For p ≥ 1 and 0 T ∞ fixed T . Consider the following type of simple processes
M
p ,0
G
0, T = {η : η
t
ω =
N −1
X
j=
ξ
t
j
ωI
[t
j
,t
j+ 1
t, ∀N ≥ 1, 0 = t
· · · t
N
= T, ξ
t
j
ω ∈ L
p G
F
t
j
, j = 0. · · · , N − 1}. For each η
t
ω = P
N −1
j=
ξ
t
j
ωI
[t
j
,t
j+ 1
t ∈ M
p ,0
G
0, T , the related Bochner’s integral is defined as Z
T
η
t
ωd t =
N −1
X
j=
ξ
t
j
ωt
j+ 1
− t
j
, and
e E
G
[η] = 1
T Z
T
E
G
[η
t
]d t = 1
T
N −1
X
j=
E
G
[ξ
t
j
ω]t
j+ 1
− t
j
. We can easily check that e
E
G
[·] : M
1,0 G
0, T → R satisfies a ∼ d in Section 2.
kηk
p
= 1
T Z
T
kη
p t
kd t
1p
=
1 T
N −1
X
j=
E
G
[|ξ
t
j
ω|
p
]t
j+ 1
− t
j
1p
.
As discussed in Remark 2.6, k · k
p
forms a norm in M
p ,0
G
0, T . Let M
p G
0, T be the completion of M
p ,0
G
0, T under this norm.
3 C
APACITY
U
NDER
G-F
RAMEWORK
3.1 G-Expectation and Related Properties
G-expectation is a kind of time consistent nonlinear expectations, which has the properties of linear expectation in Weiner space except linearity. In this section, we prove some fundamental properties
of G-expectation.
We set S
p
= {M|M : R
+
× Ω → R, Mt, ω ∈ L
p G
F
t
, ∀T 0, {M
t
}
t ∈[0,T ]
∈ M
p G
0, T }. First we will prove that an important class of random variable-bounded continuous functions belongs
to the completion space- L
1 G
F
Remark 3.1. If X ∈ L
1 G
F
T
, there is a sequence f
n
∈ L
i p
F
T
, such that f
n
converges to f in L
1 G
F
T
, then for each P
v
∈ Λ, f
n
converges to f in L
1
Ω, P
v
, and this convergence is uniform with respect to P
v
. We have E
G
[ f ] = sup
P
v
∈Λ
E
P
v
[ f ] see Proposition 2.2 in [9]. Next we will prove the tightness of Λ.
2046
Lemma 3.2. Λ is tight, that is, for any ǫ 0, there exists a compact set K ⊂ C[0, T ] ⊂ Ω, such that
for any P
v
∈ Λ, P
v
K
c
ǫ, where K
c
is the complement of K.
Proof: For any continuous function xt, t ∈ [0, T ], define
ω
x
δ = sup
|s−t|≤δ,s,t∈[0,T ]
|x
t
− x
s
|. By Arzela-Ascoli theorem and Prokhrov theorem see Theorem 4.4.11 in [4], to prove the tightness
of Λ, we only need to prove that for any α 0, for any P
v
∈ Λ, we have lim
δ→0
P
v
{x : ω
x
δ ≥ α} = 0.
For α 0, by Proposition 7.2 in the Appendix we know that lim
δ→0
P
v
{x : ω
x
δ ≥ α} ≤ lim
δ→0
E
P
v
[ω
2 x
δ] α
2
= 0, then Λ is tight.
Remark 3.3. For any fixed T, C[0, T ] is a polish space, by Prokhrov theorem, we know that Λ is weakly compact. For X
n
∈ L
i p
F
T
, X
n
↓ 0 pointwise. As in the Appendix of [9], by Dinni lemma, E
G
[X
n
] ↓ 0.
Lemma 3.4. For any fixed T 0, 0 t T , f ∈ C
b
W
t
, we have f ∈ L
1 G
F
t
.
Proof: For any bounded continuous f ∈ C
b
W
t
, | f | ≤ M, M 0, there exists a sequence of random variables f
n
∈ L
i p
F
t
, such that f
n
monotonically converges to f . As Λ is tight, for any ǫ 0, there exists a compact set K
∈ F
T
, such that sup
P
v
∈Λ
E
P
v
[I
K
c
] ǫ, where K
c
is the complement of K. Since a compact set is closed in any metric space, we know that K
c
is an open set, hence E
G
[I
K
c
] = sup
P
v
∈Λ
E
P
v
[I
K
c
] ǫ. And by Dini’s theorem on any compact set K, f
n
converges to f uniformly, lim
n →∞
E
G
[| f
n
− f |] ≤ lim
n →∞
[E
G
[| f
n
− f |I
K
] + E
G
[| f
n
− f |I
K
c
]] ≤ Mǫ, then lim
n →∞
E
G
[| f
n
− f |] = 0, note that this convergence is uniform with respect to t, so f ∈ L
1 G
F
T
.
3.2 Capacity under G-Framework
