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504

A. Sghir

Moreover (2) and the fact that H ∈ (0, 1) imply that


p
X
Var 
uj [StH,K
− StH,K
]
j
j−1
j=1

≥ Cp

p
X

u2j |tj − tj−1 |2HK − pC32 (K)CK

j=1

p
X

u2j |tj − tj−1 |2H

j=1

h

≥ Cp − pC32 (K)CK δ 2H(1−K)

p
iX

u2j |tj − tj−1 |2HK .

j=1

In addition, the elementary inequality (a + b)2 ≥ 12 a2 − b2 implies that
E[StH,K − SsH,K ]2 ≤ 2E[StHK − SsHK ]2 + 2C32 (K)E[XtKH − XsKH ]2
≤ CH,K |t − s|2HK + 2C32 (K)CK |tH − sH |2
≤ [CH,K + 2C32 (K)CK δ 2H(1−K) ]|t − s|2HK
≤ C(H, K, δ)|t − s|2HK .
Therefore, it suffices now to choose
1
2H(1−K)

Cp
˜
∧δ
δ<
pC32 (K)CK
and to consider
C=

1
[Cp − pC32 (K)CK δ˜2H(1−K) ].
C(H, K, δ)

This with Proposition 2.1 complete the proof of Proposition 2.5.
Now, we are in position to give the main result of this section.
Theorem 2.6. Assume H ∈ (0, 1) and K ∈ (0, 1) and let δ > 0 the
constant appearing in Lemma 2.4. For any integer p ≥ 2 there exists a
constant Cp > 0 such that, for any t ≥ 0, any h ∈ (0, δ), all x, y ∈ R,
and any 0 < ξ < 1−HK
2HK ,
hp(1−HK)
,
Γ(1 + p(1 − HK))
(7) E[L(t + h, y) − L(t, y) − L(t + h, x) + L(t, x)]p
(6) E[L(t + h, x) − L(t, x)]p ≤ Cp

≤ Cp |y − x|pξ

hp(1−HK(1+ξ))
.
Γ(1 + p(1 − HK(1 + ξ)))