# Strong Convergence of the Split-Stepθ-Method for Stochastic Age-Dependent Capital System with Random Jump Magnitudes

Authors: Jianguo Tan; A. Rathinasamy; Hongli Wang; Yongfeng Guo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We develop a new split-stepθ (SSθ) method for stochastic age-dependent capital system with random jump magnitudes. The main aim of this paper is to investigate the convergence of the SSθ method for a class of stochastic age-dependent capital system with random jump magnitudes. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory.


---

## Body

## 1. Introduction

Stochastic partial differential equations are becoming increasingly used to model real-world phenomena in different fields, such as economics, biology, and physics. Recently, the study of the stochastic age-dependent (vintage) capital system has received a great deal of attention. For example, Wang studied stability of solutions for stochastic investment system [1]. Zhang et al. studied the convergence and exponential stability of numerical solutions to the stochastic age-dependent capital system [2, 3].In the stochastic age-dependent capital system, due to the effects of external environment for capital system, such as innovations in technique, introduction of new products, natural disasters, and changes in laws and government policies, the size of the capital systems increases or decreases drastically. So Poisson jumps with deterministic jump magnitude have been used in stochastic age-dependent population equations. For example, Li et al. [4] studied the Euler numerical method for stochastic age-dependent population equations with Poisson jumps. L. Wang and X. Wang [5] analysed the convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps. Rathinasamy et al. [6] developed the numerical method for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching. However, the random jump magnitude is now commonly seen in financial models [7–9]. In this paper, we will consider the following stochastic age-dependent capital system with random jump magnitudes as shown in [10]: (1)
d
K
(
t
,
a
)
=
-
μ
(
t
,
a
)
K
(
t
,
a
)
d
t
+
f
(
t
,
K
(
t
,
a
)
)
d
t
+
g
(
t
,
K
(
t
,
a
)
)
d
W
(
t
)
+
h
(
t
,
K
(
t
,
a
)
,
γ
N
(
t
)
+
1
)
d
N
(
t
)
,
in
D
,
K
(
t
,
0
)
=
φ
(
t
)
=
α
(
t
)
B
(
t
)
F
(
L
(
t
)
,
∫
0
A
‍
K
(
t
,
a
)
d
a
)
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiii
in
t
∈
[
0
,
T
]
,
K
(
0
,
a
)
=
K
0
(
a
)
,
in
a
∈
[
0
,
A
]
,
ℕ
(
t
)
=
∫
0
A
‍
K
(
t
,
a
)
d
a
,
in
t
∈
[
0
,
T
]
,
where K
(
t
,
a
) denotes the stock of capital goods of age a at time t, d
K
(
t
,
a
)
=
(
∂
K
(
t
,
a
)
/
∂
t
+
∂
K
(
t
,
a
)
/
∂
a
)
d
t, φ
(
0
)
=
K
0
(
0
), and D
=
[
0
,
T
]
×
[
0
,
A
]. ℕ
(
t
) is defined as total output produced in year t; also a is the age of the capital; the investment φ
(
t
) in the new capital. The f is the appreciation (when f
≥
0) or depreciation (when f
≤
0) of the production capacity, and g represents the volatility of the capital stock. The value of h is the actual jump and γ
i is the underlying random variables of the magnitudes, and often it is called “mark" of the jump.AlsoW
(
t
) is a standard Wiener process. N
(
t
) is a scalar Poisson process with intensity λ
1. It is assumed that for some q
≥
1 there is a constant C such that 𝔼
[
|
γ
i
|
2
q
]
≤
C; that is, the 2
qth moment of the jump magnitude is bounded. The maximum physical lifetime of capital A, the planning interval of calendar time [
0
,
T
), the depreciation rate μ
(
t
,
a
) of capital, and the capital density K
0
(
a
) (the initial distribution of capital over age) are given. The α
(
t
) denotes the accumulative rate of capital at the moment of t, 0
<
α
(
t
)
<
1, and B
(
t
) is the technical progress at the moment of t. This makes that total output produced in year t be defined as ℕ
(
t
)
=
∫
0
A
‍
K
(
t
,
a
)
d
a. In each sector all the firms have an identical neoclassical technology and produce output using labor and capital. The production function F
(
L
(
t
)
,
∫
0
A
‍
K
(
t
,
a
)
d
a
) is neoclassical, where ∫
0
A
‍
K
(
t
,
a
)
d
a is the total sum of capital at time t and L
(
t
) is the labor force.The integral version of (1) is given by the equation
(2)
K
t
=
K
0
-
∫
0
t
‍
∂
K
s
∂
a
d
s
-
∫
0
t
‍
μ
(
s
,
a
)
K
s
d
s
+
∫
0
t
‍
f
(
s
,
K
s
)
d
s
+
∫
0
t
‍
g
(
s
,
K
s
)
d
W
(
s
)
+
∫
0
t
‍
h
(
s
,
K
s
,
γ
(
s
)
)
d
N
(
s
)
,
where K
t
=
K
(
t
,
a
) for fixed a.Since the system (1) does not have closed form solutions, it is necessary to develop numerical methods for (1). Recently, Zhang and Rathinasamy [10] first derived the numerical solutions for stochastic age-dependent capital system with random jump magnitudes. However, their method belongs to the classic explicit Euler method and has a lower accuracy in [10] if we do not consider the appropriate step sizes.Higham and Kloeden [11] first constructed the split-step backward Euler (SSBE) method for nonlinear stochastic differential equations with Poisson jumps. Tan and Wang [12] studied the convergence and stability of the SSBE method for linear stochastic delay integrodifferential equations. Ding et al. [13] developed the split-step θ method for solving the stochastic differential equations. Rathinasamy [14] investigated the split-step θ methods for stochastic age-dependent population equations with Markovian switching. Thus, we can construct the SSθ method for stochastic age-dependent population equations with random Poisson jumps.In this paper, we will investigate the convergence of the SSθ method for system (1). The outline of the paper is as follows. In Section 2, we will introduce some preliminary results which are essential for our analysis. Section 3 will show us the SSθ method for solving stochastic age-dependent population equations with random Poisson jumps. In Section 4, several lemmas which are useful for our main result are proved. We give the main result that the numerical solutions converge to the true solutions with strong order 1/2 in Section 5. At last, a numerical example is given to verify the results obtained from the theory.


## 2. Preliminaries

Throughout this paper, it will be denoted byL
2
(
[
0
,
A
]
) the space of functions that are square-integrable over the domain [
0
,
A
]. Let
(3)
V
=
H
1
(
[
0
,
A
]
)
≡
{
ξ
∣
ξ
∈
L
2
(
[
0
,
A
]
)
,
∂
ξ
∂
a
∈
L
2
(
[
0
,
A
]
)
}
,
where ∂
ξ
/
∂
a is generalized partial derivative with respect to age a and V is a Sobolev space. H
=
L
2
(
[
0
,
A
]
) such that V
↪
H
≡
H
′
↪
V
′. V
′
=
H
-
1
(
[
0
,
A
]
) is the dual space of V. We denote by ∥
·
∥, |
·
| and ∥
·
∥
* the norms in V, H, and V
′, respectively, by (
·
,
·
) the scalar product in H. 〈
·
,
·
〉, the duality product between V and V
′, is defined by
(4)
〈
·
,
·
〉
=
∫
0
A
‍
u
·
v
d
a
,
u
∈
V
,
v
∈
V
′
.LetW
(
t
) be a Wiener process defined on (
Ω
,
ℱ
,
ℙ
) and taking its values in the separable Hilbert space S,
(5)
W
(
t
)
=
∑
i
=
1
∞
‍
β
i
(
t
)
e
i
,
where {
e
i
}
i
≥
0 is an orthonormal set of eigenvectors of G, β
i
(
t
) are mutually independent real Wiener processes with incremental covariance λ
i
>
0, G
e
i
=
λ
i
e
i, and tr
⁡
G
=
∑
i
=
1
∞
‍
λ
i (tr denotes the trace of an operator). For an operator g
∈
ℒ
(
S
,
H
) to be the space of all bounded linear operators from S into H, it is denoted by ∥
g
∥
2 the Hilbert-Schmidt norm; that is,
(6)
∥
g
∥
2
2
=
tr
⁡
(
g
G
g
T
)
.LetC
=
C
(
[
0
,
T
]
;
H
) be the space of all continuous function from [
0
,
T
] into H with sup-norm ∥
ϕ
∥
c=sup
⁡
0
≤
s
≤
T
|
ϕ
|
(
s
), L
V
p
=
L
p
(
[
0
,
T
]
;
V
), and L
H
p
=
L
p
(
[
0
,
T
]
;
H
).Let(
Ω
,
ℱ
,
ℙ
) be a complete probability space with a filtration {
ℱ
t
}
t
≥
0, satisfying the usual conditions (i.e., it is increasing and right continuous while ℱ
0 contains all ℙ-null sets).Definition 1.
Let(
Ω
,
ℱ
,
{
ℱ
t
}
,
ℙ
) be the stochastic basis and W
t a Wiener process. Suppose that K
0 is a random variable such that 𝔼
|
K
0
|
2
<
∞. A stochastic process K
t
≡
K
(
t
,
a
) for fixed a is said to be a solution on Ω to the stochastic age-structured capital system for t
∈
[
0
,
T
] if the following conditions are satisfied:(D1)
K
t is a ℱ
t-measurable random variable;
(D2)
K
t
∈
I
p
(
[
0
,
T
]
;
V
)
∩
L
2
(
Ω
;
C
(
[
0
,
T
]
;
V
)
), p
>
1, T
>
0, where I
p
(
[
0
,
T
]
;
V
) denotes the space of all V-valued processes (
K
t
)
t
∈
[
0
,
T
] (one will write K
t for short) measurable (from [
0
,
T
]
×
Ω into V), and satisfying 𝔼
∫
0
T
‍
∥
K
t
∥
p
d
t
<
∞. Here C
(
[
0
,
T
]
;
V
) denotes the space of all continuous functions from [
0
,
T
] to V;
(D3)
it satisfies the following equation:(7)
〈
K
t
,
v
〉
+
∫
0
t
‍
〈
∂
K
s
∂
a
,
v
〉
d
s
=
〈
K
0
,
v
〉
-
∫
0
t
‍
〈
μ
(
s
,
a
)
K
s
,
v
〉
d
s
iiiiiiiiiii
+
∫
0
t
‍
〈
f
(
s
,
K
s
)
,
v
〉
d
s
iiiiiiiiiii
+
∫
0
t
‍
〈
g
(
s
,
K
s
)
,
v
〉
d
W
(
s
)
iiiiiiiiiii
+
∫
0
t
‍
〈
h
(
s
,
K
s
,
γ
N
(
s
)
+
1
)
,
v
〉
d
N
(
s
)
, for all v
∈
V, t
∈
[
0
,
T
], a.e. ω
∈
Ω, where the stochastic integrals are understood in the Itô sense.The parameterA is the maximal age of the capital, so K
(
t
,
a
)
≡
K
t
=
0, ∀
a
≥
A.As the standing hypotheses we always assume that the following conditions are satisfied.(ℍ1)
f
(
t
,
0
)
=
0
,
g
(
t
,
0
)
=
0, and h
(
t
,
0
,
γ
N
(
t
)
+
1
)
=
0, t
∈
[
0
,
T
].
(ℍ2)
(Lipschitz condition) there exists a positive constantk such that x
,
y
∈
H and ∀
t,
(i)
|
f
(
t
,
y
)
-
f
(
t
,
x
)
|
∨
∥
g
(
t
,
y
)
-
g
(
t
,
x
)
∥
2
≤
k
|
y
-
x
|,
(ii)
|
h
(
t
,
y
,
u
)
-
h
(
t
,
x
,
v
)
|
≤
k
[
|
y
-
x
|
+
|
u
-
v
|
].
(ℍ3)
μ
(
t
,
a
) is nonnegative measurable in D such that
(8)
0
≤
μ
0
≤
μ
(
t
,
a
)
≤
μ
-
<
∞
,
and B
(
t
) is nonnegative continuous in [
0
,
T
] such that
(9)
α
(
t
)
B
(
t
)
≤
η
;
η
is a non-negativeconstant in
[
0
,
T
]
.
(ℍ4)
F
(
L
,
N
)
≥
0, (
F
(
L
,
0
)
=
0
), ∂
F
/
∂
L
>
0, 0
<
∂
F
/
∂
N
≤
F
1, where F
1 is a positive constant.In an analogous way to the corresponding proof presented in [15], the following existence and uniqueness of solutions is established: under the conditions (
ℍ
1
)
–
(
ℍ
4
), (1) has a unique continuous solution K
(
t
,
a
) on (
t
,
a
)
∈
D.We note for the following jump process:(10)
γ
(
t
)
:
=
γ
N
(
t
)
+
1
=
∑
j
‍
γ
j
+
1
I
[
τ
j
,
τ
j
+
1
)
(
t
)
,
where τ
0
=
0 and I
A is the indicator function for the set A; that is, I
A
(
x
)
=
{
1
,
x
∈
A
0
,
x
∉
A.Then the following lemma can be found in [16, 17].Lemma 2.
There exists a constantB for any t
∈
[
0
,
T
] and 𝔼
[
|
γ
i
|
2
q
]
≤
B for some q
>
1 such that
(11)
𝔼
[
∫
0
t
‍
|
γ
-
(
s
)
-
γ
(
s
)
|
2
d
s
]
≤
C
Δ
1
-
(
1
/
q
)
.


## 3. The Split-Stepθ-Method

Letτ
j denote the jth jump of N
(
s
) occurrence time. Suppose, for example, that jumps arrive at distinct, ordered times τ
1
<
τ
2
<
,
…; let t
1
,
t
2
,
…
,
t
m be a deterministic grid point of [
0
,
T
]. We construct approximate solutions to models of the form (1) at a discrete set of times τ
n. This set is the superposition of the random jump times of a Poisson process on [0,T] and a deterministic grid t
1
,
t
2
,
…
,
t
m and satisfies max|
τ
i
+
1
-
τ
i
|
<
Δ. Let Δ
=
t
n
+
1
-
t
n, Δ
W
n
:
=
W
(
t
n
+
1
)
-
W
(
t
n
), and Δ
N
n
:
=
N
(
t
n
+
1
)
-
N
(
t
n
) denote the increments of the time, Brownian motion, and the Poisson processes, respectively.For system (1) the split-step θ approximate solution is defined by the iterative scheme(12a)
Q
n
*
=
Q
n
-
∂
Q
n
+
1
∂
a
+
[
[
-
μ
(
t
n
,
a
)
Q
n
*
+
f
(
t
n
,
Q
n
*
)
]
(
1
-
θ
)
[
-
μ
(
t
n
,
a
)
Q
n
+
f
(
t
n
,
Q
n
)
]
llllllllllllllllllllllllllllll
+
θ
[
-
μ
(
t
n
,
a
)
Q
n
*
+
f
(
t
n
,
Q
n
*
)
]
]
Δ
,
(12b)
Q
n
+
1
=
Q
n
*
+
g
(
t
n
,
Q
n
*
)
Δ
W
n
+
h
(
t
n
,
Q
n
*
,
γ
N
(
t
n
)
+
1
)
Δ
N
n
,with initial values Q
0
=
K
(
0
,
A
), Q
n
(
t
,
0
)
=
∫
0
A
‍
β
(
t
,
a
)
Q
n
d
a, n
>
1; Q
n is the numerical approximation of K
(
t
n
,
a
) with t
n
=
n
·
Δ; the time increment is Δ
=
T
/
N
≪
1. Where θ
∈
[
0,1
], if we give θ
=
1, the SSθ method becomes the SSBE method in [11]. If θ
=
0, the SSθ method is an explicit method.To answer the question of the existence of numerical solution, we will give the following lemma.Lemma 3.
Let conditions(
ℍ
2
) and (
ℍ
3
) hold, and let 0
<
θ
<
1 and 0
<
Δ
<
1
/
θ
(
k
+
μ
-
); then the implicit equation (12a) can be solved uniquely for Q
n
*, with probability 1.Proof.
Writing (12a) as Q
n
*
=
F
(
Q
n
*
)
=
y
+
θ
Δ
[
-
μ
(
t
n
,
a
)
Q
n
*
+
f
(
t
n
,
Q
n
*
)
], and using condition (ℍ2) and (ℍ3), we have
(13)
|
F
(
u
)
-
F
(
v
)
|
≤
θ
Δ
[
|
f
(
t
,
u
)
-
f
(
t
,
v
)
|
+
|
μ
(
t
,
a
)
(
u
-
v
)
|
]
≤
θ
Δ
(
k
+
μ
-
)
|
u
-
v
|
.
Then the result follows from the classical Banach contraction mapping theorem [18].When Lemma3 followed, we find it is convenient to use continuous-time approximation solution in our strong convergence analysis; hence for ∈
[
t
n
,
t
n
+
1
), we can define the following step functions:
(14)
Z
1
(
t
)
=
Z
1
(
t
,
a
)
=
∑
n
=
0
N
-
1
‍
Q
n
I
[
n
Δ
t
,
(
n
+
1
)
Δ
t
)
(
t
)
,
(15)
Z
2
(
t
)
=
Z
2
(
t
,
a
)
∑
n
=
0
N
-
1
‍
Q
n
*
I
[
n
Δ
t
,
(
n
+
1
)
Δ
t
)
(
t
)
,
(16)
γ
-
(
t
)
=
∑
n
=
0
N
-
1
‍
γ
(
t
n
)
I
[
n
Δ
,
(
n
+
1
)
Δ
)
(
t
)
,
where N is the largest number such that N
Δ
≤
T.Whent
∈
[
t
n
,
t
n
+
1
), Lemma 3 ensures the existence of Q
n
* by (12a); then we define
(17)
Q
t
=
Q
0
-
∫
0
t
‍
∂
Q
s
∂
a
d
s
+
∫
0
t
‍
(
1
-
θ
)
[
-
μ
(
s
,
a
)
Z
1
(
s
)
+
f
(
s
,
Z
1
(
s
)
)
]
d
s
+
∫
0
t
‍
θ
[
-
μ
(
s
,
a
)
Z
2
(
s
)
+
f
(
s
,
Z
2
(
s
)
)
]
d
s
+
∫
0
t
‍
g
(
s
,
Z
2
(
s
)
)
d
W
s
+
∫
0
t
‍
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
d
N
s
,
with initial value Q
0
=
p
(
0
,
a
), Q
(
t
,
0
)
=
∫
0
A
‍
β
(
t
,
a
)
Q
t
d
a, Q
t
=
Q
(
t
,
a
).It is easy to verify thatZ
1
(
t
n
,
a
)
=
Q
n
=
Q
(
t
n
,
a
); that is, Z
1
(
t
,
a
) and Q
(
t
,
a
) coincide with the discrete solutions at the grid points. Hence we refer to Q
(
t
,
a
) as a continuous-time extension of the discrete approximation {
Q
n
}. So our plan is to prove a strong convergence result for Q
(
t
,
a
).


## 4. Several Lemmas

In this section, we will give several lemmas which are useful for the following main result.Lemma 4.
Under the conditions(
ℍ
1
)
–
(
ℍ
4
), there are constants p
≥
2 and C
1
>
0 such that
(18)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
K
t
|
p
]
≤
C
1
.The proof is similar to that in [3].Next lemma shows the relationship between𝔼
|
Q
n
*
|
2 and 𝔼
|
Q
n
|
2.Lemma 5.
Under conditions(
ℍ
1
)
–
(
ℍ
4
), let 0
<
θ
<
1, 0
<
Δ
<
min
⁡
{
1,1
/
θ
(
k
+
μ
-
)
,
1
/
2
3
(
μ
-
2
+
k
2
)
}, and 𝔼
|
∂
Q
n
+
1
/
∂
a
|
2
<
∞; then there exist two positive constants C
2 and C
3 such that
(19)
𝔼
|
Q
n
*
|
2
≤
C
2
𝔼
|
Q
n
|
2
+
C
3
,
where Q
n
* and Q
n are produced by (12a) and (12b).Proof.
Squaring both sides of (12a) and using the elementary inequality (
a
+
b
+
c
)
2
≤
3
a
2
+
3
b
2
+
3
c
2, we find
(20)
|
Q
n
*
|
2
≤
3
|
Q
n
|
2
+
3
|
∂
Q
n
+
1
∂
a
|
2
Δ
2
+
3
[
(
1
-
θ
)
[
-
μ
(
t
n
,
a
)
Q
n
+
f
(
t
n
,
Q
n
)
]
+
θ
[
-
μ
(
t
n
,
a
)
Q
n
*
+
f
(
t
n
,
Q
n
*
)
]
]
2
Δ
2
.
Using the elementary inequality |
(
1
-
θ
)
x
+
θ
y
|
2
≤
(
1
-
θ
)
|
x
|
2
+
θ
|
y
|
2, we obtain
(21)
|
Q
n
*
|
2
≤
3
|
Q
n
|
2
+
3
|
∂
Q
n
+
1
∂
a
|
2
Δ
2
+
3
(
1
-
θ
)
[
-
μ
(
t
n
,
a
)
Q
n
+
f
(
t
n
,
Q
n
)
]
2
Δ
2
+
3
θ
[
-
μ
(
t
n
,
a
)
Q
n
*
+
f
(
t
n
,
Q
n
*
)
]
2
Δ
2
.
Due to 0
<
θ
<
1 and conditions (
ℍ
2
)
–
(
ℍ
3
), we can get
(22)
|
Q
n
*
|
2
≤
3
|
Q
n
|
2
+
3
|
∂
Q
n
+
1
∂
a
|
2
Δ
2
+
6
[
|
μ
(
t
n
,
a
)
Q
n
|
2
+
|
f
(
t
n
,
Q
n
)
|
2
]
Δ
2
+
6
[
|
μ
(
t
n
,
a
)
Q
n
*
|
2
+
|
f
(
t
n
,
Q
n
*
)
|
2
]
Δ
2
≤
3
|
Q
n
|
2
+
3
|
∂
Q
n
+
1
∂
a
|
2
Δ
2
+
6
(
α
-
2
+
k
2
)
|
Q
n
|
2
Δ
2
+
6
(
α
-
2
+
k
2
)
|
Q
n
*
|
2
]
Δ
2
.
Taking mathematical expectation for both sides, we can obtain
(23)
𝔼
|
Q
n
*
|
2
≤
3
𝔼
|
∂
Q
n
+
1
∂
a
|
2
Δ
2
+
[
3
+
6
(
α
-
2
+
k
2
)
Δ
2
]
𝔼
|
Q
n
|
2
+
6
(
α
-
2
+
k
2
)
Δ
2
𝔼
|
Q
n
*
|
2
.
Since 6
(
α
-
2
+
k
2
)
Δ
2
<
1
/
2, thus 1
-
6
(
α
-
2
+
k
2
)
Δ
2
≥
1
/
2; then by 0
<
Δ
<
1, we have
(24)
𝔼
|
Q
n
*
|
2
≤
3
+
6
(
α
-
2
+
k
2
)
Δ
2
1
-
6
(
α
-
2
+
k
2
)
Δ
2
𝔼
|
Q
n
|
2
+
3
𝔼
|
∂
Q
n
+
1
/
∂
a
|
2
Δ
2
1
-
6
(
α
-
2
+
k
2
)
Δ
2
≤
6
[
1
+
2
(
α
-
2
+
k
2
)
]
𝔼
|
Q
n
|
2
+
6
𝔼
|
∂
Q
n
+
1
∂
a
|
2
=
C
2
𝔼
|
Q
n
|
2
+
C
3
,
where C
2
=
6
+
12
(
α
-
2
+
k
2
) and C
3
=
6
𝔼
|
∂
Q
n
+
1
/
∂
a
|
2. The proof is completed.The next lemma shows that the continuous numerical solutionsQ
t have bounded second moments.Lemma 6.
Under conditions(
ℍ
1
)
–
(
ℍ
4
), let 0
<
θ
<
1, 0
<
Δ
<
min
⁡
{
1
,
1
/
θ
(
k
+
μ
-
)
,
1
/
2
3
(
μ
-
2
+
k
2
)
}; then there exists a constant C
4
>
0 such that
(25)
𝔼
[
sup
⁡
t
∈
[
0
,
T
]
|
Q
t
∧
υ
n
|
2
]
≤
C
4
,
where τ
n
:
=
inf
⁡
{
t
≥
0
,
|
K
t
|
≥
n
}, σ
n
:
=
inf
⁡
{
t
≥
0
,
|
Q
t
|
≥
n
}, and υ
n
=
τ
n
∧
σ
n.Proof.
From (17), we can get
(26)
d
Q
t
=
-
∂
Q
t
∂
a
d
t
+
(
1
-
θ
)
[
-
μ
(
t
,
a
)
Z
1
(
t
)
+
f
(
s
,
Z
1
(
t
)
)
]
d
t
+
θ
[
-
μ
(
t
,
a
)
Z
2
(
t
)
+
f
(
s
,
Z
2
(
t
)
)
]
d
t
+
g
(
s
,
Z
2
(
t
)
)
d
W
t
+
h
(
s
,
Z
2
(
t
)
,
γ
-
(
t
)
)
d
N
t
.
Applying the Ito
^ formula [19] to |
Q
t
∧
υ
n
|
2 it yields
(27)
|
Q
t
∧
υ
n
|
2
=
|
Q
0
|
2
+
2
∫
0
t
∧
υ
n
‍
〈
-
∂
Q
s
∂
a
,
Q
s
〉
d
s
-
2
∫
0
t
∧
υ
n
‍
(
Q
s
(
1
-
θ
)
μ
(
s
,
a
)
Z
1
(
s
)
iiiiiiiiiiiiiii
+
θ
μ
(
s
,
a
)
Z
2
(
s
)
,
Q
s
)
d
s
+
2
∫
0
t
∧
υ
n
‍
(
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
iiiiiiiiiiiiiiiii
+
θ
f
(
s
,
Z
2
(
s
)
)
,
Q
s
)
d
s
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
g
(
s
,
Z
s
)
)
d
W
(
s
)
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
h
(
s
,
Z
s
,
γ
-
(
s
)
)
)
d
N
(
s
)
+
∫
0
t
∧
υ
n
‍
∥
g
(
s
,
Z
s
)
∥
2
2
d
s
+
∫
0
t
∧
v
n
‍
|
h
(
s
,
Z
s
,
γ
-
(
s
)
)
|
2
d
N
(
s
)
.
Using conditions (
ℍ
1
)
–
(
ℍ
3
) and the compensated Poisson process N
~
(
t
)
:
=
N
(
t
)
-
λ
1
t, we have
(28)
|
Q
t
∧
υ
n
|
2
≤
|
Q
0
|
2
+
2
∫
0
t
∧
υ
n
‍
〈
-
∂
Q
s
∂
a
,
Q
s
〉
d
s
+
2
μ
0
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
(
1
-
θ
)
Z
1
(
s
)
+
θ
Z
2
(
s
)
|
d
s
+
2
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
iiiiiiiiiiiiiiiiiiiiiii
+
θ
f
(
s
,
Z
2
(
s
)
)
|
d
s
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
g
(
s
,
Z
2
(
s
)
)
d
W
s
)
+
∫
0
t
∧
υ
n
‍
∥
g
(
s
,
Z
2
(
s
)
)
∥
2
2
d
s
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
+
∫
0
t
∧
υ
n
‍
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
,
+
λ
1
∫
0
t
∧
υ
n
‍
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
+
2
λ
1
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
d
s
.
Note
(29)
〈
-
∂
Q
s
∂
a
,
Q
s
〉
=
-
∫
0
A
‍
Q
s
·
∂
Q
s
∂
a
d
a
=
1
2
α
2
(
s
)
A
2
(
s
)
[
F
(
L
(
s
)
,
∫
0
A
‍
Q
s
d
a
)
iiiiiiiiiiiiiiiiiiiiiiiiiii
-
∫
0
A
‍
F
(
L
(
s
)
,
0
)
]
2
≤
1
2
η
2
(
∂
F
(
L
,
N
)
∂
N
|
y
)
2
(
∫
0
A
‍
Q
s
d
a
)
≤
1
2
A
F
1
2
η
2
|
Q
s
|
2
,
where y
∈
(
0
,
∫
0
A
‍
Q
s
d
a
). Consider
(30)
2
μ
0
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
(
1
-
θ
)
Z
1
(
s
)
+
θ
Z
2
(
s
)
|
d
s
≤
μ
0
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
+
2
μ
0
∫
0
t
∧
υ
n
‍
[
|
Z
1
(
s
)
|
2
+
|
Z
2
(
s
)
|
2
]
d
s
,
2
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
+
θ
f
(
s
,
Z
2
(
s
)
)
|
d
s
≤
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
i
+
2
∫
0
t
∧
υ
n
‍
[
|
f
(
s
,
Z
1
(
s
)
)
|
2
+
|
f
(
s
,
Z
2
(
s
)
)
|
2
]
d
s
≤
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
+
2
k
2
∫
0
t
∧
υ
n
‍
[
|
Z
1
(
s
)
|
2
+
|
Z
2
(
s
)
|
2
]
d
s
,
2
λ
1
∫
0
t
∧
υ
n
‍
|
Q
s
|
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
d
s
≤
λ
1
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
+
λ
1
∫
0
t
∧
υ
n
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
‍
d
s
≤
λ
1
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
+
λ
1
k
2
∫
0
t
∧
υ
n
‍
|
Z
2
(
s
)
|
2
+
|
γ
-
(
s
)
-
γ
(
s
)
|
2
d
s
.
Taking (29)-(30) into (28), we compute that for some positive constants K
1
=
A
2
F
1
2
η
2
+
1
+
μ
0
+
λ
1, K
2
=
2
(
k
2
+
μ
0
), K
3
=
3
k
2
+
2
λ
1
k
2
+
2
μ
0,
(31)
|
Q
t
∧
υ
n
|
2
≤
|
Q
0
|
2
+
K
1
∫
0
t
∧
υ
n
‍
|
Q
s
|
2
d
s
+
K
2
∫
0
t
∧
υ
n
‍
|
Z
1
(
s
)
|
2
d
s
+
K
3
∫
0
t
∧
υ
n
‍
|
Z
2
(
s
)
|
2
d
s
+
λ
1
K
2
∫
0
t
∧
υ
n
‍
|
γ
-
(
s
)
-
γ
(
s
)
|
2
d
s
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
g
(
s
,
Z
2
(
s
)
)
d
W
s
)
+
2
∫
0
t
∧
υ
n
‍
(
Q
s
,
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
+
∫
0
t
∧
υ
n
‍
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
.
Now, it follows that for any t
1
∈
[
0
,
T
] and by Lemma 5
(32)
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
∧
υ
n
|
2
]
≤
𝔼
|
Q
0
|
2
+
(
K
1
+
K
2
+
K
3
C
2
)
∫
0
t
1
∧
υ
n
‍
𝔼
[
sup
⁡
0
≤
s
≤
t
|
Q
s
|
2
]
d
t
+
K
3
C
3
T
+
λ
1
k
2
C
T
+
2
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
(
Q
s
,
g
(
s
,
Z
2
(
s
)
)
d
W
s
)
]
+
2
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
(
Q
s
,
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
]
+
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
]
.
By Burkholder-Davis-Gundy inequality, we have
(33)
2
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
(
Q
s
,
g
(
s
,
Z
2
(
s
)
)
d
W
s
)
]
≤
1
4
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
|
2
]
+
K
4
∫
0
t
1
∧
υ
n
‍
∥
g
(
t
,
Z
2
(
t
)
)
∥
2
2
d
t
≤
1
4
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
|
2
]
+
K
4
k
2
∫
0
t
1
∧
υ
n
‍
𝔼
∥
Z
2
(
t
)
∥
2
d
t
,
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
]
≤
4
λ
1
𝔼
∫
0
t
1
∧
υ
n
‍
|
h
(
t
,
Z
2
(
t
)
,
γ
-
(
s
)
)
|
2
d
t
≤
K
5
k
2
∫
0
t
1
∧
υ
n
‍
𝔼
∥
Z
2
(
t
)
∥
2
+
𝔼
|
γ
-
(
s
)
-
γ
(
s
)
|
2
d
t
≤
K
5
k
2
∫
0
t
1
∧
υ
n
‍
𝔼
∥
Z
2
(
t
)
∥
2
d
t
+
K
5
k
2
C
T
,
2
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
∫
0
t
∧
υ
n
‍
(
Q
s
,
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
]
≤
1
4
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
|
2
]
+
K
6
∫
0
t
1
∧
υ
n
‍
|
h
(
t
,
Z
2
(
t
)
,
γ
-
(
s
)
)
|
2
d
t
≤
1
4
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
|
2
]
+
K
6
k
2
∫
0
t
1
∧
υ
n
‍
𝔼
∥
Z
2
(
t
)
∥
2
d
t
+
K
6
k
2
C
T
,
for some positive constants K
4, K
5, K
6
>
0. Substituting (33) into (32) and then by Lemma 5, we can obtain that
(34)
𝔼
[
sup
⁡
t
∈
[
0
,
t
1
]
|
Q
t
∧
υ
n
|
2
]
≤
2
𝔼
|
Q
0
|
2
+
2
(
K
1
+
K
2
+
K
3
C
2
)
∫
0
t
1
∧
υ
n
‍
𝔼
[
sup
⁡
0
≤
s
≤
t
|
Q
s
|
2
]
d
t
+
2
K
3
C
3
T
+
2
(
K
4
+
K
5
+
K
6
)
k
2
∫
0
t
1
∧
υ
n
‍
𝔼
∥
Z
2
(
t
)
∥
2
d
t
+
2
(
λ
1
+
K
5
+
K
6
)
k
2
C
T
≤
2
[
K
1
+
K
2
+
K
3
C
2
+
(
K
4
+
K
5
+
K
6
)
k
2
C
2
]
×
∫
0
t
1
∧
υ
n
‍
𝔼
[
sup
⁡
0
≤
s
≤
t
|
Q
s
|
2
]
d
t
+
2
𝔼
|
Q
0
|
2
+
2
K
3
C
3
T
+
2
(
K
4
+
K
5
+
K
6
)
k
2
C
3
T
+
2
(
λ
1
+
K
5
+
K
6
)
k
2
C
T
:
=
K
7
+
K
8
∫
0
t
1
∧
υ
n
‍
𝔼
[
sup
⁡
0
≤
s
≤
t
|
Q
s
|
2
]
d
t
,
where
(35)
K
7
=
2
𝔼
|
Q
0
|
2
+
2
K
3
C
3
T
+
2
(
K
4
+
K
5
+
K
6
)
k
2
C
3
T
+
2
(
λ
1
+
K
5
+
K
6
)
k
2
C
T
,
K
8
=
2
[
K
1
+
K
2
+
K
3
C
2
+
(
K
4
+
K
5
+
K
6
)
k
2
C
2
]
.
Applying the continuous Gronwall inequality, we can easy get
(36)
𝔼
[
sup
⁡
t
∈
[
0
,
T
]
|
Q
t
∧
υ
n
|
2
]
≤
C
4
,
with C
4
=
K
7
exp
⁡
(
K
8
T
).The next lemma shows that the continuous-time approximationQ
t in (17) remains close to the step functions Z
1
(
t
) and Z
2
(
t
) in the mean square sense.Lemma 7.
Under conditions(
ℍ
1
)
–
(
ℍ
4
), 𝔼
|
∂
Q
s
/
∂
a
|
2
<
∞, and let 0
<
θ
<
1, 0
<
Δ
<
min
⁡
{
1
,
1
/
θ
(
k
+
μ
-
)
,
1
/
2
3
(
μ
-
2
+
k
2
)
}; then there exist two positive constants C
5 and C
6 that are independent of Δ, such that
(37)
𝔼
|
Q
t
-
Z
1
(
t
)
|
2
≤
C
5
Δ
,
(38)
𝔼
|
Q
t
-
Z
2
(
t
)
|
2
≤
C
6
Δ
,
where Z
1
(
t
), Z
2
(
t
), and Q
t are defined by (14), (15), and (17), respectively.Proof.
Considert
∈
[
n
Δ
,
(
n
+
1
)
Δ
]
⊆
[
0
,
T
]; we have
(39)
Q
t
-
Z
1
(
t
)
=
Q
t
-
Q
n
=
-
∫
n
Δ
t
‍
∂
Q
s
∂
a
d
s
+
∫
n
Δ
t
‍
[
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
+
θ
f
(
s
,
Z
2
(
s
)
)
]
d
s
-
∫
n
Δ
t
‍
μ
(
s
,
a
)
[
(
1
-
θ
)
Z
1
(
s
)
+
θ
Z
2
(
s
)
]
d
s
+
∫
n
Δ
t
‍
g
(
s
,
Z
2
(
s
)
)
d
W
s
+
∫
n
Δ
t
‍
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
d
N
s
.
Squaring both sides and using the element inequality (
a
+
b
+
c
+
e
+
f
)
2
≤
5
|
a
|
2
+
5
|
b
|
2
+
5
|
c
|
2
+
5
|
d
|
2
+
5
|
e
|
2, we have
(40)
|
Q
t
-
Z
1
(
t
)
|
2
≤
5
|
∫
n
Δ
t
‍
∂
Q
s
∂
a
d
s
|
2
+
5
|
∫
n
Δ
t
‍
[
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
+
θ
f
(
s
,
Z
2
(
s
)
)
]
d
s
|
2
+
5
|
∫
n
Δ
t
‍
μ
(
s
,
a
)
[
(
1
-
θ
)
Z
1
(
s
)
+
θ
Z
2
(
s
)
]
d
s
|
2
+
5
|
∫
n
Δ
t
‍
g
(
s
,
Z
2
(
s
)
)
d
W
(
s
)
|
2
+
5
|
∫
n
Δ
t
‍
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
d
N
(
s
)
|
2
.
Now, the Cauchy-Schwarz inequality, condition (ℍ
3), and the compensated Poisson process give
(41)
|
Q
t
-
Z
1
(
t
)
|
2
≤
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
5
μ
-
2
Δ
∫
n
Δ
t
‍
|
(
1
-
θ
)
Z
1
(
s
)
+
θ
Z
2
(
s
)
|
2
d
s
+
5
Δ
∫
n
Δ
t
‍
|
(
1
-
θ
)
f
(
s
,
Z
1
(
s
)
)
+
θ
f
(
s
,
Z
2
(
s
)
)
|
2
d
s
+
5
|
∫
n
Δ
t
‍
g
(
s
,
Z
2
(
s
)
)
d
W
(
s
)
|
2
+
10
|
∫
n
Δ
t
‍
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
d
N
~
(
s
)
|
2
+
10
λ
1
2
|
∫
n
Δ
t
‍
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
d
s
|
2
.
Taking mathematical expectation, by element inequality (
a
+
b
)
2
≤
2
|
a
|
2
+
2
|
b
|
2, and using the martingale isometry, we have
(42)
𝔼
|
Q
t
-
Z
1
(
t
)
|
2
≤
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
5
μ
-
2
Δ
∫
n
Δ
t
‍
𝔼
(
|
Z
1
(
s
)
|
2
+
|
Z
2
(
s
)
|
2
)
d
s
+
10
Δ
∫
n
Δ
t
‍
𝔼
|
f
(
s
,
Z
1
(
s
)
)
|
2
+
𝔼
|
f
(
s
,
Z
2
(
s
)
)
|
2
d
s
+
5
∫
n
Δ
t
‍
𝔼
|
g
(
s
,
Z
2
(
s
)
)
|
2
d
s
+
10
λ
1
∫
n
Δ
t
‍
𝔼
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
+
10
λ
1
2
Δ
∫
n
Δ
t
‍
𝔼
|
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
.
By conditions (ℍ
1) and (ℍ
2), we get
(43)
𝔼
|
Q
t
-
Z
1
(
t
)
|
2
≤
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
5
μ
-
2
Δ
∫
n
Δ
t
‍
𝔼
(
|
Z
1
(
s
)
|
2
+
|
Z
2
(
s
)
|
2
)
d
s
+
10
Δ
k
2
∫
n
Δ
t
‍
𝔼
|
Z
1
(
s
)
|
2
+
𝔼
|
Z
2
(
s
)
|
2
d
s
+
5
k
2
∫
n
Δ
t
‍
𝔼
|
Z
2
(
s
)
|
2
d
s
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
∫
n
Δ
t
‍
𝔼
|
Z
2
(
s
)
|
2
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
+
𝔼
|
γ
-
(
s
)
-
γ
(
s
)
|
2
d
s
.
=
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
(
10
Δ
k
2
+
5
μ
-
2
Δ
)
∫
n
Δ
t
‍
𝔼
|
Z
1
(
s
)
|
2
d
s
+
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
∫
n
Δ
t
‍
𝔼
|
Z
2
(
s
)
|
2
d
s
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
Δ
.
Since Z
1
(
t
)
≡
Q
n and Z
2
(
t
)
≡
Q
n
* on [
n
Δ
t
,
(
n
+
1
)
Δ
t
), we have
(44)
𝔼
|
Q
t
-
Z
1
(
t
)
|
2
≤
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
(
10
Δ
k
2
+
5
μ
-
2
Δ
)
∫
n
Δ
t
‍
𝔼
|
Q
n
|
2
d
s
+
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
∫
n
Δ
t
‍
𝔼
|
Q
n
*
|
2
d
s
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
Δ
.
Then by Lemmas 5 and 6, we can derive
(45)
𝔼
|
Q
t
-
Z
1
(
t
)
|
2
≤
[
10
Δ
k
2
+
5
μ
-
2
Δ
i
+
C
2
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
]
∫
n
Δ
t
‍
𝔼
|
Q
n
|
2
d
s
+
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
3
Δ
+
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
Δ
≤
[
10
Δ
k
2
+
5
μ
-
2
Δ
+
C
2
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
]
C
4
Δ
+
(
5
+
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
3
Δ
+
5
Δ
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
Δ
≤
C
5
Δ
,
where
(46)
C
5
=
[
10
k
2
+
5
μ
-
2
+
C
2
(
5
+
10
λ
1
+
10
λ
1
2
)
k
2
]
C
4
+
(
10
λ
1
+
10
λ
1
2
Δ
)
k
2
C
+
(
5
+
10
λ
1
+
10
λ
1
2
)
k
2
C
3
+
5
∫
n
Δ
t
‍
|
∂
Q
s
∂
a
|
2
d
s
.
Thus we can prove (37), and a similar analysis gives the proof of (38).Lemma 8.
Under conditions(
ℍ
1
)
–
(
ℍ
4
), for any p
≥
2, there exists a positive constant C
7 such that
(47)
𝔼
[
sup
⁡
t
∈
[
0
,
T
]
|
Q
t
|
p
]
≤
C
7
.Proof .
The proof is similar to that of Lemma4.


## 5. Main Result

Now we use the above lemmas to prove a strong convergent result.Theorem 9.
Under condition(
ℍ
1
)
–
(
ℍ
4
), let 0
<
θ
<
1, 0
<
Δ
<
min
⁡
{
1,1
/
θ
(
k
+
μ
-
)
,
1
/
2
3
(
μ
-
2
+
k
2
)
}; then there exists a constant C
8
>
0 such that
(48)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
K
t
∧
υ
n
-
Q
t
∧
υ
n
|
2
]
≤
C
8
Δ
1
-
(
1
/
q
)
.Proof.
From (2) and (17), we have
(49)
K
t
-
Q
t
=
-
∫
0
t
‍
∂
(
K
s
-
Q
s
)
∂
a
d
s
-
∫
0
t
‍
μ
(
s
,
a
)
[
(
1
-
θ
)
(
K
s
-
Z
1
(
s
)
)
iiiiiiiiiiiiiiiiiii
+
θ
(
K
s
-
Z
2
(
s
)
)
]
d
s
+
∫
0
t
‍
[
(
1
-
θ
)
(
f
(
s
,
K
s
)
-
f
(
s
,
Z
1
(
s
)
)
)
iillii
+
θ
(
f
(
s
,
K
s
)
-
f
(
s
,
Z
2
(
s
)
)
)
]
d
s
+
∫
0
t
‍
[
g
(
s
,
K
s
)
-
g
(
s
,
Z
2
(
s
)
)
]
d
W
s
+
∫
0
t
‍
[
h
(
s
,
K
s
,
γ
(
s
)
)
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
]
d
N
s
.
Using the generalized Ito
^ formula it yields
(50)
|
K
t
-
Q
t
|
2
=
2
∫
0
t
‍
〈
K
s
-
Q
s
,
-
∂
(
K
s
-
Q
s
)
∂
a
〉
d
s
-
2
∫
0
t
‍
(
K
s
-
Q
s
,
μ
(
s
,
a
)
iiiii
×
[
(
1
-
θ
)
(
K
s
-
Z
1
(
s
)
)
+
θ
(
K
s
-
Z
2
(
s
)
)
]
d
s
)
+
∫
0
t
‍
(
K
s
-
Q
s
,
[
(
1
-
θ
)
(
f
(
s
,
K
s
)
-
f
(
s
,
Z
1
(
s
)
)
)
iiiiiiiiiiiiiiiiiiiiiiiii
+
θ
(
f
(
s
,
P
s
)
-
f
(
s
,
Z
2
(
s
)
)
)
]
d
s
)
+
∫
0
t
‍
∥
g
(
s
,
K
s
)
-
g
(
s
,
Z
2
(
s
)
)
∥
2
2
d
s
+
2
∫
0
t
‍
(
K
s
-
Q
s
,
[
g
(
s
,
K
s
)
-
g
(
s
,
Z
2
(
s
)
)
]
d
W
s
)
+
2
∫
0
t
‍
(
iiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
K
s
-
Q
s
,
h
(
s
,
K
s
,
γ
(
s
)
)
iiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
+
2
λ
1
∫
0
t
‍
(
K
s
-
Q
s
,
h
(
s
,
K
s
,
γ
(
s
)
)
iiiiiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
s
+
∫
0
t
‍
|
h
(
s
,
K
s
,
γ
(
s
)
)
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
+
λ
1
∫
0
t
‍
|
h
(
s
,
K
s
,
γ
(
s
)
)
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
.
By Cauchy-Schwartz inequality and condition (ℍ
2), we have
(51)
|
K
t
-
Q
t
|
2
≤
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
∫
0
t
‍
|
K
s
-
Q
s
|
2
d
s
+
2
(
K
2
+
μ
0
)
∫
0
t
‍
|
K
s
-
Z
1
(
s
)
|
2
d
s
+
2
λ
1
K
2
∫
0
t
‍
|
γ
(
s
)
-
γ
-
(
s
)
|
2
d
s
+
(
3
K
2
+
2
λ
1
K
2
+
2
μ
0
)
∫
0
t
‍
|
K
s
-
Z
2
(
s
)
|
2
d
s
+
2
∫
0
t
‍
(
K
s
-
Q
s
,
[
g
(
s
,
K
s
)
-
g
(
s
,
Z
2
(
s
)
)
]
d
W
s
)
+
2
∫
0
t
‍
(
iiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
K
s
-
Q
s
,
h
(
s
,
K
s
,
γ
(
s
)
)
iiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
)
d
N
~
s
+
∫
0
t
‍
|
h
(
s
,
K
s
,
γ
(
s
)
)
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
N
~
s
.
Hence for any t
∈
[
0
,
T
]
(52)
𝔼
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
≤
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
i
×
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
K
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
2
(
k
2
+
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
1
(
s
)
∥
c
2
d
s
+
2
λ
1
k
2
C
Δ
1
-
(
1
/
q
)
+
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
1
(
s
)
∥
c
2
d
s
+
2
𝔼
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
(
K
r
-
Q
r
,
[
(
r
,
Z
2
(
r
)
)
g
(
r
,
K
r
)
iiiiiiiiiiiiiiiiiiiiiiiiilliiiiiiiiiiiii
-
g
(
r
,
Z
2
(
r
)
)
]
d
W
r
)
+
2
𝔼
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
(
h
(
s
,
Z
2
(
r
)
,
γ
-
(
r
)
)
)
K
r
-
Q
r
,
h
(
r
,
K
r
,
γ
(
r
)
)
iiiiiiiiiiiiiiiiiiiiiiiiiiii
-
h
(
s
,
Z
2
(
r
)
,
γ
-
(
r
)
)
)
d
N
~
r
+
𝔼
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
|
h
(
r
,
K
r
,
γ
(
r
)
)
IIIIIIIIIIIIIIIIIIII
-
h
(
r
,
Z
2
(
r
)
,
γ
-
(
r
)
)
|
2
d
N
~
r
.
Now using the Burkholder-Davis-Gundy inequality and Lemma 2, we have
(53)
2
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
(
K
r
-
Q
r
,
[
g
(
r
,
K
r
)
-
g
(
r
,
Z
2
(
r
)
)
]
d
W
r
)
]
≤
8
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
(
∫
0
t
∧
υ
n
‍
∥
g
(
s
,
K
s
)
-
g
(
t
,
Z
2
(
s
)
)
∥
2
2
d
s
)
1
/
2
lllllllllllllllllllll
×
(
∫
0
t
∧
υ
n
‍
∥
g
(
s
,
K
s
)
-
g
(
t
,
Z
2
(
s
)
)
∥
2
2
d
s
)
1
/
2
]
≤
1
4
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
]
+
k
1
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
2
(
s
)
∥
2
d
s
,
2
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
(
K
r
-
Q
r
,
(
h
(
r
,
K
r
)
-
h
(
s
,
Z
2
(
r
)
)
)
)
d
N
~
r
]
≤
8
λ
1
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
iiiiii
×
(
∫
0
t
∧
υ
n
‍
|
h
(
s
,
K
s
,
γ
(
s
)
)
iiiiiiiiiiiiiiiiiiiiii
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
∫
0
t
∧
υ
n
‍
)
1
/
2
∫
0
t
∧
υ
n
‍
]
≤
1
4
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
]
+
k
2
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
2
(
s
)
∥
2
+
𝔼
|
γ
(
s
)
-
γ
-
(
s
)
|
2
d
s
≤
1
4
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
]
+
k
2
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
2
(
s
)
∥
2
d
s
+
k
2
C
Δ
1
-
(
1
/
q
)
,
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
∫
0
s
∧
υ
n
‍
|
h
(
r
,
K
r
,
γ
(
r
)
)
-
h
(
r
,
Z
2
(
r
)
,
γ
-
(
r
)
)
|
2
d
N
~
r
]
≤
4
λ
1
∫
0
t
∧
υ
n
‍
𝔼
|
h
(
s
,
K
s
,
γ
(
s
)
)
-
h
(
s
,
Z
2
(
s
)
,
γ
-
(
s
)
)
|
2
d
s
≤
4
λ
1
k
2
∫
0
t
∧
υ
n
‍
𝔼
|
K
s
-
Z
2
(
s
)
|
2
+
𝔼
|
γ
(
s
)
-
γ
-
(
s
)
|
2
d
s
≤
k
3
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
2
(
s
)
∥
2
d
s
+
k
3
C
Δ
1
-
(
1
/
q
)
,
where k
1
,
k
2
,
k
3 are positive constants. Therefore, inserting (53) into (52) it has
(54)
𝔼
sup
⁡
s
∈
[
0
,
t
]
|
K
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
≤
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
I
×
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
K
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
2
(
k
2
+
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
1
(
s
)
∥
c
2
d
s
+
2
λ
1
k
2
C
Δ
1
-
(
1
/
q
)
+
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Z
2
(
s
)
∥
c
2
d
s
+
(
k
1
+
k
2
+
k
3
)
∫
0
t
∧
υ
n
‍
𝔼
∥
K
s
-
Q
s
∥
c
2
d
s
+
(
k
2
+
k
3
)
C
Δ
1
-
(
1
/
q
)
+
1
2
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
P
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
]
.
Then using Lemma 7, we have
(55)
𝔼
sup
⁡
s
∈
[
0
,
t
]
|
P
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
≤
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
I
×
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
P
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
4
(
k
2
+
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
∥
P
s
-
Q
s
∥
c
2
d
s
+
4
(
k
2
+
μ
0
)
C
5
Δ
+
2
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
×
∫
0
t
∧
υ
n
‍
𝔼
∥
P
s
-
Q
s
∥
c
2
d
s
+
2
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
C
6
Δ
+
(
2
λ
1
k
2
+
k
2
+
k
3
)
C
Δ
1
-
(
1
/
q
)
+
1
2
𝔼
[
sup
⁡
s
∈
[
0
,
t
]
|
P
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
]
.
That is,
(56)
𝔼
sup
⁡
s
∈
[
0
,
t
]
|
P
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
≤
2
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
×
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
P
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
8
(
k
2
+
μ
0
)
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
P
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
4
(
3
k
2
+
2
λ
1
K
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
×
∫
0
t
∧
υ
n
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
P
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
+
8
(
k
2
+
μ
0
)
C
5
Δ
+
4
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
C
6
Δ
+
2
(
2
λ
1
k
2
+
k
2
+
k
3
)
C
Δ
1
-
(
1
/
q
)
≤
d
1
Δ
1
-
(
1
/
q
)
+
d
2
∫
0
t
‍
𝔼
sup
⁡
r
∈
[
0
,
s
]
|
P
r
∧
υ
n
-
Q
r
∧
υ
n
|
2
d
s
,
where
(57)
d
1
=
8
(
k
2
+
μ
0
)
C
5
+
4
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
C
6
+
2
(
2
λ
1
k
2
+
k
2
+
k
3
)
C
Δ
1
-
(
1
/
q
)
,
d
2
=
2
(
A
2
F
1
2
η
2
+
μ
0
+
λ
1
+
1
)
+
8
(
k
2
+
μ
0
)
+
4
(
3
k
2
+
2
λ
1
k
2
+
2
μ
0
+
k
1
+
k
2
+
k
3
)
.
Using the Gronwall inequality, we have
(58)
𝔼
sup
⁡
s
∈
[
0
,
t
]
|
P
s
∧
υ
n
-
Q
s
∧
υ
n
|
2
≤
d
1
exp
⁡
(
d
2
T
)
Δ
:
=
C
8
Δ
1
-
(
1
/
q
)
.
Since for any t
∈
[
0
,
T
], thus we have
(59)
𝔼
sup
⁡
t
∈
[
0
,
T
]
|
P
t
∧
υ
n
-
Q
t
∧
υ
n
|
2
≤
C
8
Δ
1
-
(
1
/
q
)
.
The proof is completed.Theorem 10.
Under conditions(
ℍ
1
)
–
(
ℍ
4
), let 0
<
θ
<
1, 0
<
Δ
<
min
⁡
{
1,1
/
θ
(
k
+
μ
-
)
,
1
/
2
3
(
μ
-
2
+
k
2
)
}; then there exists a constant C
9
>
0 such that
(60)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
K
t
-
Q
t
|
2
]
≤
C
9
Δ
t
+
o
(
Δ
)
.Proof.
Lete
(
t
)
=
K
t
-
Q
t; it is easy to see that
(61)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
]
=
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
>
T
,
σ
n
>
T
}
]
+
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
≤
T
or
σ
n
≤
T
}
]
=
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
υ
n
>
T
}
]
+
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
≤
T
or
σ
n
≤
T
}
]
≤
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
∧
υ
n
)
|
2
]
+
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
≤
T
or
σ
n
≤
T
}
]
.
Recall the Young inequality: for (
1
/
p
)
+
(
1
/
q
)
=
1
(
p
,
q
>
0
), we have
(62)
a
b
≤
a
δ
1
/
p
b
δ
1
/
p
≤
(
a
δ
1
/
p
)
p
p
+
b
q
q
δ
q
/
p
=
a
p
δ
p
+
b
q
q
δ
q
/
p
,
∀
a
,
b
,
δ
>
0
.
Let p
=
2
,
δ
=
Δ; we have
(63)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
≤
T
or
σ
n
≤
T
}
]
≤
Δ
2
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
4
]
+
1
2
Δ
P
{
τ
n
≤
T
or
σ
n
≤
T
}
.
Note
(64)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
4
]
≤
8
(
𝔼
[
sup
⁡
0
≤
t
≤
T
|
P
t
|
4
]
+
𝔼
[
sup
⁡
0
≤
t
≤
T
|
Q
t
|
4
]
)
≤
8
(
C
1
+
C
7
)
.
By Lemma 4, then
(65)
P
{
τ
n
≤
T
}
=
𝔼
[
I
{
τ
n
≤
T
}
|
P
τ
n
|
p
n
4
]
≤
1
n
4
𝔼
[
sup
⁡
0
≤
t
≤
T
|
P
t
|
4
]
≤
C
1
n
4
;
similarly, the result can be derived for σ
n
(66)
P
{
σ
n
≤
T
}
=
𝔼
[
I
{
τ
n
≤
T
}
|
Q
τ
n
|
p
n
4
]
≤
1
n
4
𝔼
[
sup
⁡
0
≤
t
≤
T
|
Q
t
|
4
]
≤
C
7
n
4
,
so that
(67)
P
{
σ
d
≤
T
or
ν
d
≤
T
}
≤
P
{
σ
d
≤
T
}
+
P
{
ν
d
≤
T
}
≤
(
C
1
+
C
7
)
n
4
.
Then
(68)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
I
{
τ
n
≤
T
or
σ
n
≤
T
}
]
≤
4
Δ
(
C
1
+
C
7
)
+
(
C
1
+
C
7
)
2
Δ
n
4
.
By Theorem 9, (61) becomes
(69)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
]
≤
C
8
Δ
+
o
(
Δ
)
+
4
Δ
(
C
1
+
C
7
)
+
(
C
1
+
C
7
)
2
Δ
n
4
.
Let n
≥
2
(
Δ
2
)
-
1
/
4; then
(70)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
e
(
t
)
|
2
]
≤
C
8
Δ
+
o
(
Δ
)
+
4
Δ
(
C
1
+
C
7
)
+
Δ
(
C
1
+
C
7
)
:
=
C
9
Δ
+
o
(
Δ
)
,
where C
9
=
5
(
C
1
+
C
7
)
+
C
8. The proof is completed.Theorem 11.
Under the conditions(
ℍ
1
)
-
(
ℍ
4
) there exists a constant C such that 𝔼
[
|
γ
i
|
2
q
]
≤
C for some q
>
1, the numerical approximate solution (1) will converge to the exact solution to (1) in the sense
(71)
lim
⁡
Δ
→
0
𝔼
[
sup
⁡
0
≤
t
≤
T
|
K
t
-
Q
t
|
2
]
=
0
.Proof.
The proof is easily deduced from Theorem10.


## 6. Numerical Example

In this section, we consider the following stochastic age-dependent capital system with random jump magnitudes age-dependent in [10]
(72)
d
K
(
t
,
a
)
=
-
1
(
1
-
a
)
2
K
d
t
+
2
t
K
d
t
-
t
K
d
W
(
t
)
+
γ
-
(
t
)
K
d
N
(
t
)
,
in
D
,
K
(
t
,
0
)
=
t
2
(
1
-
t
)
2
∫
0
1
‍
K
(
t
,
a
)
d
a
,
in
t
∈
[
0
,
T
]
,
K
(
0
,
a
)
=
e
-
1
/
(
1
-
a
)
,
in
a
∈
(
0
,
A
)
,
ℕ
(
t
)
=
∫
0
1
‍
K
(
t
,
a
)
d
a
,
in
t
∈
[
0
,
T
]
,
where W
(
t
) is a scalar Brownian motion and N
(
t
) is a scalar Poisson process with intensity λ
1
=
1. Let γ
-
(
t
) be defined in (16). Take T
=
1, A
=
1, and γ
- to be a lognormal random variable such that 𝔼
[
γ
-
]
=
0.2 and 𝔼
[
γ
-
2
]
=
0.25.It is easy to verify that the conditions(
ℍ
1
)
–
(
ℍ
4
) are satisfied. Then the approximate solution will converge to the true solution of (1) for any (
t
,
a
)
∈
(
0,1
)
×
(
0,1
) in the sense of Theorem 11. Obviously, K
(
t
,
a
) in (72) cannot be solved explicitly. It is necessary to know numerical approximation Q
(
t
,
a
) of K
(
t
,
a
).Now we fix step sizesΔ
=
0.005 and Δ
a
=
0.05 and change the parameter θ in all figures. Then we compare the expectation of the numerical solution, where 𝔼
[
Q
(
t
,
a
)
]
=
(
1
/
1000
)
∑
n
=
1
1000
‍
Q
n
(
t
,
a
).In Figure1, we show the expectation of numerical solution of the system (72) by Euler method in [10] and split-step θ-method with θ = 0.2, respectively. From the figure, we can see that the split-step θ-method also reveals the age-dependent capital system tendency.Expectation of numerical solution with 1000 tests, whereΔ
=
0.005 and Δ
a
=
0.05.
(a)
(b)In Figures2 and 3, the four pictures are simulation for numerical solution Q
n
(
t
,
a
) of the system (72) by split-step θ-method with θ = 0.3, 0.4, 0.5, and 0.6, respectively.Expectation of numerical solution with 1000 tests, whereΔ
=
0.005 and Δ
a
=
0.05.
(a)
(b)Expectation of numerical solution with 1000 tests, whereΔ
=
0.005 and Δ
a
=
0.05.
(a)
(b)


## 7. Conclusion

Due to the complexity of this model, the comparison between the Euler approximation and split-step methods is not possible in this paper. Similarly, error analysis is also not obtained in this paper due to unavailability of closed form solutions. They will be discussed in the future researches. But the visualization of Figures1, 2, and 3 provides the information about the split-step methods which coincide with the Euler approximation. The comparison between Euler approximation and split-step methods in the stochastic differential equations in [11–13] shows that the split-step methods are better than the Euler approximation. Similarly, we believe that this new approximation of the system (1) in this paper is a good approximation when compared with the Euler approximation.



---

*Source: 791048-2014-04-15.xml*