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An Extension of Sub-Fractional Brownian Motion

505

Proof: We will prove only (7), the proof of (6) is similar. It follows
from (4) that for any x, y ∈ R, t, t + h ≥ 0 and for any integer p ≥ 2,
E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
Z
Z Y
p
1
=
[e−iyuj − e−ixuj ]
(2π)p [t,t+h]p Rp j=1
p
p
Pp
Y
Y
i
u S H,K
× E e j=1 j sj
duj
dsj .
j=1


1−ξ

Using the elementary inequality |1 − e | ≤ 2
and any θ ∈ R, we obtain

j=1

ξ

|θ| for all 0 < ξ < 1

(8) E[L(t+h, y)−L(t, y)−L(t+h, x)+L(t, x)]p ≤ (2ξ π)−p p!|x−y|pξ



Z
Z Y
p
p
p
p
Y
Y
X
 duj tj ,
|uj |ξ Eexpi uj StH,K
×
j
t<t1 <···<tp <t+h

Rp j=1

j=1

j=1

j=1

where in order to apply the LND property of StH,K , we replaced the
integration over the domain [t, t + h] by over the subset t < t1 < · · · <
tp < t + h. We deal now with the inner multiple integral over the u’s.
Change the variables of integration by mean of the transformation
uj = vj − vj+1 , j = 1, . . . , p − 1; up = vp .
Then, the linear combination in the exponent in (8) is transformed according to
p
p
X
X
uj StH,K
=
vj (StH,K
− StH,K
),
j
j
j−1
j=1

j=1
H,K

where t0 = 0. Since S
is a Gaussian process, the characteristic
function in (8) has the form



p
X
1
vj (StH,K
− StH,K
) .
(9)
exp − Var 
j
j−1
2
j=1
Since |x − y|ξ ≤ |x|ξ + |y|ξ for all 0 < ξ < 1, it follows that
(10)

p
Y
j=1

|uj |ξ ≤

p−1
Y

(|vj |ξ + |vj+1 |ξ )|vp |ξ .

j=1

p−1
Moreover, the
is at most equal to a finite
terms
Qplast product
Ppsum of 2
ξεj
of the form j=1 |xj | , where εj = 0, 1 or 2 and j=1 εj = p.