucial for the double-sum method. It turns out that the case c ≤ r may
not be easily solved in general; new techniques will be explored for it elsewhere.
In the framework of ruin theory, see, e.g., [4, 11, 18, 24, 25], [11] given that ruin happens one wants to know when it
happens. With this motivation, we are interested to know when the first passages occur given that X1 , X2 both ever
pass a threshold u > 0 on [0, 1].
Define the first passage times of X1 , X2 to the threshold u by
τ1 (u) = inf{s ≥ 0, X1 (s) > u} and τ2 (u) = inf{t ≥ 0, X2 (t) > u},
(8)
respectively (here we use the common assumption that inf{∅} = ∞). Further, define τ1∗ (u), τ2∗ (u), u > 0 in the same
probability space such that
d
(τ1∗ (u), τ2∗ (u)) =
(9)
(τ1 (u), τ2 (u))(τ1 (u) ≤ 1, τ2 (u) ≤ 1),
d
where = stands for equality of distribution functions. With motivation from the aforementioned contributions, our
second principle result is concerned with the distributional approximation of the random vector (τ1∗ (u), τ2∗ (u)), as
d
u → ∞. Let Ei , i = 1, 2 be two independent unit exponential random variables, and denote by → the convergence in
distribution.
Theorem 1.2. Under the assumptions of Theorem 1.1 we have as u → ∞
d
2(1 + r)
2(1 + r)
2
∗
2
∗
E1 ,
E2 .
(10)
u (1 − τ1 (u)), u (1 − τ2 (u)) →
α1
α2
Remark: Let M1 , M2 be given as in Remark b) above. By the self-similarity of the fBm’s we have for any x1 , x2 ≥
0, u > 0
n
o
o
n
P
sup
X
(S
s)
>
u,
sup
X
(S
t)
>
u
1,u
2,u
s∈[0,1] 1
t∈[0,1] 2
x2 x1
,
P M1 > u + , M2 > u + M1 > u, M2 > u =
u
u
P {M1 > u, M2 > u}
where Si,u = (1 + xi u−2 )−2/αi , i = 1, 2, u > 0. Therefore, by a similar argument as in the proof of Theorem 1.2 we
conclude that
n
o
x1
x2 lim P M1 > u + , M2 > u + M1 > u, M2 > u
=
u→∞
u
u
(11)
x1 + x2
exp −
.
1+r
In view of Theorem 4.1 in [20] (see also Section 4.1 in [21])
n
o
x1
x2 lim P X1 (1) > u + , X2 (1) > u + X1 (1) > u, X2 (1) > u
u→∞
u
u
=
x1 + x2
exp −
1+r
holds for any x1 , x2 ∈ [0, ∞), from which we see that (11) is not surprising since the minimum of the function
h(s, t), (s, t) ∈ (0, 1]2 given in (19) is attained at the unique point (1, 1), at which the processes usually contribute most
to the asymptotics.
Organization of the rest of the paper: In Section 2 we present some preliminary results including the Borell-TIS
inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields. The proofs of Theorems 1.1 and
1.2 are given in Section 3, while proofs of other results are relegated to Appendix.
4
ENKELEJD HASHORVA AND LANPENG JI
2. Preliminaries
In the asymptotic theory of Gaussian processes, two of the important inequalities are the Borell-TIS inequality (cf.
[1, 36]) and the Piterbarg inequality (cf. [36]). Let {Z(t), t ∈ K} be a centered Gaussian process with a.s. continuous
sample paths, and let K ⊂ R be a compact set with Lebesgue measure mes(K) > 0. The Borell-TIS inequality, which
was proved by [6] and [41] independently, states that
(u − µ)2 2
P sup Z(t) > u ≤ exp −
τm
(12)
2
t∈K
2
holds for any u ≥ µ := E {supt∈K Z(t)}, with τm
:= inf t∈K (VarZ(t))
−1
∈ (0, ∞).
The upper bound in (12) might not be precise enough for various applications due to the appearance of the constant
µ. V.I. Piterbarg obtained an upper bound under a global Hölder condition on the Gaussian process, which eliminates
the constant µ; see e.g., Theorem 8.1 in [36] or Theorem 8.1 in [37]. Specifically, if there are some positive constants
γ and G such that E (Z(t) − Z(t0 ))2 ≤ G|t − t0 |γ for all t, t0 ∈ K, then
2 2
u 2
(13)
P sup Z(t) > u ≤ C mes(K) u γ −1 exp − τm
2
t∈K
holds for any u large enough, with some positive constant C not depending on u. The last inequality is commonly
referred to as the Piterbarg inequality; see e.g., Proposition 3.2 in [40] for the case of chi-processes.
Next, let V ⊂ R2 be a compact set, and let {(Z1 (t), Z2 (t)), t ≥ 0} be a 2-dimensional centered vector Gaussian
process with components which have a.s. continuous sample paths. Motivated by the findings of [13, 38], we present in
Theorem 2.1 and Theorem 2.2 generalizations of the Borell-TIS and Piterbarg inequalities for 2-dimensional Gaussian
random fields {(Z1 (s), Z2 (t)), (s, t) ∈ V}. As it will be seen from the proof of Theorem 1.1, the generalized Borell-TIS
and Piterbarg inequalities are very powerful tools.
Theorem 2.1. Let {Zi (t), t ≥ 0}, i = 1, 2 be two centered Gaussian processes with a.s. continuous sample paths,
variance functions σi (t), i = 1, 2 being further jointly Gaussian with cross-correlation function r(s, t) ∈ (−1, 1). Then
there exists a constant µ such that for u ≥ µ


 [

(u − µ)2 2
P
{Z1 (s) > u, Z2 (t) > u} ≤ exp −
τm ,
(14)


2
(s,t)∈V
2
where τm
= inf (s,t)∈V σ 2 (s, t) > 0 with (below I(·) stands for the indicator function)
(c(s, t) − r(s, t))2
σ1 (s) σ2 (t)
1
2
(15) σ (s, t) =
1+
I(r(s, t) < c(s, t)) , c(s, t) = min
,
.
min(σ12 (s), σ22 (t))
1 − r2 (s, t)
σ2 (t) σ1 (s)
In particular, if r(s, t) < c(s, t) for all (s, t) ∈ V, then (14) holds with
(16)
2
τm
= inf
(s,t)∈V
σ12 (s) + σ22 (t) − 2σ1 (s)σ2 (t)r(s, t)
σ12 (s)σ22 (t)(1 − r2 (s, t))
and further, if r(s, t) ≥ c(s, t) for all (s, t) ∈ V, then (14) holds with
2
τm
= inf
(s,t)∈V
1
.
min(σ12 (s), σ22 (t))
Theorem 2.2. Let {Zi (t), t ≥ 0}, i = 1, 2 be as in Theorem 2.1. Assume that σ1 (s), σ2 (t), r(s, t), (s, t) ∈ V are all
twice continuously differentiable with respect to their arguments. If there exist some positive constants γ and L such
that the following global Hölder condition
(17)
E (Zi (vi ) − Zi (wi ))2 ≤ L|vi − wi |γ ,
i = 1, 2
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
5
holds for all (v1 , v2 ), (w1 , w2 ) ∈ V, then for all u large


2  [

4
u 2
P
{Z1 (s) > u, Z2 (t) > u} ≤ C mes(V) u γ −1 exp − τm
(18)
,


2
(s,t)∈V
2
where τm
is given as in Theorem 2.1, and C is some positive constant not depending on u.
S
Remark 2.3. Assume that G = {(s, t) ∈ V : (s, t) = arg inf σ(s, t)} is a finite set. Define Gε =
([s − ε, s + ε] ×
(s,t)∈G
[t − ε, t + ε] ∩ V) for any small positive ε. In view of the proof of Theorem 8.1 in [36], the claim (18) still holds if (17)
is valid for all (v1 , v2 ), (w1 , w2 ) ∈ Gε for some small positive ε.
Now, we come back to our principle problem of finding the exact asymptotics of Pr (u) as u → ∞. In view of the
2
findings of [3, 13, 38] we deduce that the constant τm
given in (16) (restricted to fBm’s case) should play a crucial
role in the exact asymptotics of Pr (u). Thus, we need to analyze the following function
α1
tα2 + sα1 − 2rs 2 t
h(s, t) =
sα1 tα2 (1 − r2 )
(19)
α2
2
,
s, t ∈ (0, 1].
The function h(s, t), s, t ∈ (0, 1] attains its minimum at the unique point (s0 , t0 ) = (1, 1) and further h(1, 1) =
2
1+r .
Let (ŝ0 , t̂0 ) := (ŝ0 (u), t̂0 (u)), u > 0 be a family of points in [0, 1]2 satisfying 1 − ŝ0 ≤ (ln u)2 /u2 and 1 − t̂0 ≤ (ln u)2 /u2 .
For the use of the double-sum method, we need to deal with the asymptotics of the following joint survival function


 [

RΛ1 ,Λ2 (u) := P
{X1 (s) > u, X2 (t) > u} ,
as u → ∞,


(s,t)∈Ku
where Ku = (ŝ0 , t̂0 ) + (u−2/α1 Λ1 , u−2/α2 Λ2 ) with Λi , i = 1, 2 two compact sets in R. Here in our notation, for any
Λ ∈ R aΛ := {ax : x ∈ Λ}, and for any Λ ∈ R2 (x1 , x2 ) + Λ := {(x1 , x2 ) + (y1 , y2 ) : (y1 , y2 ) ∈ Λ}.
The following lemma can be seen as a generalization of Pickands and Piterbarg lemmas (cf. [22, 33, 35, 36]) for
2-dimensional Gaussian random fields. Its proof is presented in Appendix.
Lemma A. Let {Xi (t), t ≥ 0}, i = 1, 2 be two standard fBm’s with Hurst indexes αi /2 ∈ (0, 1/2], i = 1, 2, respectively.
Assume further that the joint correlation function of them is a constant r ∈ (−1, 1). Then as u → ∞
2
3
(1 + r) 2 −2
u
RΛ1 ,Λ2 (u) = Qα1 [Λ1 ]Qα2 [Λ2 ] √
(20)
u exp − h(ŝ0 , t̂0 ) (1 + o(1)),
2
2π 1 − r
where h(·, ·) is given as in (19), and

α2 
i
1
0

,
if αi ∈ (0, 1),
 Hαi Λi √2(1+r)
Qαi [Λi ] =
2 
−(1+r)
1

Λi √2(1+r)
, if αi = 1,
 H1
i = 1, 2.
3. Proofs of Theorems 1.1 and 1.2
In this section, we first present the proof of Theorem 1.1 which is based on a tailored double-sum method as in [3];
see the classical monograph [36] for a deep explanation on the double-sum method. Then we present the proof of
Theorem 1.2.
Proof of Theorem 1.1: Let δ(u) = (ln u)2 /u2 , and set Du = {(s, t) ∈ [0, 1]2 : 1 − s ≤ δ(u), 1 − t ≤ δ(u)}. With these
notation we have
P1,r (u) := P



[
(s,t)∈Du
{X1 (s) > u, X2 (t) > u}



≤
Pr (u)
6
ENKELEJD HASHORVA AND LANPENG JI
≤
P1,r (u) + P


[

(s,t)∈[0,1]2 /Du
{X1 (s) > u, X2 (t) > u}



=: P1,r (u) + P2,r (u).
Next, we shall derive the exact asymptotics of P1,r (u) as u → ∞, and show that
(21)
P2,r (u) = o(P1,r (u)),
u→∞
implying thus
Pr (u) = P1,r (u)(1 + o(1))
u → ∞.
Next, we derive an upper bound for P2,r (u) by utilising the generalized Borell-TIS and Piterbarg inequalities. Choose
some small ε ∈ (0, 1) such that
ĉ(s, t) := min
(22)
Clearly, for any u positive


P2,r (u) ≤ P

[
(s,t)∈[0,1]2 /[1−ε,1]
tα2 /2 sα1 /2
,
sα1 /2 tα2 /2
> r, ∀(s, t) ∈ [1 − ε, 1]2 .




{X1 (s) > u, X2 (t) > u} + P


2
[
(s,t)∈[1−ε,1]2 /Du


{X1 (s) > u, X2 (t) > u} .

It follows from the Borell-TIS inequality in Theorem 2.1 that for all u large




[
(u − µ)2
P
{X1 (s) > u, X2 (t) > u} ≤ exp −
(23)
inf
f (s, t) ,


2
(s,t)∈(0,1]2 /[1−ε,1]2
2
2
(s,t)∈[0,1] /[1−ε,1]
where µ ∈ (0, ∞) is some constant and
1
f (s, t) =
min (sα1 , tα2 )
(ĉ(s, t) − r)2
1+
I(r < ĉ(s, t)) , (s, t) ∈ (0, 1]2 /[1 − ε, 1]2 .
1 − r2
Further, straightforward calculations yield that (recall (19) for the expression of h(·, ·))
inf
(s,t)∈(0,1]2 /[1−ε,1]2
f (s, t) > h(1, 1) =
2
.
1+r
Moreover, in view of (22) we have from the Piterbarg inequality in Theorem 2.2 and its remark that, for all u large


2


[
4
u
−1
P
inf 2
h(s, t) ,
{X1 (s) > u, X2 (t) > u} ≤ Cu min(α1 ,α2 ) exp −


2 (s,t)∈[1−ε,1] /Du
2
(s,t)∈[1−ε,1] /Du
with C > 0 not depending on u. In addition from the Taylor expansion of h(s, t) around the point (1, 1) we have
h(s, t) = h(1, 1) +
1
(α1 (1 − s) + α2 (1 − t)) (1 + o(1)).
1+r
Hence, for the chosen small enough ε > 0 there exists some positive constant C1 such that
h(s, t) ≥ h(1, 1) + C1 δ(u)
for any (s, t) ∈ [1 − ε, 1]2 /Du , implying thus, for all u large




[
4
C1
−1
(24)
P
{X1 (s) > u, X2 (t) > u} ≤ Cu min(α1 ,α2 ) ψr (u) exp − (ln u)2 ,


2
2
(s,t)∈[1−ε,1] /Du
where we set
2
u
u2
ψr (u) := exp − h(1, 1) = exp −
.
2
1+r
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
7
Consequently, from (23) and (24) we obtain the following upper bound for P2,r (u) when u is large
4
(u − µ)2
C1
−1
2
min(α1 ,α2 )
P2,r (u) ≤ exp −
f
(s,
t)
+
Cu
ψ
(u)
exp
−
(25)
inf
(ln
u)
.
r
2
2
(s,t)∈[0,1]2 /[1−ε,1]2
From now on we focus on the asymptotics of P1,r (u) as u → ∞. Let T1 , T2 be two positive constants. For αi ≤ 1, i =
1, 2, we can split the rectangle Du into several sub-rectangles of side lengths T1 u−2/α1 and T2 u−2/α2 . Specifically, let
[
4k,l = 41k × 42l = [sk+1 , sk ] × [tl+1 , tl ], k, l ∈ N {0},
with sk = 1 − kT1 u−2/α1 and tl = 1 − lT2 u−2/α2 , and further set
k
j
2
−2
+ 1,
Ni (u) = Ti−1 (ln u)2 u αi
i = 1, 2.
Here b·c denotes the ceiling function. Thus
N1 (u)−1 N2 (u)−1
(26)
[
[
k=0
l=0
N1 (u) N2 (u)
4k,l ⊂ Du ⊂
[
[
k=0
l=0
4k,l .
In what follows, we deal with only three cases (distinguished by αi ’s):
Case i) α1 ∈ (0, 1) and α2 ∈ (0, 1). Applying the Bonferroni inequality in Lemma B (given in Appendix) we obtain
N1 (u) N2 (u)
P1,r (u) ≤
X X
k=0
(
)
sup X1 (s) > u, sup X2 (t) > u
P
s∈41k
l=0
t∈42l
and
N1 (u)−1 N2 (u)−1
P1,r (u) ≥
(27)
X
X
k=0
l=0
(
)
sup X1 (s) > u, sup X2 (t) > u
P
s∈41k
− Σ1 (u) − Σ2 (u),
t∈42l
where




Σ1 (u) =
P sup X1 (s) > u, sup X2 (t) > u, sup X2 (t) > u ,
s∈41

t∈42l
t∈42l
k
k=0 0≤l1 <l2 ≤N2 (u)
1
2


N2 (u)


X
XX
Σ2 (u) =
P
sup X1 (s) > u, sup X1 (s) > u, sup X2 (t) > u .
s∈41

s∈41
t∈42
N1 (u)
X
XX
l=0 0≤k1 <k2 ≤N1 (u)
k1
k2
l
Further, in view of Lemma A
"
P1,r (u) ≤
0
Hα
1
×
[−T1 , 0]
1
√
2(1 + r)
α2 #
1
"
0
Hα
2
[−T2 , 0]
1
√
2(1 + r)
α2 #
2
2
N1 (u) N2 (u)
3
(1 + r) 2 −2 X X
u
√
u
exp − h(sk , tl ) (1 + o(1))
2
2π 1 − r
k=0 l=0
as u → ∞. Since by Taylor expansion
h(sk , tl ) = h(1, 1) +
1
(α1 (1 − sk ) + α2 (1 − tl )) (1 + o(1)),
1+r
u→∞
we have
N1 (u) N2 (u)
X X
k=0
l=0
u2
exp − h(sk , tl )
2
2
2
2 Z ∞
u α1 + α2 −4 Y
αj
= ψr (u)
exp −
x dx (1 + o(1)).
T1 T2
2(1 + r)
0
j=1
Therefore, as u → ∞
1
(28)
P1,r (u) ≤
1
7
2
2
0
0
[0, b1 T1 ] Hα
[0, b2 T2 ] α2 + α2 −6
21− α1 − α2 (1 + r) 2 − α1 − α2 Hα
1
2
√
u 1 2 ψr (u)(1 + o(1)),
b1 T1
b2 T2
πα1 α2 1 − r
8
ENKELEJD HASHORVA AND LANPENG JI
√
2/αi
where bi = 1/( 2(1 + r))
, i = 1, 2. The same arguments yield that
N1 (u)−1 N2 (u)−1
X
X
k=0
l=0
(
sup X1 (s) > u, sup X2 (t) > u
P
s∈41k
1
1
)
=
t∈42l
7
2
2
0
0
[0, b1 T1 ] Hα
[0, b2 T2 ]
21− α1 − α2 (1 + r) 2 − α1 − α2 Hα
1
2
√
b1 T1
b2 T2
πα1 α2 1 − r
2
2
×u α1 + α2 −6 ψr (u)(1 + o(1))
(29)
as u → ∞. Next, we consider the estimates of Σi (u), i = 1, 2. To this end, we define, for any T, T0 ∈ (0, ∞)
(
)
Z ∞
√
√
0
α
α
Hα ([0, T ], [T0 , T0 + T ]) =
2Bα (t) − |t| > x,
2Bα (t) − |t| > x dx, α ∈ (0, 2)
exp(x)P sup
sup
−∞
t∈[0,T ]
t∈[T0 ,T0 +T ]
and denote, for any n ≥ 1
0
0
Hα
(n; T ) = Hα
([0, T ], [nT, (n + 1)T ]).
It follows from Lemma 3 in [3] or Lemmas 6 and 7 in [23] that
P∞
0
n=1 Hα (n; T )
(30)
= 0.
lim
T →∞
T
Since




P sup X1 (s) > u, sup X2 (t) > u, sup X2 (t) > u
s∈41

t∈42l
t∈42l
k
1
2








= P sup X1 (s) > u, sup X2 (t) > u + P sup X1 (s) > u, sup X2 (t) > u
s∈41

s∈41

t∈42l
t∈42l
k
k
2
1




−P sup X1 (s) > u,
sup
X
(t)
>
u
2
S 2
s∈41

t∈42
4
k
l1
l2
similar arguments as in the derivation of (28) imply that
Σ1 (u) ≤
1
1
7
2
2
∞
0
0
[0, b1 T1 ] X Hα
[n; b2 T2 ]
21− α1 − α2 (1 + r) 2 − α1 − α2 Hα
1
2
√
b1 T1
b2 T2
πα1 α2 1 − r
n=1
2
2
×u α1 + α2 −6 ψr (u)(1 + o(1)).
(31)
Similarly
Σ2 (u) ≤
1
7
2
2
1
∞
0
0
[n; b1 T1 ] Hα
[0, b2 T2 ]
21− α1 − α2 (1 + r) 2 − α1 − α2 X Hα
1
2
√
b
T
b
πα1 α2 1 − r
1
1
2 T2
n=1
2
2
×u α1 + α2 −6 ψr (u)(1 + o(1)).
(32)
Consequently, from (28-32) by letting T1 , T2 → ∞ we obtain
1
(33)
P1,r (u)
=
1
7
2
2
2
2
21− α1 − α2 (1 + r) 2 − α1 − α2
√
Hα1 Hα2 u α1 + α2 −6 ψr (u)(1 + o(1))
πα1 α2 1 − r
Case ii) α1 ∈ (0, 1) and α2 = 1. Applying the Bonferroni inequality we have
N1 (u)
P1,r (u) ≤
X
(
P
k=0
)
sup X1 (s) > u, sup X2 (t) > u
N1 (u) N2 (u)
+
X X
k=0
t∈420
s∈41k
l=1
(
P
)
sup X1 (s) > u, sup X2 (t) > u
s∈41k
t∈42l
as u → ∞.
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
and
N1 (u)−1
X
P1,r (u) ≥
(
)
sup X1 (s) > u, sup X2 (t) > u
P
s∈41k
k=0
− Σ3 (u),
t∈420
where




P
sup X1 (s) > u, sup X1 (s) > u, sup X2 (t) > u .
s∈41

s∈41
t∈420
XX
Σ3 (u) =
0≤k1 <k2 ≤N1 (u)
k1
k2
By Lemma A
N1 (u)(or N1 (u)−1)
X
(
sup X1 (s) > u, sup X2 (t) > u
P
s∈41k
k=0
t∈420
2
N1 (u)
3
u
(1 + r) 2 −2 X
√
u
exp − h(sk , 1) (1 + o(1))
2
2π 1 − r
k=0
=
0
Hα
[0, b1 T1 ]H11+r [0, b2 T2 ]
1
=
0
2
[0, b1 T1 ] 1+r
2− α1 (1 + r) 2 − α1 Hα
1
√
H1 [0, b2 T2 ]u α1 −4 ψr (u)(1 + o(1))
b1 T1
πα1 1 − r
5
1
(34)
)
2
as u → ∞, where bi , i = 1, 2 are the same as in (28). Similarly
(
)
N1 (u) N2 (u)
X X
P sup X1 (s) > u, sup X2 (t) > u
k=0
(35)
=
l=1
s∈41k
t∈42l
2
5
1
∞
0
X
2
[0, b1 T1 ] 0
T2 l
2− α1 (1 + r) 2 − α1 Hα
1
√
H1 [0, b2 T2 ]
exp −
u α1 −4 ψr (u)(1 + o(1))
b1 T1
2(1 + r)
πα1 1 − r
l=1
as u → ∞. Moreover, it follows with similar arguments as in (31) that
(36)
Σ3 (u) ≤
5
2
1
∞
0
2
[n; b1 T1 ] 1+r
2− α1 (1 + r) 2 − α1 X Hα
1
√
H1 [0, b2 T2 ]u α1 −4 ψr (u)(1 + o(1))
b1 T1
πα1 1 − r
n=1
as u → ∞. Consequently, letting T1 , T2 → ∞ from (34-36) we have
1
2
3
2
2− α1 (2 + r)(1 + r) 2 − α1
√
Hα1 u α1 −4 ψr (u)(1 + o(1))
P1,r (u) =
πα1 1 − r
as u → ∞,
where we used the fact that H11+r := limT →∞ H11+r [ΛT ] = (2 + r)/(1 + r); see e.g., [14] or [22].
Case iii) α1 ∈ (0, 1) and α2 ∈ (1, 2). Since α2 > 1, it follows that δ(u) ⊂ 420 . Thus
(
)
N1 (u)
X
P1,r (u) ≤
P sup X1 (s) > u, sup X2 (t) > u
s∈41k
k=0
t∈420
and further
N1 (u)−1
X
P1,r (u) ≥
(
P
k=0
)
sup X1 (s) > u, X2 (1) > u
− Σ4 (u),
s∈41k
where
Σ4 (u) =
XX




P
sup X1 (s) > u, sup X1 (s) > u, X2 (1) > u .
s∈41

s∈41
0≤k1 <k2 ≤N1 (u)
k1
k2
Using the same technique as in the proof of Lemma A (or let T2 → 0 therein), we can show that
(
)
"
α2 #
1
1
0
P sup X1 (s) > u, X2 (1) > u
= Hα1 [−T1 , 0] √
2(1 + r)
s∈41k
2
3
u
(1 + r) 2 −2
× √
u exp − h(sk , 1) (1 + o(1))
2
2π 1 − r
9
10
ENKELEJD HASHORVA AND LANPENG JI
as u → ∞, implying
(
)
5
2
1
N1 (u)−1
0
X
[0, b1 T1 ] α2 −4
2− α1 (1 + r) 2 − α1 Hα
1
√
(37)
P sup X1 (s) > u, X2 (1) > u =
u 1 ψr (u)(1 + o(1)),
b1 T1
πα1 1 − r
s∈41k
k=0
u → ∞.
Moreover
5
1
2
∞
0
[n; b1 T1 ] α2 −4
2− α1 (1 + r) 2 − α1 X Hα
1
√
Σ4 (u) ≤
u 1 ψr (u)(1 + o(1))
b
T
πα1 1 − r
1
1
n=1
(38)
as u → ∞. Consequently, letting T1 → ∞, T2 → 0 we conclude from (34), (37) and (38) that
1
5
2
2
2− α1 (1 + r) 2 − α1
√
P1,r (u) =
Hα1 u α1 −4 ψr (u)(1 + o(1))
πα1 1 − r
as u → ∞.
With all the techniques used in the proofs of Cases i)-iii) we see that the other cases for the possible choices of α1 and
α2 can be shown similarly without any further difficulty, thus the detailed proofs are omitted. Moreover, it follows
from (25) and the asymptotics of P1,r (u) in any of the remaining cases that (21) holds, and thus the proof is complete.
Proof of Theorem 1.2: First note that, for any x1 , x2 ≥ 0, u > 0
n
o
P
sup
X
(s)
>
u,
sup
X
(t)
>
u
1
2
s∈[0,T1,u ]
t∈[0,T2,u ]
n
o ,
P u2 (1 − τ1∗ (u)) > x1 , u2 (1 − τ2∗ (u)) > x2 =
P sups∈[0,1] X1 (s) > u, supt∈[0,1] X2 (t) > u
with Ti,u = 1 − xi u−2 , i = 1, 2. Further, we write
(
P
sup
X1 (s) > u,
s∈[0,T1,u ]
sup
)
X2 (t) > u
(
= P
t∈[0,T2,u ]
)
f
f
sup X1 (s) > u, sup X2 (t) > u ,
s∈[0,1]
t∈[0,1]
fu (s, t) := h(T1,u s, T2,u t), (s, t) ∈ (0, 1]2 , with h(·, ·) given as in (19). It
fi (t) := Xi (Ti,u t), t ∈ [0, 1]. Define h
where X
f1 , X
f2 , without any other changes
follows from a slight modification of the proof of Lemma A that (20) holds for X
fu (·, ·). With this modification of Lemma A, by a similar argument as in the
apart from that h(·, ·) is replaced by h
proof of Theorem 1.1 we conclude that, as u → ∞
)
(
2
3
+ r) 2
u f
f2 (t) > u = (1√
f1 (s) > u, sup X
(39) P sup X
Υ1 (u)Υ2 (u)u−2 exp − h
(1,
1)
(1 + o(1)),
u
2
2π 1 − r
t∈[0,1]
s∈[0,1]
where Υi (u), i = 1, 2 are given as in Theorem 1.1. Consequently, from the last formula and Theorem 1.1, for any
x1 , x2 ≥ 0
P u2 (1 − τ1∗ (u)) > x1 , u2 (1 − τ2∗ (u)) > x2
2
u f
exp − (h
(1,
1)
−
h(1,
1))
(1 + o(1))
u
2
α1
α2
→ exp −
x1 +
x2
, u→∞
2(1 + r)
2(1 + r)
=
establishing thus the claim, and hence the proof is complete.
4. Appendix
Below we present the proofs of Theorem 2.1, Theorem 2.2 and Lemma A. We also state and prove Lemma B which
is of some interest on its own.
Proof of Theorem 2.1: Denote
A(s, t) = σ12 (s) + σ22 (t) − 2σ1 (s)σ2 (t)r(s, t),
(s, t) ∈ V.
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
Next, we introduce two nonnegative functions a(s, t), b(s, t), (s, t) ∈ V as follows
 2
σ (t)−σ1 (s)σ2 (t)r(s,t)

, if c(s, t) > r(s, t),
 2
A(s,t)

a(s, t) =
1,
if c(s, t) ≤ r(s, t) and σ1 (s) ≤ σ2 (t),


 0,
otherwise,
11
(s, t) ∈ V
and
b(s, t) =




σ12 (s)−σ1 (s)σ2 (t)r(s,t)
,
A(s,t)
if c(s, t) > r(s, t),
if c(s, t) ≤ r(s, t) and σ2 (t) < σ1 (s),
1,


 0,
(s, t) ∈ V.
otherwise,
Since a(s, t) + b(s, t) = 1, (s, t) ∈ V, it follows that


 [ n
o
P
Z1 (s) > u, Z2 (t) > u


(s,t)∈V


 [

≤ P
{a(s, t)Z1 (s) + b(s, t)Z2 (t) > a(s, t)u + b(s, t)u}


(s,t)∈V
(
)
(40)
= P
sup Y (s, t; a, b) > u
(s,t)∈V
where
Y (s, t; a, b) = a(s, t)Z1 (s) + b(s, t)Z2 (t),
(s, t) ∈ V.
Since further
(E (Y (s, t; a, b))2 )−1
=
=
1
a2 (s, t)σ12 (s) + b2 (s, t)σ22 (t) + 2a(s, t)b(s, t)σ1 (s)σ2 (t)r(s, t)
σ12 (s) + σ22 (t) − 2σ1 (s)σ2 (t)r(s, t)
I(c(s, t) > r(s, t))
σ12 (s)σ22 (t)(1 − r2 (s, t))
1
I(c(s, t) ≤ r(s, t))
+
2
min(σ1 (s), σ22 (t))
the claim follows from the Borell-TIS inequality for one-dimensional Gaussian random fields (e.g., [1]) with
(
)
µ=E
sup Y (s, t; a, b)
<∞
(s,t)∈V
and thus the proof is complete.
Proof of Theorem 2.2: We use the same notation as in the proof of Theorem 2.1. In the light of (40) and Theorem
8.1 in [36], it suffices to show that
(41)
E (Y (s, t; a, b) − Y (s0 , t0 ; a, b))2 ≤ L1 (|s − s0 |γ + |t − t0 |γ ), ∀(s, t), (s0 , t0 ) ∈ V
holds for some positive constants L1 and γ, which can be confirmed by some straightforward calculations, and thus
the claim follows.
Proof of Lemma A: Using the classical technique, see e.g., [3, 23, 36], we have for any u > 0


Z ∞Z ∞  [
1
x
y
(42)
P
{X1 (s) > u, X2 (t) > u}|X1 (ŝ0 ) = u − , X2 (t̂0 ) = u −
RΛ1 ,Λ2 (u) =
u2 −∞ −∞ 
u
u
(s,t)∈Ku
x
y
×fX1 (ŝ0 ),X2 (t̂0 ) u − , u −
dxdy,
u
u
12
ENKELEJD HASHORVA AND LANPENG JI
where
1
x
1
y
α1 /2 α2 /2
α2 2
α1 2
q
exp − α1 α2
fX1 (ŝ0 ),X2 (t̂0 ) u − , u −
=
t̂
x
+
ŝ
y
−
2rŝ
t̂
xy
0
0
0
0
u
u
2ŝ0 t̂0 (1 − r2 )u2
1 α2
2
2π ŝα
0 t̂0 (1 − r )
2
u
1
α1 /2 α2 /2
α1
α2
y
+
2rŝ
t̂
(x
+
y)
exp
−
x
−
2ŝ
−2
t̂
× exp − α1 α2
h(ŝ
,
t̂
)
, x, y ∈ R,
0 0
0
0
0
0
2
2ŝ0 t̂0 (1 − r2 )
where h(·, ·) is defined as in (19). Set for x, y ∈ R
2
2
ξu (s) = u(X1 (ŝ0 + u− α1 s) − u) + x, ηu (t) = u(X2 (t̂0 + u− α2 t) − u) + y.
The probability in the integrand of (42) can be rewritten as




[
pu (x, y) = P
{ξu (s) > x, ηu (t) > y}|ξu (0) = 0, ηu (0) = 0 .


(s,t)∈Λ1 ×Λ2
Next, we calculate the expectation and covariance of the conditional random vector (ξu (s), ηu (t))|(ξu (0), ηu (0)). We
have
(
E
)
(
)
ξu (s)
ξu (s) ξu (0)
=E
+A
ηu (t)
ηu (t) ηu (0)
ξu (0) − E {ξu (0)}
!
,
ηu (0) − E {ηu (0)}
where
ξu (s)
A = Cov
!
ηu (t)
,
ξu (0)
!!
× Cov
ηu (0)
!−1
ξu (0)
ηu (0)
and further
Cov
!
ξ (0)
u
= Cov
ηu (t1 ) − ηu (s1 ) ηu (0)
ξu (t) − ξu (s)
ξu (t) − ξu (s)
!
ξu (0)
+ B Cov
ηu (t1 ) − ηu (s1 )
!−1
B>,
ηu (0)
where
b11 (u) b12 (u)
B=
b21 (u) b22 (u)
!
= Cov
!
ξu (t) − ξu (s)
ηu (t1 ) − ηu (s1 )
,
ξu (0)
!!
.
ηu (0)
Further
(43)
ξu (0)
Cov
!−1
ηu (0)
u−2
= α2 α1
t̂0 ŝ0 (1 − r2 )
2
t̂α
0
α2
α2
α1
−rt̂02 ŝ02
α1
!
1
ŝα
0
−rt̂02 ŝ02
and
A=





×



1
×
− r2 )
α1 α2
− α2
− α2
α1
α1
1
1 α2
α
−2 α1
α
−2 α1
2
2
1 s) 1 − u
1 s) 1 − u
t̂
ŝ
+
(ŝ
+
u
s
−
−
r
t̂
ŝ
ŝ
+
(ŝ
+
u
s
+
0
0
0
0
0
0
0
2
2
α21
α21
α2 α2
2
2
2
2
1
−r2 t̂α
ŝ0 + u− α1 s
+rt̂02 ŝα
ŝ0 + u− α1 s
0 ŝ0
0
α2
α1 − α2
− α2
α2
1 α1
α
−2 α2
α
−2 α2
2
2 t) 2 − u
2 t) 2 − u
− 21 rt̂02 ŝ02 t̂α
ŝ
t̂
+
(
t̂
+
u
t
+
t
−
0
0 + (t̂0 + u
0
0
2
α22
α22
α1
α2
2
2
2
1
+rŝ02 t̂α
t̂0 + u α2 t
−r2 t̂02 ŝα
t̂0 + u− α2 t
0
0
1
t̂0α2 ŝα
0 (1
Set next
e1 (u)
e2 (u)
!
(
:= E
)
ξu (s) ξu (0) = 0
=
ηu (t) ηu (0) = 0
x − u2
y − u2
!
+A
u2 − x
u2 − y
!
.
It follows that
e1 (u) =
1
α2 α1
t̂0 ŝ0 (1 − r2 )
1 α2 1 α22 α21
−
t̂0 − rt̂0 ŝ0
|s|α1 + λ1 (u)u2 + λ2 (u)x + λ3 (u)y ,
2
2





.



EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
where
α1 α1
α1 α1 α1
α1
2
2
1 α22 α22 (ŝ0 + u− α1 s) 2 + ŝ02 − 2r2 ŝ02 − rŝ02 (ŝ0 + u− α1 s) 2 − ŝ02
t̂0
t̂0
2
α1 α1
2
× (ŝ0 + u− α1 s) 2 − ŝ02
α1
α1 α1
α1
α1
2
2
1 2
ŝ02 − (ŝ0 + u− α1 s) 2
λ2 (u) = t̂α
(ŝ0 + u− α1 s) 2 + ŝ02 − 2r2 ŝ02
0
2
2
α1
2
1 α22 α21 α21
λ3 (u) = rt̂0 ŝ0
ŝ0 − (ŝ0 + u− α1 s) 2
.
2
λ1 (u) =
Further
1
e2 (u) = α2 α1
t̂0 ŝ0 (1 − r2 )
1 α1 1 α22 α21
α2
2
ŝ − rt̂0 ŝ0
−
|t| + δ1 (u)u + δ2 (u)x + δ3 (u)y ,
2 0
2
where
α2
α2 α2 α2 α2
α2
2
2
1 α21 α21 ŝ0
ŝ0
(t̂0 + u− α2 t) 2 + t̂02 − 2r2 t̂02 − rt̂02 (t̂0 + u− α2 t) 2 − t̂02
2
α2 α2
2
× (t̂0 + u− α2 t) 2 − t̂02
2
α2
2
1 α2 α1 α2
δ2 (u) = rt̂02 ŝ02 t̂02 − (t̂0 + u− α2 t) 2
2
α2
α2 α2
α2
α2
2
2
1 α1 δ3 (u) = ŝ0 (t̂0 + u− α2 t) 2 + t̂02 − 2r2 t̂02
t̂02 − (t̂0 + u− α2 t) 2 .
2
δ1 (u) =
Thus we have
(
(44)
lim e1 (u) =
1
− 2(1+r)
|s|α1 ,
1
− 2(1+r)
|s|
u→∞
+
if α1 ∈ (0, 1),
1
2s
if α1 = 1
and
(
(45)
lim e2 (u) =
u→∞
1
− 2(1+r)
|t|α2 ,
1
|t|
− 2(1+r)
+
if α2 ∈ (0, 1),
1
2t
if α2 = 1.
Similarly
α1 α1 2
2
1 α1
− ŝ0 + u− α1 s
s − tα1 + u2 ŝ0 + u− α1 t
,
2
α21 α21
α2
2
2
ŝ0 + u− α1 t
− ŝ0 + u− α1 s
,
b12 (u) = u2 rt̂02
b11 (u) =
α1
b21 (u) = u2 rŝ02
b22 (u) =
2
t̂0 + u− α2 t1
α22
α22 2
− t̂0 + u− α2 s1
,
α2 α2 1 α2
− α2
− α2
2
2
2 t
2 s
s1 − tα
+
u
t̂
+
u
−
t̂
+
u
,
0
1
0
1
1
2
which together with (43) gives that
B Cov
ξu (0)
ηu (0)
!−1
B> =
o(1)
o(1)
o(1)
o(1)
!
as u → ∞. Further
Cov(ξu (t) − ξu (s), ξu (t) − ξu (s)) = |t − s|α1 , Cov(ηu (t1 ) − ηu (s1 ), ηu (t1 ) − ηu (s1 )) = |t1 − s1 |α2 ,
α1
α1
2
2
Cov(ξu (t) − ξu (s), ηu (t1 ) − ηu (s1 )) = u2 r (ŝ0 + u− α1 t) 2 − (ŝ0 + u− α1 s) 2
α2
α2
2
2
× (t̂0 + u− α2 t1 ) 2 − (t̂0 + u− α2 s1 ) 2 = o(1),
13
14
ENKELEJD HASHORVA AND LANPENG JI
as u → ∞. Therefore,
!
ξ (0) = 0
u
=
ηu (t1 ) − ηu (s1 ) ηu (0) = 0
ξu (t) − ξu (s)
Cov
!
|t − s|α1
o(1)
o(1)
|t1 − s1 |α2
,
as u → ∞.
Consequently, using similar arguments as in [3] (see also [10], [23] or [36]) we obtain
lim pu (x, y) = P sup χ1 (s) > x P sup χ2 (t) > y
u→∞
s∈Λ1
t∈Λ2
for any x, y ∈ R, where χ1 and χ2 are two independent stochastic processes given by
(
1
− 2(1+r)
|s|α1 ,
if α1 ∈ (0, 1),
b
χ1 (s) = Bα1 (s) +
s∈R
1
1
− 2(1+r) |s| + 2 s if α1 = 1,
and
(
eα (t) +
χ2 (t) = B
2
1
− 2(1+r)
|t|α2 ,
1
− 2(1+r)
|t|
+
if α2 ∈ (0, 1),
1
2t
t ∈ R.
if α2 = 1
bα and B
eα are two independent fBm’s defined on R with Hurst indexes α1 /2 and α2 /2 ∈ (0, 1), respectively.
Here B
1
2
Similar arguments as in [3] and [23] show that the limit (letting u → ∞) can be passed under the integral sign in (42).
It follows then that
RΛ1 ,Λ2 (u) =
2
1
u
exp − h(ŝ0 , t̂0 )
2
2π 1 − r2 u2
!
Z
2
∞
Y
x
×
exp
P sup χi (s) > x dx , u → ∞.
1+r
s∈Λi
−∞
i=1
(1 + o(1))
√
Since
Z
∞
exp
−∞

α2 
i
 (1 + r)H0 Λ1 √ 1
,
if αi ∈ (0, 1),

αi
2(1+r)
x
P sup χi (s) > x dx =
2 
1+r
s∈Λi
 (1 + r)H−(1+r) Λi √ 1
, if αi = 1

1
2(1+r)
the claim follows.
Lemma B. Let (Ω, F, P) be a probability space and A1 , · · · , An and B1 , · · · , Bm be n + m events in F for n, m ≥ 2.
Then



n X
m
 X
[
P {Ak ∩ Bl }
(Ak ∩ Bl ) ≥
P




k=1,··· ,n
 k=1 l=1




l=1,··· ,m
−
(46)
n
X
X
P {Ak ∩ Bl1 ∩ Bl2 } −
k=1 1≤l1 <l2 ≤m
m
X
X
P {Ak1 ∩ Ak2 ∩ Bl } .
l=1 1≤k1 <k2 ≤n
Proof of Lemma B: The proof relies on the following Bonferroni inequality; see e.g., Lemma 2 in [31].
( n
)
n
n
X
[
X
X
P {Ak } ≥ P
Ak ≥
P {Ak } −
P {Ak1 ∩ Ak2 } .
k=1
k=1
1≤k1 <k2 ≤n
k=1
Since further






)
( n

m

[
[
[
P
(Ak ∩ Bl ) = P
(Ak ∩ ( Bl ))




l=1
k=1
k=1,··· ,n

l=1,··· ,m
≥
n
X
k=1
(
P Ak ∩ (
m
[
l=1
)
Bl )
(
−
X
1≤k1 <k2 ≤n
P Ak1 ∩ Ak2 ∩ (
m
[
l=1
)
Bl )
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FBM’S
≥
n X
m
X
P {Ak ∩ Bl } −
k=1 l=1
n
X
X
P {Ak ∩ Bl1 ∩ Bl2 } −
k=1 1≤l1 <l2 ≤m
m
X
X
15
P {Ak1 ∩ Ak2 ∩ Bl }
l=1 1≤k1 <k2 ≤n
the proof is complete.
Acknowledgement: We would like to thank both the Editor, associate Editor and the Referees for their kind
comments and corrections which significantly improved the manuscript. We are thankful to Krzysztof Dȩbicki, Yimin
Xiao and Yuzhen Zhou for various discussions related to the topic of this paper. This work was support from the
Swiss National Science Foundation Grant 200021-140633/1 and the project RARE -318984 (an FP7 Marie Curie IRSES
Fellowship).
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Enkelejd Hashorva, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015 Lausanne, Switzerland
E-mail address: [email protected]
Lanpeng Ji, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015 Lausanne, Switzerland
E-mail address: [email protected]