# Existence of Solutions for G-SFDEs with Cauchy-Maruyama Approximation Scheme

Authors: Faiz Faizullah
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We present the Cauchy-Maruyama (CM) approximation scheme and establish the existence theory of stochastic functional differential equations driven by G-Brownian motion (G-SFDEs). Several useful properties of Cauchy-Maruyama (CM) approximate solutionsX
k of G-SFDEs are given. We show that the unique solution of G-SFDEs gets convergence from Cauchy-Maruyama (CM) approximate solutions. The existence theorem for G-SFDEs is developed with the above mentioned scheme.


---

## Body

## 1. Introduction

A significant role is played by stochastic differential equations (SDEs) in a broad range of applied disciplines, including biology, economics, finance, chemistry, physics, microelectronics, and mechanics. In many applications, one assumes that the future state of the system is independent of the past states. However, under close scrutiny, it becomes apparent that a more realistic model would contain some of the past state of the system and one needs stochastic functional differential equations to formulate such systems [1]. Also see [2, 3]. The motion theory of G-Brown, corresponding to Itô’s calculus and stochastic differential equations driven by G-Brownian motion (G-SDEs), was introduced by Peng [4, 5]. Then many authors paid much attention to this theory and G-SDEs [6–10]. The existence theory for solutions of stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) was introduced by Ren et al. via Picard approximation [11]. However, like the classical stochastic differential equations, the convergence of Picard approximate solutions to the unique solution of G-SFDEs under the linear growth and Lipschitz condition is still open [1]. In contrast, here we introduce the Cauchy-Maruyama approximation scheme for G-SFDEs and show that the unique solution X
(
t
) of G-SFDEs gets convergence from CM approximate solutions X
k
(
t
), k
≥
1. Furthermore, using CM approximation scheme, it is shown that the G-SFDE has a unique solution.Some mathematical preliminaries are given in subsequent section. In Section3, the CM approximation scheme for G-SFDEs is developed. In Section 4, some properties of the CM approximate solutions are presented. In Section 5, the theory of existence for the solutions of G-SFDEs with the above stated method is established.


## 2. Basic Notions

Some basic definitions and notions are presented in this section [4, 5, 11–13]. Suppose that if Λ is a basic space that is nonempty and H is a space of linear real valued mappings stated on the space Λ such as b
∈
H, where b is any arbitrary constant and if Y
1
,
Y
2
,
…
,
Y
n
∈
H, φ
(
Y
1
,
Y
2
,
…
,
Y
n
)
∈
H for every φ
∈
C
l
.
Lip
(
R
n
). Furthermore,
(1)
C
l
.
Lip
(
R
n
)
=
{
φ
:
R
n
⟶
R
∣
∃
C
∈
R
+
,
(
1
+
|
y
|
q
+
|
z
|
q
)
i
i
q
∈
Z
+
s
.
t
.
|
φ
(
y
)
-
φ
(
z
)
|
ii
≤
C
(
1
+
|
y
|
q
+
|
z
|
q
)
|
y
-
z
|
}
,
for y, z
∈
R
n; that is, C
l
.
Lip
(
R
n
) is a space containing the linear mappings φ. Here H represents the space of random variables.Definition 1.
A functionalE
:
H
→
R is named a sublinear expectation, if for every Y, Z
∈
H, θ
≥
0, and b
∈
R it holds the following monotonic, subadditivity, constant preserving and positive homogeneity properties, respectively:(i)
Z
≥
Y implies E
[
Z
]
≥
E
[
Y
].
(ii)
E
[
Z
+
Y
]
≤
E
[
Z
]
+
E
[
Y
].
(iii)
E
[
b
]
=
b.
(iv)
E
[
θ
Y
]
=
θ
E
[
Y
].The space given by triple(
Λ
,
H
,
E
) is named sublinear expectation space. The functional E is known as a nonlinear expectation if it holds only the first two properties.Definition 2.
Assume two random vectorsY and Z, which are n-dimensional and are, respectively, defined on the spaces (
Λ
,
H
,
E
) and (
Λ
~
,
H
~
,
E
~
). These are distributed identically, represented as Y
~
Z if
(2)
E
[
φ
(
Y
)
]
=
E
~
[
φ
(
Z
)
]
,
∀
φ
∈
C
l
.
Lip
(
R
n
)
.Definition 3.
Suppose(
Λ
,
H
,
E
) is a sublinear expectation space with Y
∈
H and
(3)
σ
¯
2
=
E
[
Y
2
]
,
σ
_
2
=
-
E
[
-
Y
2
]
.
Then Y is called N
(
0
;
[
σ
¯
2
,
σ
_
2
]
)-distributed or G-distributed, if for every c, d
≥
0 we get
(4)
c
Y
+
d
Z
~
c
2
+
d
2
Y
,
for every Z
∈
H, where Z
~
Y and it is independent of YTo define the G-Brownian motion, we suppose thatΛ
=
C
0
(
R
+
) represents the space of all real valued incessant paths (
w
t
)
t
∈
R
+ such that w
0
=
0 having norm
(5)
ρ
(
w
1
,
w
2
)
=
∑
k
=
1
∞
1
2
k
(
max
⁡
t
∈
[
0
,
k
]
⁡
|
w
t
1
-
w
t
2
|
∧
1
)
.
Consider for t
∈
[
0
,
∞
) and w
∈
Λ the process B
t
(
w
)
=
w
t, which is commonly known as the canonical process. Then for every rigid T
∈
[
0
,
∞
) we get that
(6)
L
i
r
(
Λ
T
)
=
{
φ
(
B
t
1
,
B
t
2
,
…
,
B
t
n
)
:
t
1
,
…
,
t
n
∈
[
0
,
T
]
,
n
∈
Z
+
,
φ
∈
C
l
.
Lip
(
R
n
)
}
,
for t
≤
T, L
i
r
(
Λ
t
)
⊆
L
i
r
(
Λ
T
), and L
i
r
(
Λ
)
=
∪
m
=
1
∞
L
i
r
(
Λ
m
). Take {
ξ
k
}
k
=
1
∞ which is a sequence of n-dimensional random vectors on the space (
Λ
,
H
,
E
). Also, for every k
=
1,2
,
…
,
n
-
1
,
ξ
k
+
1 is independent of (
ξ
1
,
ξ
2
,
…
,
ξ
k
) and ξ
k is N
(
0
;
[
σ
¯
2
,
σ
_
2
]
)-distributed. Then we represent a sublinear expectation E
[
·
] stated on L
i
r
(
Λ
) as follows. For every Y
=
φ
(
B
t
1
-
B
t
0
,
B
t
2
-
B
t
1
,
…
,
B
t
n
-
B
t
n
-
1
)
∈
L
i
r
(
Λ
)
,
0
=
t
0
<
t
1
<
⋯
<
t
n
<
∞, and φ
∈
C
l
.
Lip
(
R
n
) we have
(7)
E
[
φ
(
B
t
1
-
B
t
0
,
B
t
2
-
B
t
1
,
…
,
B
t
n
-
B
t
n
-
1
)
]
=
E
~
[
φ
(
t
1
-
t
0
ξ
1
,
…
,
t
n
-
t
n
-
1
ξ
n
)
]
.Definition 4.
The expectation mentioned above, that is,E
:
L
i
r
(
Λ
)
→
R, is known as a G-expectation and the related process {
B
t
,
t
≥
0
} is said to be a G-Brownian motion.The completion ofL
i
r
(
Λ
) is presented with the norm ∥
X
∥
r
=
(
E
[
|
X
|
r
]
)
1
/
r for r
≥
1 by L
G
r
(
Λ
). For 0
≤
t
≤
T
<
∞ we have L
G
r
(
Λ
t
)
⊆
L
G
r
(
Λ
T
)
⊆
L
G
r
(
Λ
). Also, F
t
=
σ
{
B
s
,
0
≤
s
≤
t
}, F
=
{
F
t
}
t
≥
0, denotes the filtration caused due to the stated process {
B
t
,
t
≥
0
}. To define G-Itô’s integral, we consider, for every T
∈
R
+, a partition of [
0
,
T
] which is a finite ordered subset π
T
=
{
t
0
,
t
1
,
…
,
t
N
} where T
=
t
N
>
t
N
-
1
>
⋯
>
t
1
>
t
0
=
0. Assume r
≥
1 to be fixed. For the above mentioned given partition π
T we represent
(8)
M
G
r
,
0
(
0
,
T
)
=
{
η
t
(
w
)
=
∑
k
=
0
N
-
1
‍
ξ
k
(
w
)
I
[
t
k
,
t
k
+
1
]
(
t
)
:
ξ
k
∈
L
G
r
(
F
t
k
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiii
t
k
<
t
k
+
1
,
k
=
0,1
,
…
,
N
-
1
∑
k
=
0
N
-
1
‍
}
.
Remember that L
G
r
(
F
t
)
=
{
ξ
∈
L
G
1
(
F
t
)
;
E
[
|
ξ
|
r
]
<
∞
}. We use M
G
r
(
0
,
T
) to denote the completion of M
G
r
,
0
(
0
,
T
) under the norm ∥
η
∥
=
{
∫
0
T
‍
E
[
|
η
v
|
r
]
d
v
}
1
/
r where for 1
≤
r
≤
p, M
G
r
(
0
,
T
)
⊃
M
G
p
(
0
,
T
).Definition 5.
For everyη
t
∈
M
G
2,0
(
0
,
T
), G-Itô’s integral is denoted by I
(
η
t
) and is defined by
(9)
I
(
η
)
=
∑
k
=
0
N
-
1
‍
ξ
k
(
B
t
k
+
1
-
B
t
k
)
=
∫
0
T
‍
η
v
d
B
v
.Let a sequence of partition of[
0
,
t
] be given by π
T
N
=
{
t
0
N
,
t
1
N
,
…
,
t
N
N
}, N
=
1,2
,
3
,
…, where π
t
N
=
max
⁡
1
≤
k
≤
N
|
t
k
N
-
t
k
-
1
N
|.Definition 6.
The process{
〈
B
〉
t
,
t
≥
0
}, known as quadratic variation process of {
B
t
,
t
≥
0
}, is a continuous increasing process having 〈
B
〉
0
=
0 and is stated by
(10)
〈
B
〉
t
=
B
t
2
-
2
∫
0
t
‍
B
v
d
B
v
=
lim
⁡
π
t
N
→
∞
⁡
∑
k
=
0
N
-
1
‍
ξ
k
(
B
t
k
+
1
-
B
t
k
)
2
.Definition 7.
SupposeB
(
Λ
) is the Borel σ-algebra of Λ and P is the (weakly compact) combination of measures for probability P that is stated on (
Λ
,
B
(
Λ
)
). Afterwards, the capacity c
^
(
·
) attached to P is given by
(11)
c
^
(
D
)
=
sup
⁡
P
∈
P
⁡
P
(
D
)
,
D
∈
B
(
Λ
)
.
In case of the capacity of a set D to be zero then the set D is called polar, that is, c
^
(
D
)
=
0 and a property carries quasi-surely (in short q
.
s.) if it is satisfied outside a polar set.Lemma 8 (Gronwall inequality).
Letκ and c
<
d be real numbers, g
(
t
)
≥
0 for t
∈
[
c
,
d
], and h
(
t
) be a real valued continuous function on [
c
,
d
] such that h
(
t
)
≤
κ
+
∫
c
d
‍
g
(
v
)
h
(
v
)
d
v; then
(12)
h
(
t
)
≤
κ
e
∫
c
d
‍
g
(
v
)
d
v
,
t
∈
[
c
,
d
]
.


## 3. The Cauchy-Maruyama Approximation Scheme for G-SFDEs

Let0
=
t
0
≤
t
≤
T
<
∞, a
∈
[
-
τ
,
0
], and τ
>
0. Represent by B
C
(
[
-
τ
,
0
]
;
R
d
) the space of bounded incessant R
d-valued mappings φ defined on [
-
τ
,
0
] and having norm ∥
φ
∥
=
sup
⁡
a
≤
0
|
φ
(
a
)
|. Suppose α
:
[
t
0
,
T
]
×
B
C
(
[
-
τ
,
0
]
;
R
d
)
→
R
d, β
:
[
t
0
,
T
]
×
B
C
(
[
-
τ
,
0
]
R
d
)
→
R
d
×
m, and γ
:
[
t
0
,
T
]
×
B
C
(
[
-
τ
,
0
]
;
R
d
)
→
R
d
×
m are Borel measurable. Assume the following d-dimensional G-SFDE [5]:
(13)
d
Y
(
t
)
=
α
(
t
,
Y
(
t
+
a
)
)
d
t
+
β
(
t
,
Y
(
t
+
a
)
)
d
〈
B
,
B
〉
(
t
)
+
γ
(
t
,
Y
(
t
+
a
)
)
d
B
(
t
)
,
t
∈
[
t
0
,
T
]
,
with initial data
(14)
Y
t
0
=
ξ
=
{
ξ
(
a
)
:
-
τ
≤
a
≤
0
}
is
F
0
-measurable
,
B
C
(
[
-
τ
,
0
]
;
R
d
)
-value
random
variable
iiiiiiiiiiiiiiii
so
that
ξ
∈
M
G
2
(
[
-
τ
,
T
]
;
R
d
)
.
Also, {
〈
B
,
B
〉
(
t
)
:
t
≥
0
} represent the process of quadratic variation of G-Brownian motion {
B
(
t
)
:
t
≥
0
}. The coefficients α, β, and γ are given mappings satisfying α
(
·
,
y
), β
(
·
,
y
), γ
(
·
,
y
)
∈
M
G
2
(
[
t
0
-
τ
,
T
]
;
R
d
) for all y
∈
R
d. The integral form of (13) with initial data (14) is given as the following:
(15)
Y
(
t
)
=
ξ
(
0
)
+
∫
t
0
t
‍
α
(
v
,
Y
(
v
+
a
)
)
d
v
+
∫
t
0
t
‍
β
(
v
,
Y
(
v
+
a
)
)
d
〈
B
,
B
〉
(
v
)
+
∫
t
0
t
‍
γ
(
v
,
Y
(
v
+
a
)
)
d
B
(
v
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
t
∈
[
t
0
,
T
]
,
where ξ
(
0
)
∈
R
d is a given initial condition.Definition 9.
AnR
d-valued stochastic process Y
(
t
) defined on t
0
-
τ
≤
t
≤
T is said to be the solution of (13) having initial data (14) if it holds the properties given below:(i)
the solutionY
(
t
) is continuous and it is F
t-adapted, for all 0
=
t
0
≤
t
≤
T;
(ii)
α
(
t
,
Y
t
)
∈
L
1
(
[
0
,
T
]
;
R
d
) and β
(
t
,
Y
t
)
,
γ
(
t
,
Y
t
)
∈
L
2
(
[
0
,
T
]
;
R
d
×
m
);
(iii)
Y
t
0
=
ξ and for every t
∈
[
t
0
,
T
]
(16)
Y
(
t
)
=
ξ
(
0
)
+
∫
t
0
t
‍
α
(
v
,
Y
(
v
+
a
)
)
d
v
+
∫
t
0
t
‍
β
(
v
,
Y
(
v
+
a
)
)
d
〈
B
,
B
〉
(
v
)
+
∫
t
0
t
‍
γ
(
v
,
Y
(
v
+
a
)
)
d
B
(
v
)
q
.
s
.The Cauchy-Maruyama approximation scheme for G-SFDE (15) with initial data (14) is shown as the following. For any integer k
≥
1, we define Y
k
(
t
) on [
t
0
-
τ
,
T
] as follows:
(17)
Y
k
(
t
0
+
a
)
=
ξ
(
a
)
for
-
τ
≤
a
≤
0
,
(18)
Y
k
(
t
)
=
Y
k
(
t
0
+
n
k
)
+
∫
t
0
+
(
n
/
k
)
t
‍
α
(
v
,
Y
k
(
t
0
+
n
k
+
a
)
)
d
v
+
∫
t
0
+
(
n
/
k
)
t
‍
β
(
v
,
Y
k
(
t
0
+
n
k
+
a
)
)
d
〈
B
,
B
〉
(
v
)
+
∫
t
0
+
(
n
/
k
)
t
‍
γ
(
v
,
Y
k
(
t
0
+
n
k
+
a
)
)
d
B
(
v
)
,
for t
0
+
(
n
/
k
)
<
t
≤
(
t
0
+
(
(
n
+
1
)
/
k
)
)
∧
T, n
=
0,1
,
2
,
…. Obviously, Y
k
(
·
) can be determined by stepwise iterations over the interval (
t
0
,
t
0
+
(
1
/
n
)
]
,
(
t
0
+
(
1
/
n
)
,
t
0
+
(
2
/
n
)
]
,
and
so
forth
.Furthermore, if we define Y
k
(
t
0
+
a
)
=
Y
~
k
(
t
0
) and Y
k
(
t
0
+
(
n
/
k
)
+
a
)
=
Y
~
k
(
t
) for t
0
+
(
n
/
k
)
<
t
≤
(
t
0
+
(
(
n
+
1
)
/
k
)
)
∧
T, n
=
0,1
,
2
,
…, then (18) yields
(19)
Y
k
(
t
)
=
ξ
(
0
)
+
∫
t
0
t
‍
α
(
v
,
Y
~
k
(
v
)
d
v
)
+
∫
t
0
t
‍
β
(
v
,
Y
~
k
(
v
)
)
d
〈
B
,
B
〉
(
v
)
+
∫
t
0
t
‍
γ
(
v
,
Y
~
k
(
v
)
)
d
B
(
v
)
,
for t
0
≤
t
≤
T. We will use the following linear growth and Lipschitz conditions, respectively. Suppose two positive numbers K
^ and K
~ such that(i)
for everyt
∈
[
t
0
,
T
],
(20)
|
α
(
t
,
z
)
|
2
+
|
β
(
t
,
z
)
|
2
+
|
γ
(
t
,
z
)
|
2
≤
K
~
(
1
+
|
z
|
2
)
,
(ii)
for ally, z
∈
(
[
t
0
-
τ
,
T
]
R
d
) and t
∈
[
t
0
,
T
],
(21)
|
α
(
t
,
y
)
-
α
(
t
,
z
)
|
2
+
|
β
(
t
,
y
)
-
β
(
t
,
z
)
|
2
+
|
γ
(
t
,
y
)
-
γ
(
t
,
z
)
|
2
≤
K
^
|
y
-
z
|
2
.


## 4. Properties of Cauchy-Maruyama Approximate Solutions

Here we show that the Cauchy-Maruyama approximate solutionsY
k
(
t
) satisfy some very useful properties, which are given in the form of Lemmas (10) and (12).Lemma 10.
Let the linear growth condition (20) hold. Then for all k
≥
1,
(22)
sup
⁡
t
0
-
τ
≤
t
≤
T
E
[
|
Y
k
(
t
)
|
2
]
≤
C
,
f
o
r
a
n
y
T
>
0
,
where C
=
E
[
∥
ξ
∥
2
]
+
K
1
e
4
K
~
K
2
T, K
1
=
4
(
1
+
T
K
~
K
2
)
E
[
∥
ξ
∥
2
]
+
4
K
~
K
2
T, K
2
=
C
1
T
+
C
2
T
+
C
3, and C
1, C
2, and C
3 are arbitrary positive constants.Proof.
Using the basic inequality(
a
+
b
+
c
+
d
)
2
≤
4
a
2
+
4
b
2
+
4
c
2
+
4
d
2 then from (19) we have
(23)
|
X
k
(
t
)
|
2
≤
4
|
ξ
(
0
)
|
2
+
4
|
∫
t
0
t
‍
α
(
v
,
Y
~
k
(
v
)
)
d
v
|
2
+
4
|
∫
t
0
t
‍
β
(
v
,
Y
~
k
(
v
)
)
d
〈
B
,
B
〉
(
v
)
|
2
+
4
|
∫
t
0
t
‍
γ
(
v
,
Y
~
k
(
v
)
)
d
B
(
v
)
|
2
,
for t
∈
[
t
0
,
T
]. Applying the Burkholder-Davis-Gundy (BDG) inequalities [9] and the linear growth condition (20) and taking sublinear expectation, it is implied that
(24)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
|
2
]
≤
4
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
‍
(
1
+
E
[
|
Y
~
k
(
v
)
|
2
]
)
d
v
≤
4
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
T
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
‍
E
[
|
Y
~
k
(
v
)
|
2
]
d
v
.
Remembering the notion of Y
~
k
(
t
), we proceed as
(25)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
|
2
]
≤
4
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
T
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
E
[
sup
⁡
t
0
-
τ
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
d
v
.
Noting the fact that
(26)
sup
⁡
t
0
-
τ
≤
t
≤
T
|
Y
k
(
t
)
|
2
≤
∥
ξ
∥
2
+
sup
⁡
t
0
≤
t
≤
T
|
Y
k
(
t
)
|
2
,
E
[
sup
⁡
t
0
-
τ
≤
t
≤
T
|
Y
k
(
t
)
|
2
]
≤
E
[
∥
ξ
∥
2
]
+
E
[
sup
⁡
t
0
≤
t
≤
T
|
Y
k
(
t
)
|
2
]
,
we have
(27)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
|
2
]
≤
4
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
T
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
(
E
[
∥
ξ
∥
2
]
+
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
)
d
v
≤
4
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
T
+
4
K
~
T
(
C
1
T
+
C
2
T
+
C
3
)
E
[
∥
ξ
∥
2
]
+
4
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
(
E
~
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
)
d
v
=
K
1
+
4
K
~
K
2
×
∫
t
0
t
(
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
)
d
v
,
where K
1
=
4
(
1
+
T
K
~
K
2
)
E
[
∥
ξ
∥
2
]
+
4
K
~
K
2
T and K
2
=
C
1
T
+
C
2
T
+
C
3. Thus the Gronwall inequality gives
(28)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
|
2
]
≤
K
1
e
4
K
~
K
2
t
.
Letting t
=
T, we have
(29)
E
[
sup
⁡
t
0
≤
u
≤
T
|
Y
k
(
u
)
|
2
]
≤
K
1
e
4
K
~
K
2
T
.
Consequently,
(30)
E
[
sup
⁡
t
0
-
τ
≤
u
≤
T
|
Y
k
(
u
)
|
2
]
≤
E
[
∥
ξ
∥
2
]
+
E
[
sup
⁡
t
0
≤
u
≤
T
|
Y
k
(
u
)
|
2
]
≤
E
[
∥
ξ
∥
2
]
+
K
1
e
4
K
~
K
2
T
=
C
,
which is the required result (22).Remark 11.
The above lemma shows that, for allk
≥
1, Y
k
(
t
)
∈
M
G
2
(
[
t
0
-
τ
,
T
]
;
R
d
). Similarly, as the previous lemma, one can show that
(31)
sup
⁡
t
0
-
τ
≤
t
≤
T
E
[
|
Y
(
t
)
|
2
]
≤
C
,
for
any
T
>
0
.Lemma 12.
For allk
≥
1 and t
0
≤
s
<
t
≤
T,
(32)
E
[
|
Y
k
(
t
)
-
Y
k
(
s
)
|
2
]
≤
K
3
(
t
-
s
)
,
where K
3
=
3
K
~
K
2
(
1
+
E
∥
ξ
∥
2
+
K
*
) and K
*
=
K
1
e
4
K
~
K
2
T.Proof.
Obviously, one can observe that, fork
≥
1 and t
0
≤
s
<
t
≤
T,
(33)
Y
k
(
t
)
-
Y
k
(
s
)
=
∫
s
t
‍
α
(
v
,
Y
~
k
(
v
)
)
d
v
+
∫
s
t
‍
β
(
v
,
Y
~
k
(
v
)
)
d
〈
B
,
B
〉
(
v
)
+
∫
s
t
‍
γ
(
v
,
Y
~
k
(
v
)
)
d
B
(
v
)
.
Thus using the linear growth condition (20) and the BDG inequalities [9] and taking sublinear expectation, we obtain
(34)
E
[
sup
⁡
s
≤
r
<
u
≤
t
|
Y
k
(
u
)
-
Y
k
(
r
)
|
2
]
≤
3
E
[
sup
⁡
s
≤
r
<
u
≤
t
|
∫
r
u
‍
α
(
v
,
Y
~
k
(
v
)
)
d
v
|
2
]
+
3
E
[
sup
⁡
s
≤
r
<
u
≤
t
|
∫
r
u
‍
β
(
v
,
Y
~
k
(
v
)
)
d
〈
B
,
B
〉
(
v
)
|
2
]
+
3
E
[
sup
⁡
s
≤
r
<
u
≤
t
|
∫
r
u
‍
γ
(
v
,
Y
~
k
(
v
)
)
d
B
(
v
)
|
2
]
≤
3
C
1
(
t
-
s
)
∫
s
t
‍
E
[
|
α
(
v
,
Y
~
k
(
v
)
)
|
2
]
d
v
+
3
C
2
(
t
-
s
)
∫
s
t
‍
E
[
|
β
(
v
,
Y
~
k
(
v
)
)
|
2
]
d
v
+
3
C
3
∫
s
t
‍
E
[
|
γ
(
v
,
Y
~
k
(
v
)
)
|
2
]
d
v
≤
3
K
~
(
C
1
(
t
-
s
)
+
C
2
(
t
-
s
)
+
C
3
)
×
∫
s
t
(
1
+
E
[
|
Y
~
k
(
v
)
|
2
]
)
d
v
≤
3
K
~
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
s
t
(
1
+
E
[
|
Y
~
k
(
v
)
|
2
]
)
d
v
=
3
K
~
K
2
(
t
-
s
)
+
3
K
~
K
2
×
∫
s
t
E
[
|
sup
⁡
t
0
-
τ
≤
u
≤
v
Y
k
(
u
)
|
2
]
d
v
≤
3
K
~
K
2
(
t
-
s
)
+
3
K
~
K
2
×
∫
s
t
(
E
[
∥
ξ
∥
2
]
+
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
)
d
u
≤
3
K
~
K
2
(
t
-
s
)
+
3
K
~
K
2
E
[
∥
ξ
∥
2
]
(
t
-
s
)
+
3
K
~
K
2
∫
s
t
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
|
2
]
d
u
.
From (29) taking K
*
=
K
1
e
4
K
~
K
2
T we get
(35)
E
[
|
Y
k
(
t
)
-
Y
k
(
s
)
|
2
]
≤
3
K
~
K
2
(
t
-
s
)
+
3
K
~
K
2
E
[
∥
ξ
∥
2
]
×
(
t
-
s
)
+
3
K
~
K
2
K
*
(
t
-
s
)
≤
3
K
~
K
2
[
1
+
E
∥
ξ
∥
2
+
K
*
]
(
t
-
s
)
=
K
3
(
t
-
s
)
,
which is the required result (32).Remark 13.
In a similar way as the previous lemma one can prove(36)
E
[
|
Y
(
t
)
-
Y
(
s
)
|
2
]
≤
K
3
(
t
-
s
)
.


## 5. Existence of Solutions for G-SFDEs

Firstly, this section shows that the unique solution of G-SFDEs (15) gets convergence from Cauchy-Maruyama (CM) approximate solutions and then it determines that the G-SFDE (15) has a unique solution with the Cauchy-Maruyama approximation scheme. Suppose if there is a constant θ that is positive such as
(37)
E
[
|
ξ
(
a
2
)
-
ξ
(
a
1
)
|
2
]
≤
θ
(
a
2
-
a
1
)
if
-
τ
≤
a
1
<
a
2
≤
0
;
that is, the initial data ξ
=
{
ξ
(
a
)
:
-
τ
≤
a
≤
0
} is uniformly Lipschitz L
2-continuous.Theorem 14.
Suppose that the coefficientsα, β, and γ satisfy the linear growth and Lipschitz conditions (20) and (21), respectively. Let the initial data satisfy condition (37). Then
(38)
E
[
sup
⁡
t
0
≤
t
≤
T
|
Y
k
(
t
)
-
Y
(
t
)
|
2
]
≤
C
*
k
,
where C
*
=
6
K
^
K
2
[
C
2
T
+
(
C
2
∨
θ
)
τ
]
e
6
K
^
K
2
T.Proof.
Using the BDG inequalities [9], the Lipschitz condition (21), and inequality (32), we have
(39)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
≤
3
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
k
(
v
)
-
Y
(
v
+
a
)
|
2
]
d
v
≤
3
K
^
K
2
∫
t
0
t
‍
E
[
-
Y
~
(
v
)
+
Y
~
(
v
)
|
2
]
|
Y
~
k
(
v
)
-
Y
(
v
+
a
)
iiiiiiiiiiiiiiiiiiiiiiiiii
-
Y
~
(
v
)
+
Y
~
(
v
)
|
2
]
d
v
≤
6
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
k
(
v
)
-
Y
~
(
v
)
|
2
]
d
v
+
6
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
(
v
)
-
Y
(
v
+
a
)
|
2
]
d
v
=
6
K
^
K
2
∫
t
0
t
‍
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
d
v
+
I
,
where I
=
6
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
(
v
)
-
Y
(
v
+
a
)
|
2
]
d
v. Then by Gronwall’s inequality
(40)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
≤
I
e
6
K
^
K
2
T
.
Using (36) we estimate I as follows:
(41)
I
=
6
K
^
K
2
×
∑
k
≥
0
∫
t
0
+
(
n
/
k
)
(
t
0
+
(
(
n
+
1
)
/
k
)
)
∧
T
E
[
-
Y
(
v
+
a
)
Y
(
t
0
+
n
k
+
a
)
|
2
]
|
Y
(
t
0
+
n
k
+
a
)
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
-
Y
(
v
+
a
)
Y
(
t
0
+
n
k
+
a
)
|
2
]
d
v
≤
6
K
^
K
2
C
2
(
T
-
t
0
)
1
k
+
6
K
^
K
2
×
∑
k
≥
0
∫
t
0
+
(
n
/
k
)
(
t
0
+
(
(
n
+
1
)
/
k
)
)
∧
τ
E
[
-
Y
(
v
+
a
)
|
Y
(
t
0
+
n
k
+
a
)
|
2
|
Y
(
t
0
+
n
k
+
a
)
|
Y
(
t
0
+
n
k
+
a
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
-
Y
(
v
+
a
)
|
Y
(
t
0
+
n
k
+
a
)
|
2
|
Y
(
t
0
+
n
k
+
a
)
]
d
v
≤
6
K
^
K
2
C
2
T
1
k
+
6
K
^
K
2
(
C
2
∨
θ
)
τ
1
k
=
6
K
^
K
2
[
C
2
T
+
(
C
2
∨
θ
)
τ
]
1
k
.
Thus substituting the value of I in (40), we obtain
(42)
E
[
sup
⁡
t
0
≤
u
≤
t
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
≤
6
K
^
K
2
[
C
2
T
+
(
C
2
∨
θ
)
τ
]
1
k
e
6
K
^
K
2
T
=
C
*
k
,
where C
*
=
6
K
^
K
2
[
C
2
T
+
(
C
2
∨
θ
)
τ
]
e
6
K
^
K
2
T, which is the required result (38).Theorem 15.
Suppose that the coefficientsα, β, and γ satisfy the linear growth and Lipschitz conditions (20) and (21), respectively. Let the initial data satisfy condition (37). Then {
Y
k
,
k
≥
1
} defined by (18) is a Cauchy sequence in M
G
2
(
t
0
-
τ
,
T
;
R
n
) and it converges to the unique solution Y
(
t
) of the G-SFDE (15).Proof.
Firstly, it is proved that the sequence{
Y
k
,
k
≥
1
} is Cauchy. Assume l
>
k; then for t
∈
[
t
0
,
T
],
(43)
|
Y
l
(
t
)
-
Y
k
(
t
)
|
2
≤
3
|
∫
t
0
t
‍
[
α
(
v
,
Y
~
l
(
v
)
)
-
α
(
v
,
Y
~
k
(
v
)
)
]
d
v
|
2
+
3
|
∫
t
0
t
‍
[
β
(
v
,
Y
~
l
(
v
)
)
-
β
(
v
,
Y
~
k
(
v
)
)
]
d
〈
B
,
B
〉
(
v
)
|
2
+
3
|
∫
t
0
t
‍
[
γ
(
v
,
Y
~
l
(
v
)
)
-
γ
(
v
,
Y
~
k
(
v
)
)
]
d
B
(
v
)
|
2
.
By using the BDG inequalities [9], Lipschitz condition (21), and Lemma (10), we get
(44)
sup
⁡
t
0
≤
t
≤
T
E
[
|
Y
l
(
t
)
-
Y
m
(
t
)
|
2
]
≤
3
K
^
(
C
1
T
+
C
2
T
+
C
3
)
×
∫
t
0
t
‍
E
[
|
Y
~
l
(
v
)
-
Y
~
k
(
v
)
|
2
]
d
v
≤
6
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
l
(
v
)
-
Y
~
(
v
)
|
2
]
d
v
+
6
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
(
v
)
|
2
-
Y
~
k
(
v
)
]
d
v
≤
6
K
^
K
2
∫
t
0
t
‍
sup
⁡
t
0
≤
u
≤
v
E
[
|
Y
l
(
u
)
-
Y
(
u
)
|
2
]
d
v
+
6
K
^
K
2
∫
t
0
t
‍
sup
⁡
t
0
≤
u
≤
v
E
[
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
d
v
.
Using (38), it yields
(45)
sup
⁡
t
0
≤
t
≤
T
E
[
|
Y
l
(
t
)
-
Y
k
(
t
)
|
2
]
≤
6
K
^
K
2
∫
t
0
t
‍
C
*
l
d
v
+
6
K
^
K
2
∫
t
0
t
‍
C
*
k
d
v
≤
6
K
^
K
2
T
C
*
(
1
l
+
1
k
)
.
Consequently, we have
(46)
sup
⁡
t
0
≤
t
≤
T
E
[
|
Y
l
(
t
)
-
Y
k
(
t
)
|
2
]
≤
C
1
*
(
1
k
+
1
l
)
⟶
0
,
when
k
,
l
⟶
∞
,
where C
1
*
=
6
K
^
K
2
T
C
*. Thus the sequence {
X
k
(
t
)
,
k
≥
1
} is a Cauchy sequence in M
G
2
(
[
t
0
-
τ
,
T
]
;
R
n
). Denoting its limits by X
(
t
) and letting k
→
∞ in (46) yield
(47)
sup
⁡
t
0
-
τ
≤
t
≤
T
E
[
|
Y
l
(
t
)
-
Y
(
t
)
|
2
]
≤
C
1
*
l
.
Now we show that Y
(
t
) satisfies the G-SFDE (15). Let t
∈
[
t
0
,
T
]; then using the Lipschitz condition (21), Lemma 10, and the BDG inequalities [9], we get
(48)
E
[
|
∫
t
0
t
‍
[
α
(
v
,
Y
~
k
(
v
)
)
-
α
(
v
,
Y
(
v
+
a
)
)
]
d
v
|
2
]
+
E
[
IIIIII
-
β
(
v
,
Y
(
v
+
a
)
)
β
(
v
,
Y
~
k
(
v
)
)
]
d
〈
B
,
B
〉
(
v
)
∫
t
0
t
‍
|
2
]
|
∫
t
0
t
‍
[
β
(
v
,
Y
~
k
(
v
)
)
IIIIII
-
β
(
v
,
Y
(
v
+
a
)
)
β
(
v
,
Y
~
k
(
v
)
)
]
d
〈
B
,
B
〉
(
v
)
∫
t
0
t
‍
|
2
]
+
E
[
-
γ
(
v
,
X
(
v
+
a
)
)
γ
(
v
,
X
~
k
(
v
)
)
]
d
B
(
v
)
∫
t
0
t
‍
|
2
|
∫
t
0
t
‍
]
|
∫
t
0
t
‍
[
γ
(
v
,
X
~
k
(
v
)
)
IIIIII
i
i
-
γ
(
v
,
X
(
v
+
a
)
)
γ
(
v
,
X
~
k
(
v
)
)
]
d
B
(
v
)
∫
t
0
t
‍
|
2
|
∫
t
0
t
‍
]
≤
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
k
(
v
)
-
Y
(
v
+
a
)
|
2
]
d
v
≤
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
k
(
v
)
-
Y
~
(
v
)
|
2
]
d
v
+
K
^
K
2
∫
t
0
t
‍
E
[
|
Y
~
(
v
)
-
Y
(
v
+
a
)
|
2
]
d
v
.
≤
K
^
K
2
∫
t
0
t
‍
E
[
sup
⁡
t
0
≤
u
≤
v
|
Y
k
(
u
)
-
Y
(
u
)
|
2
]
d
v
+
I
6
.
≤
K
^
K
2
T
C
*
k
+
K
^
K
2
[
C
2
T
+
(
C
2
∨
θ
)
τ
]
1
k
=
K
^
K
2
[
T
C
*
+
C
2
T
+
(
C
2
∨
θ
)
τ
]
1
k
=
C
1
*
k
⟶
0
as
k
⟶
∞
,
where C
1
*
=
K
^
K
2
[
T
C
*
+
C
2
T
+
(
C
2
∨
θ
)
τ
]. Thus we get Y
(
t
) as a solution of (15). To show the uniqueness, assume that the G-SFDE (15) has two solutions, say Y
(
t
) and Z
(
t
). Then in a similar fashion as above one can prove that
(49)
E
[
sup
⁡
t
0
≤
t
≤
T
|
Z
(
t
)
-
Y
(
t
)
|
2
]
=
0
,
which gives Z
(
t
)
=
Y
(
t
) for t
∈
[
t
0
,
T
]. Hence for all t
∈
[
t
0
-
τ
,
T
], Z
(
t
)
=
Y
(
t
) q.s.



---

*Source: 809431-2014-09-11.xml*