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An Extension of Sub-Fractional Brownian Motion

503

is sufficient for the local time L(t, u) of X to exist on [0, 1] a.s. and to
be square integrable as a function of u. For any [a, b] ⊂ [0, ∞), and for
I = [a0 , b0 ] ⊂ [a, b] such that |b0 − a0 | < δ, according to (5), we have,
Z Z
Z Z
−1/2
E[S H,K (t) − S H,K (s)]2
ds dt < C
|t − s|−HK ds dt.
I

I

I

I

The last integral is finite because 0 < HK < 1. Then S H,K possesses,
on any interval I ⊂ [a, b] with length |I| < δ, a local time which is
square integrable as function of u. Finally, since [a, b] is a finite interval,
we can obtain the local time on [a, b] by a patch-up procedure, i.e. we
partition [a, b]Pinto ∪ni=1 [ai−1 , ai ], such that |ai − ai−1 | < δ, and define
n
L([a, b], x) = i=1 L([ai−1 , ai ], x), where a0 = a and an = b.
Proposition 2.5. Assume H ∈ (0, 1) and K ∈ (0, 1). Then S H,K is LND
on [0, 1].
Proof: By virtue of (3), we have
[StHK − SsHK ] = [StH,K − SsH,K ] + C3 (K)[XtKH − XsKH ].
Therefore, the elementary inequality (a + b)2 ≤ 2a2 + 2b2 implies that




p
X
1
Var 
uj [StH,K
−StH,K
] ≥ Var
uj [StHK
− StHK
]
j
j−1
j
j−1
2
j=1
j=1


p
X
− C32 (K) Var 
uj [XtKH − XtKH ] .
p
X



j

j−1

j=1

According to Remark 2.2, the sfBm S HK is LND on [0, 1], then there
exist two constants δ > 0 and 0 < Cp < +∞ such that for any t0 = 0 <
t1 < t2 < · · · < tp < 1 with tp − t1 < δ, we have

Var 

p
X


uj [StH,K
−StH,K
] ≥ Cp
j
j−1

j=1

p
X



u2j Var StHK
− StHK
j−1
j

j=1



pC32 (K)

p
X



u2j Var XtKH − XtKH
.
j

j=1

j−1