4.1 Path analysis
In this section, we give some path properties of quadratic variation process 〈B〉
t
and the related stochastic integral
Z
t
ηsd B
s
, η ∈ M
2 G
0, T and Z
t
η
1
sd〈B〉
s
, η
1
s ∈ M
1 G
0, T . From the definition of E
G
[·], we know that the canonical process B
t
is a quadratic integrable mar- tingale under each P
v
. So they have a universal version of “quadratic variation process of B
t
, and by the definition of stochastic integral with respect to G-Brownian motion, for any η
∈ M
2 G
0, T , Z
T
η
s
d B
s
is well defined, which means Z
T
η
s
d B
s
is a P
v
local martingale. Similar arguments can be found in Lemma 2.10 in [10]. Then, due to the Doob’s martingale inequality for each P
v
, the following result holds.
Lemma 4.2. For any η ∈ M
2 G
0, T , Z
t
ηsd B
s
is quasi-sure continuous.
Proof: If η ∈ M
2,0 G
0, T , then the result is true. If η ∈ M
2 G
0, T , then there exist {η
n
} ⊂ M
2,0 G
0, T such that
Z
T
E
G
[|ηs − η
n
s|
2
]ds −→ 0. For we have
sup
P
v
∈Λ
E
P
v
[ sup
≤t≤T
| Z
t
η − η
n
d B
s
|
2
] ≤ K sup
P
v
∈Λ
E
P
v
[ Z
T
η − η
n 2
d 〈B〉
s
] ≤ K sup
P
v
∈Λ
E
P
v
[ Z
T
η − η
n 2
ds] ≤ K
Z
T
sup
P
v
∈Λ
E
P
v
[η − η
n 2
]ds −→ 0.
Hence, Z
t
η
n
sd B
s
uniformly converges to Z
t
ηsd B
s
q.s. Therefore, Z
t
ηsd B
s
is continuous q.s.
By similar argument we can get the following lemma.
Lemma 4.3. For any η ∈ M
1 G
0, T , Z
t
ηsd〈B〉
s
and Z
t
ηsds are quasi-surely continuous.
4.2 Representation theorem
In this section, we are concerned with the G-martingale when the corresponding G-heat equation is uniformly parabolic σ
0. In the following, for any X ∈ L
i p
F
T
, we will give X a repre- sentation in terms of stochastic integral. For this part, Peng gives the conjecture for representation
theorem of G-martingales. Soner et al [21] prove this theorem for a large class of G-martingales by BSDE method. Here for the ease of exposition, we prove the theorem separately for some special
martingales.
First we prove a lemma. 2051
Lemma 4.4. Suppose σ
0, if u is the solution of G-heat equation, then we have ur
, B
t −r
= ut, 0 + Z
t −r
u
x
B
v
d B
v
+ Z
t −r
u
x x
B
v
d〈B〉
v
− Z
t −r
u
+ x x
− σ
2
u
− x x
d v.
Proof: The proof follows that of Itô’s formula. Due to the regularity of parabolic equation, see [18], we know u, u
x
, u
x x
are all uniformly continuous. Since Lipschitz continuous functions are dense in uniform continuous functions, we assume that u, u
x
, u
x x
are Lipschitz continuous. Let δ
n
= t
− r n
, we have ur
, B
t −r
− ut, 0 =
n −1
X
k=
ut
− k + 1δ
n
, B
k+1δ
n
− ut − kδ
n
, B
k δ
n
=
n −1
X
k=
ut
− k + 1δ
n
, B
k+1δ
n
− ut − kδ
n
, B
k+1δ
n
+
n −1
X
k=
ut
− kδ
n
, B
k+1δ
n
− ut − kδ
n
, B
k δ
n
=
n −1
X
k=
−u
t
t − kδ
n
, B
k δ
n
δ
n
+ u
x
t − kδ
n
, B
k δ
n
B
k+1δ
n
− B
k δ
n
+ 1
2 u
x x
t − kδ
n
, B
k δ
n
B
k+1δ
n
− B
k δ
n
2
− ξ
n
+ η
n
, where
η
n
= 1
2
n −1
X
k=
[u
x x
t − kδ
n
, B
k δ
n
+ θ
1
B
k+1δ
n
− B
k δ
n
− u
x x
t − kδ
n
, B
k δ
n
]B
k+1δ
n
− B
k δ
n
2
, ξ
n
=
n −1
X
k=
u
t
t − kδ
n
+ θ
2
δ
n
, B
k+1δ
n
− u
t
t − kδ
n
, B
k+1δ
n
δ
n
, +
n −1
X
k=
u
t
t − kδ
n
, B
k+1δ
n
− u
t
t − kδ
n
, B
k δ
n
δ
n
, and θ
1
, θ
2
are constants in [0, 1], which depend on ω, t and n. Hence, E
G
[|η
n
|] ≤
n −1
X
k=
E
G
|u
x x
t − kδ
n
, B
k δ
n
+ θ B
k+1δ
n
− B
k δ
n
−u
x x
t − kδ
n
, B
k δ
n
|B
k+1δ
n
− B
k δ
n
2
≤ K
n −1
X
k=
E
G
[|B
k+1δ
n
− B
k δ
n
|
3
] ≤ K
n −1
X
k=
δ
32 n
−→ 0. Here, K is the Lipschitz constant of u
x x
, and by similar argument we get E
G
[|ξ
n
|] −→ 0 as n → ∞.
2052
Then, we have E
G
[|
n
X
k= 1
−u
t
t − kδ
n
, B
k δ
n
I
[kδ
n
,k+1δ
n
v − u
t
t − v, B
v
|] ≤ Cδ
n
+ δ
12 n
−→ 0. Therefore,
n −1
X
k=
−u
t
t − kδ
n
, B
k δ
n
δ
n
−→ Z
t −r
−u
t
t − v, B
v
d v. Similarly we get
n −1
X
k=
u
x
t − kδ
n
, B
k δ
n
B
k+1δ
n
− B
k δ
n
−→ Z
t −r
u
x
t − v, B
v
d B
v
. Note
E
G
[|
n −1
X
k=
u
x x
t − kδ
n
, B
k δ
n
B
k+1δ
n
− B
k δ
n
2
−
n −1
X
k=
u
x x
t − kδ
n
, B
k δ
n
〈B〉
k+1δ
n
− 〈B〉
k δ
n
|
2
] ≤
n −1
X
k=
E
G
[|u
x x
t − kδ
n
, B
k δ
n
|
2
|B
k+1δ
n
− B
k δ
n
2
− 〈B〉
k+1δ
n
− 〈B〉
k δ
n
|
2
] +
2 X
j 6=i
E
G
[u
x x
t − jδ
n
, B
j δ
n
u
x x
t − iδ
n
, B
i δ
n
B
j+1δ
n
− B
j δ
n
2
− 〈B〉
j+1δ
n
− 〈B〉
j δ
n
B
i+1δ
n
− B
i δ
n
2
− 〈B〉
i+1δ
n
− 〈B〉
i δ
n
] ≤
n −1
X
k=
E
G
[c + c|B
k δ
n
|
2
| Z
k+1δ
n
k δ
n
B
v
− B
k δ
n
d B
v
|
2
] ≤
n −1
X
k=
C E
G
[| Z
k+1δ
n
k δ
n
B
v
− B
k δ
n
d B
v
|
2
] ≤ C
n −1
X
k=
δ
2 n
−→ 0, and
n −1
X
k=
u
x x
t − kδ
n
, B
k δ
n
〈B〉
k+1δ
n
− 〈B〉
k δ
n
−→ Z
t −r
u
x x
t − v, B
v
d〈B〉
v
. Since u solves G-heat equation, we get
ur , B
t −r
− ut, 0 = Z
t −r
−u
t
t − v, B
v
d v + Z
t −r
u
x
t − v, B
v
d B
v
+ Z
t −r
u
x x
t − v, B
v
d〈B〉
v
= Z
t −r
u
x
t − v, B
v
d B
v
+ Z
t −r
u
x x
t − v, B
v
d〈B〉
v
− 1
2 Z
t −r
u
+ x x
− σ
2
u
− x x
d v.
2053
Theorem 4.5. When σ
0, then for any X ∈ L
i p
F
T
, we have X = E
G
[|ϕ·|] + Z
t
m
Z
s
d B
s
+ Z
t
m
ηsd〈B〉
s
− Z
t
m
η
+
− σ
2
η
−
ds.
Proof: When m = 1, for the regularity of the u, lim
r →0
ur , B
t −r
ω = ϕB
t
ω. By Lemma 4.4, and path analysis in Section 3, we know
ϕB
t
= ut
, 0 + Z
t
u
x
t − v, B
v
d B
v
+ Z
t
u
x x
t − v, B
v
d〈B〉
v
− Z
t
u
+ x x
− σ
2
u
− x x
d v. Due to the definition of G-expectation, E
G
[ϕB
t
] = ut, 0, so the result holds when m = 1. By similar argument, we get
ϕB
T
− B
t
= uT
− t, 0 + Z
T t
u
x
T − v, B
v
− B
t
d B
v
+ Z
T t
u
x x
T − v, B
v
− B
t
d〈B〉
v
− Z
T t
u
+ x x
− σ
2
u
− x x
d v. When m = 2, for each x
ϕx, B
T
− B
t
= uT
− t, x, 0 + Z
T t
u
y
T − v, x, B
v
− B
t
d B
v
+ Z
T t
u
y y
T − v, x, B
v
− B
t
d〈B〉
v
− Z
T t
u
+ y y
T − v, x, B
v
− B
t
− σ
2
u
− y y
T − v, x, B
v
− B
t
d v. By continuous dependence estimate theorem in [13], we know for each fixed t, uT
− t, x, 0 is lipschitz continuous and bounded with respect to x, then there exist η
∈ M
1 G
0, T and z ∈ M
2 G
0, T , such that
uT − t, B
t
, 0 = E
G
[|uT − t, B
t
, 0 |] +
Z
t
z
s
d B
s
+ Z
t
η
s
d 〈B〉
s
− Z
t
η
+ s
− σ
2
η
− s
ds. 4.2
That is ϕB
t
, B
T
− B
t
= E
G
[ϕB
t
, B
T
− B
t
] + Z
T
z
s
d B
s
+ Z
T
η
s
d 〈B〉
s
− Z
T
η
+ s
− σ
2
η
− s
ds. Here η and z are different from 4.2.
Then by induction, we know the result is true for any X ∈ L
i p
F
T
. 2054
4.3 Properties for the Symmetric Martingale
Kategori