The Truncated EM Method for Stochastic Differential Equations Driven
by Fractional Brownian Motion
Abstract: -We mainly focus on the numerical method of fractional Brownian motion in this paper. On the basis
of the numerical method of general SDEs, an approximation scheme is obtained for the stochastic differential
equations about fractional noise. And we get it by using the Lipschitz condition and combining with the truncation
function f∆and g∆. Furthermore, we also prove the moment boundedness and convergence of the solution by
some lemma. At last, we apply this method to the Gilpin-Ayala model. The orbital image of the solution and the
form of numerical solution are given. The error of solution also has been simulated by MATLAB.
Key-Words: -fractional Brownian motion, truncated EM method method, numerical method, convergence,
moment boundedness, Gilpin-Ayala model
Received: March 29, 2023. Revised: October 28, 2023. Accepted: November 29, 2023. Published: February 22, 2024.
1 Introduction
In recent years, SDEs driven by fractional Brow-
nian motion have attracted more attention and widely
applied in many fields. FBM has some better prop-
erties than general ones such as self-similarity and
long-term memory, which can describe the random
phenomena. Therefore, many scholars pay attention
to it. We know that fractional Brownian motion does
not meet the conditions of a semi-martingale, so the
usual Itˆoformula is not suitable, in [1], gave the Itˆo
formula for fractions, Itˆorepresentation formula and
Girsanov theorem. When the fractional noise instead
of a general one, the classical theory of random inte-
gration is no longer applicable. Therefore, in [2], [3],
the author comprehensively introduced the definition
of FBM random integral, gave some theoretical ap-
plications, and focused on the relationship between
different research approaches. For the Hearst index,
the concept is not clear in some literature, and in [4]
it is explained that fractional Brownian motion differs
from the Gauss-Markov process(H=1
2) in that the
increment of one is stationary and correlated, while
the other is nonstationary and uncorrelated. More-
over, in [5], the uniqueness and existence of the solu-
tion to the neutral pulse random delay equation driven
by FBM has been given.
The problem of the numerical solution has also at-
tracted the attention of many scholars (see, [6], [7],
[8], [9], [10]), but the numerical methods of the frac-
tional Brownian motion are relatively few. In [11],
the author derived some approximation schemes of
the scalar SDEs, and get the exact rate of conver-
gence of it. He showed that the error of the Euler
method converges to a random variable a.s. The EM
method of backward SDEs had been discussed in [12].
It is focus on stochastic Markovian neural networks
with jump. The general mean-square stability of it
has been obtained and there are sufficient conditions
which guarantee the stability of the method. In addi-
tion, an accurate calculation scheme for solving FBM-
driven stochastic differential equations is proposed
in [13], and this discretization method is based on the
quadratic interpolation technique, its error and con-
vergence are analyzed for better application. A class
of stochastic fractional integro-differential equations
has been concerned by [14]. This type of equation has
weakly singular kernels. The author proposed a mod-
ified Euler−Maruyama (EM) method and then anal-
ysed the strong convergence of it. In [15], numerical
schemes for multi-dimensional fBms with Hurst pa-
rameter movtivating stochastic differential equations
are investigated. The author provide the order con-
ditions of Runge-Kutta method to achieve the opti-
mal rate of convergence, which based on the continu-
ous dependency of numerical solutions with the driv-
ing noises and introducing an Runge-Kutta methods.
Finally by applying this method, simpler step-Euler
schemes with a strong convergence rate are devel-
oped, and the rate is confirmed by numerical exper-
iment. Sometimes the drift term will be special and it
satisfies locally Lipschitz but not bounded in neigh-
borhood of the origin, therefore the author developed
an implicit Euler schemen which can maintain posi-
tivity in [16], , then obtained rate of convergence.
In this paper, we give a new truncated EM method
for nonliner SDEs which is explicit. Suppose the
equation is of the form
dxt=f(xt)dt +g(xt)dBH
t.(1)
The coefficients meets local Lipschitz condition but
SUXIN WANG, LE IYANG
College of Science,
Civil Aviation University of China,
Tianjin 300300,
CHINA
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unfortunately, they don’t grow linearly.
This paper is organized as follows. In the first sec-
tion, the research background of the numerical solu-
tion and fractional Brownian motion are introduced.
Section 2 provides some definitions and theorems re-
quired for the next proof. Section 3 obtains the spe-
cific form of the truncated EM method and proves the
convergence of the numerical solution. And in Sec-
tion 4, the error of the Gilpin Ayala model between
truncation EM method and actual solution be simu-
lated. The last section summarizes the paper.
2 Preliminary
This section we will give some definitions and the-
orems needed in this paper, which are important for
the proof. This paper only pay attention to the situa-
tion of 1
2< H < 1.
Defintion 2.1. ( [17]) Let BH
tis a continuous Gaus-
sian process, His a Hurst index, 0< H < 1. If BH
t
satisfies the following three conditions:
1.BH
0= 0.
2.E(BH
t+∆t−BH
t) = 0, for any t > 0and ∆t > 0
.
3.For different tand s, their covariance function
is
E[BH
uBH
v] = 1
2(|u|2H+|v|2H−|u−v|2H), t, s ≥0,
then BH
t, t ≥0is named fractional Brownian motion.
From the above definition, we can see the three
facts:
1.When H=1
2,BH
tis standard Brownian motion.
2.We can know that it has stationary increments, that
is E(BH
t−BH
s)2=|t−s|2H.
3.FBM has the incremental autocorrelation. If H <
1
2, there is a negative correlation between the incre-
ments of FBM; if H > 1
2, there is a positive correla-
tion between the increments of FBM.
We have learned that BH
t, t ≥0 (H=1
2)is not
a semi-martingale, so the properties of Brownian mo-
tion are no longer valid. But we can establish the re-
lationship between them.
We have
BH
t=t
0
KH(t, s)dBs
where KH(t, s)is a square integrable kernel,
KH(t, s) = cHsH−1
2t
s
(w−s)H−3
2wH−1
2dw,
cH=H(2H−1)
β(2 −2H, H −1
2)1/2
.
For more details, refer to [18]. Suppose 1
2< H < 1,
we denote ϕ:R×R→R
ϕ(u, v) = H(2H−1)|u−v|2H−2, s, t ∈R.
Then L2
ϕis a Hilbert space, the inner product is de-
noted by
< f, g >ϕ=∞
0∞
0
f(s)g(t)ϕ(s, t)dsdt.
Then f∈L2
ϕ(R+)if
||f||2
ϕ:= ∞
0∞
0
f(u)f(v)ϕ(u, v)dudv < ∞.
Defintion 2.2. ( [19]) For a random variable F∈
Lp. We defined
DΦgG(α) = lim
ε→0
1
εG(α+ε·
0
(Φg)(u)du)−G(α)
as the ϕ-derivative in the orientation of Φg. if the limit
exists in Lp.
Moreover, Fis said to be ϕ-differentiable if there
has a process(DϕFs, s ≥0) makes
DΦgF=∞
0
DϕFsgsds a.s.
for all g∈L2
ϕ. If f:R→Ris smooth and F:
Ω→Ris ϕ-differentiable, we can say that f(F)is
ϕ-differentiable, then we have
DΦgf(F) = f′(F)DΦgF
and
Dϕ
sf(F) = f′(F)Dϕ
sF.
The rules are as follows:
Dϕ
s∞
0
fudBH
u=∞
0
ϕ(u, v)fudu = (Φf)(s);
Dϕ
sδ(f) = δ(f)∞
0
ϕ(u, s)fudu =δ(f)(Φf)(s).
Theorem 2.1. ( [19]) Fractional Itô formula(H>1
2)
Let L(0, T )be a family of stochastic process on [0, T ].
If E|F|2
ϕ<∞, then F∈ L(0, T )and Fis ϕ-
differentiable. Ft, Gtare process that satisfy the fol-
lowing assumptions:
1.There is an β > 1−Hsuch that
E|Fa−Fb|2≤C|a−b|2β
where |a−b| ≤ ε,ε > 0.
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2.
lim
0≤a,b≤t,|a−b|→0
E|Dϕ
a(Fa−Fb)|2= 0.
3.ET
0|FtDϕ
tµt|ds < ∞,Esup0≤t≤T|Gt|<
∞. Denote µt=ζ+t
0Gudu +t
0FudBH
u, ζ ∈
Rfor t∈[0, T ]. Let f:R+×R→Rbe
a function having the first continuous derivative in
its first variable as well as its second one. Let
∂f
∂x (s, µs)Fs, s ∈[0, T ]∈ L(0, T ). For t∈
[0, T ], we have
f(t, µt) = f(0, ζ) + t
0
∂f
∂s (s, µs)ds
+t
0
∂f
∂x (s, µs)Gsds +t
0
∂f
∂x (s, µs)FsdBH
s
+t
0
∂2f
∂x2(s, µs)FsDϕ
sµsds
(2)
Theorem 2.2. ( [20])Gronwall’s inequality
For T > 0and c≥0. Suppose u(·)is function
which is Borel measurable bounded and nonnegative
on [0, T ], and v(·)is nonnegation integrable on [0, T ].
We have
u(t)≤cexp t
0
v(s)ds, t ∈[0, T ],
if
u(t)≤c+t
0
v(s)u(s)ds.
3 The truncated EM method about
FBM and convergence
3.1 Description of the method
We will give the form of truncated EM method in
this subsection.
First, let’s make two assumptions for preparation.
Assumption 3.1. There is the local Lipschitz condi-
tion for coefficients: If L > 0, then
|f(x1)−f(x2)|∨|g(x1)−g(x2)| ≤ KL|x1−x2|,(3)
is hold. For x, y ∈R,|x| ∨ |y| ≤ L, where KL>0
is a constant and |·|is the Euclidean norm.
Assumption 3.2. The coefficients of equations sat-
isfy the inequality below
xTf(x)+(m−1)g(x)Dϕ
sx(t)≤Q(1 + |x|2).(4)
where m > 2and Q > 0are constants.
Now consider a SDE BH
t.
dxt=f(xt)dt +g(xt)dBH
t(5)
where t≥0, and x(0) = x0. The condition
xTf(x) + |g(x)|Dϕ
sx(t)≤Q(1 + |x|2),(6)
can guarantee the global solution. Following lemma
proves the existence and uniqueness of it.
Lemma 3.1. Suppose assumption 3.1 and Assump-
tion 3.2 are satisfied.
(i) The SDE (5) has an unique global solution x(t).
(ii)
sup
0≤t≤T
E|xt|m<∞,∀T > 0.(7)
Proof. First, since Assumption 3.1 hold. We
know that coefficients satisfy the local Lipschitz con-
dition, the equation has an unique local solution on
t∈[0, µ∞],µ∞is an explosion time (see, [21], The-
orem 3.1). We only need to proof that µ∞=∞a.s.
τlis a stopping time for l≥1,
τl=µ∞∧inf{t∈[0, µ∞] : |xt| ≥ l},
where inf ∅=∞. Clearly, τl’s are increasing so
τ∞=limk→∞ τland τ∞≤µ∞a.s. By the Theo-
rem 2.1 and the condition (6), we can claim that
E|xt∧τl|2=|x0|2
+Et∧τl
0
2|xs|f(x)ds +Et∧τl
0
2|xs|g(x)dBH
s
+Et∧τl
0
2g(x)Dϕ
sxsds
≤|x0|2+ 2Et∧τl
0
Q(1 + |xs|2)ds
≤C+ 2Qt + 2Qt
0
E|xt∧τl|2ds.
According to Theorem 2.2,
E|xt∧τl|2≤(C+ 2Qt)e2Qt.
Where Cis a constant. Define ρ:R+→R+by
ρ(r) = inf
|x|≥r,0≤t≤∞ |xt|2, for r ≥0.
Apparently,
lim
|x|→∞ inf
0≤t≤∞ |xt|2=∞.(8)
We can see ρ(|xt|)≤ |xt|2from the definition of ρ.
And lim0≤t<∞|xt|2=∞combine with condition
(8), then
lim
r→∞ ρ(r) = ∞.
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Based on the above analyses, we can obtain
Eρ(|xt∧τl|)≤E|xt∧τl|2≤(C+ 2Qt)e2Qt,
it follows that
(C+ 2Qt)e2Qt ≥Eρ(|xt∧τl|)≥ρ(l)P(τl≤t).
Setting ltends to infinity and when t→ ∞, there is
P(τ∞<∞) = 0.
Immediately, there is τ∞=∞a.s. Therefore, µ∞=
∞a.s.
Next, the certificate of (ii) is resemble to the theo-
rem 5.1.1 in [22].
In order to get this method, we select a continuous
η:R+→R+which is strictly increasing and when
r→ ∞, we have η(r)→ ∞. In addition, it’s also
satisfied
sup|x|≤r(|f|∨|g|)≤η(r),∀r≥0.(9)
Then η−1is also has the similar properties from
[η(0),∞)to R+.
Select a ∆∗∈(0,1]. There is a function k:
(0,∆∗]→(0,∞)which strictly decreasing and it
meets
k(∆∗)≥η(2),lim
∆→0k(∆) = ∞,
∆H/2k(∆) ≤1,∀∆∈(0,1).
(10)
For any ∆∈(0,1), we define
f∆(x) = f(|x| ∧ η−1(k(∆))) x
|x|,
g∆(x) = g(|x| ∧ η−1(k(∆))) x
|x|
(11)
for x∈R. They called truncated functions. When
x= 0, we have x
|x|= 0. By (9), we get
|f∆|∨|g∆| ≤ η(η−1(k(∆))) = k(∆), x ∈R.(12)
It means that, even while fand gmight not be
bounded, both f∆and g∆are.
Moreover, as stated in the following lemma, we
will demonstrate f∆and g∆also keep the condition
(4) for ∆∈(0,∆∗].
Lemma 3.2. We suppose that Assumption 3.2 is ture.
For any ∆∈(0,∆∗], we gain that
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)≤2Q(1 + |x|2),
∀x∈Rd.
(13)
Proof. Due to kis a strictly decreasing function,
from (10), we deduced that
η−1(k(∆)) ≥η−1(k(∆∗)) ≥2,∀∆∈(0,∆∗].
(14)
Choose a ∆∈(0,∆∗]. For x∈Rd,
(i) If |x| ≤ η−1(k(∆)), by (4), there have
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)
=xTf(x)+(m−1)g(x)Dϕ
sx(s)≤2Q(1 + |x|2)
so the (13) hold.
(ii) If |x|> η−1(k(∆)), we can use the (4),
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)
=xTfη−1(k(∆)) x
|x|
+(m−1)gη−1(k(∆)) x
|x|Dϕ
sx(s).
Insert an intermediate term to construct the following
form,
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)
=η−1(k(∆))xT
|x|fη−1(k(∆)) x
|x|
+(m−1)gη−1(k(∆)) x
|x|Dϕ
sx(s)
+|x|
η−1(k(∆)) −1η−1(k(∆))xT
|x|fη−1(k(∆)) x
|x|
≤Q(1 + [η−1(k(∆))]2)
+|x|
η−1(k(∆)) −1η−1(k(∆))xT
|x|fη−1(k(∆)) x
|x|.
In other hand, we note (4) xTf(x)≤Q(1 + |x|2)by
(4) for any x∈R.
For convenience, let η−1(k(∆)) = M. Then the
above equation becomes
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)
≤Q(1 + M2) + |x|
M−1Q(1 + M2)
=Q(1 + M2)|x|
M
≤Q|x|M+1
M.
By (14), we know M≥2, therefore we can obtain
xTf∆(x)+(m−1)g∆(x)Dϕ
sx(s)
≤Q|x|(1
2+η−1(k(∆)))
≤Q(1 + |x|)2≤2Q(1 + |x|2).
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The proof is completed.
Through the above theoretical preparation, the
form of the truncated EM X∆(tl)≈x(tl)can now
developed. By taking X∆(0) = x0,tl=l∆, we can
get that
X∆(tl+1) = X∆(tl) + f∆(X∆(tl))∆
+g∆(X∆(tl))∆BH
l
(15)
where l= 0,1, ..., where ∆BH
l=BH
tl+1 −BH
tland
in this form, time is discrete.
The truncated EM solutions with continuous time
defined as
ˆx∆(t) =
∞
l=0
X∆(tl)I[tl,tl+1)(t), t ≥0.(16)
and
x∆(t) = x0+t
0
f∆(ˆx∆(s))ds+t
0
g∆(ˆx∆(s))dBH
s
(17)
for t≥0.
By the definition above, x∆(tl) = ˆx∆(tl) =
X∆(tl)can be seen for all l≥0. Furthermore, x∆(t)
has the form of Itô differential
dx∆(t) = x0+f∆(ˆx∆(t))dt +g∆(ˆx∆(t))dBH
t.
(18)
Following, we will demonstrate the convergence.
3.2 The moment bound of solution
We wii show the numerical solutions will converge
in LP.Via (12),
sup
0≤t≤T
E|x∆(t)|m<∞,∀T > 0.
can be seen easily. Nevertheless, obtaining the fol-
lowing inequality is difficult,
sup
0≤t≤∆∗
sup
0≤t≤T
E|x∆(t)|m<∞,∀T > 0.(19)
In this subsection, we will establish this. First, We
will illustrates that x∆(t)close to ˆx∆(t).
Lemma 3.3. Given a ∆∈(0,∆∗], m ≥2, then
E|x∆(t)−ˆx∆(t)|m≤cm,∆,H ,∀t≥0.(20)
where cm,∆,H >0is a constant which dependent on
m,∆and H. Thus,
lim
∆→0
E|x∆(t)−ˆx∆(t)|m= 0,∀t≥0.(21)
Proof. The generic positive real constants cm,∆,H
which are only dependent on m,∆,Hand whose val-
ues might differ between occurrences, will be used in
what follows. For t≥0, Fix a ∆∈(0,∆∗]. there is a
l≥0enables tl≤t≤tl+1. By (12) and the fraction
Itô integra (see,e.g., [23]). Afterward, we infer from
(17),
E|x∆(t)−ˆx∆(t)|m=E|x∆(t)−x∆(tl)|m
≤cmE|t
tl
f∆(ˆx∆(s))ds|m+E|t
tl
g∆(ˆx∆(s))dBH
s|m
≤cm∆m−1Et
tl
|f∆(ˆx∆(s))|mds +E|t
tl
g∆(ˆx∆(s))dBH
s|m
≤cm∆m(k(∆))m+E|t
tl
g∆(ˆx∆(s))dBH
s|m.
We also have (see, e.g., [18])
E|t
tl
g∆(ˆx∆(s))dBH
s|m
≤c(H, m)||g∆(ˆx∆(s))||m
L1/H(t,tl)
=c(H, m)tl
t
|g∆(ˆx∆(s))|1
HdtmH
≤c(H, m)(k(∆))mtl
t
1dtmH
≤c(H, m)(k(∆))m∆mH .
Therefore,
E|x∆(t)−ˆx∆(t)|m≤cm,∆,H .
(20) and (21) are proofed immediately.
Next, we give proof of (19), which is also an im-
portant part.
Lemma 3.4. If Assumptions 3.1 and 3.2 are ture. The
inequality (19) is hold. Here, Cmight vary between
occurrences, and represents generic positive real con-
stants going forward, which are they are dependent on
T,m,Q,x0but independent of ∆.
Proof. For any ∆∈(0,∆∗)and T≥0. From
(17), we can infer using the Theorem 2.1, for 0≤t≤
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T.
E|x∆(t)|m≤ |x0|m+E|t
0
m|x∆(t)|m−1f∆(ˆx∆(s))|ds
+E|t
0
m(m−1)|x∆(t)|m−2g∆(ˆx∆(s))Dϕ
sx∆(s)|ds
=|x0|m+E|t
0
m|x∆(s)|m−2(ˆx∆(s)f∆(ˆx∆(s))) |ds
+E|t
0
m|x∆(s)|m−2(m−1)g∆(ˆx∆(s))Dϕ
sx∆(s)|ds
+E|s
0
m|x∆(t)|m−2(x∆(s)−ˆx∆(s))Tf∆(ˆx∆(s))|ds.
we can determined
E|x∆(t)|m≤ |x0|m
+Et
0
Qm|x∆(s)|m−21 + |ˆx∆(s)|2ds
+Et
0
m|x∆(s)|m−2(x∆(s)−ˆx∆(s))f∆(ˆx∆(s))ds
≤|x0|m+Et
0
Qm|x∆(s)|m−21 + |ˆx∆(s)|2ds
+(m−2)Et
0
|x∆(s)|mds
+2Et
0
|x∆(s)−ˆx∆(s)|m
2|f∆(ˆx∆(s))|m
2ds
≤P1+P2t
0
(E|x∆(s)|m+E|ˆx∆(s)|m)ds
+2Et
0
|x∆(s)−ˆx∆(s)|m
2|f∆(ˆx∆(s))|m
2ds
which based on the Young inequality and Lemma 3.2
and
am−2b≤m−2
mam+2
mbm
2,∀a, b ≥0,
Here, P1and P2can able to change along the progress
of this proof. Lemma 3.3 and inequalities (12) and
(10) provide us
Et
0
|x∆(s)−ˆx∆(s)|m
2|f∆(ˆx∆(s))|m
2ds
≤(k(∆))m
2T
0
E(|x∆(s)−ˆx∆(s)|m
2)ds
≤cm,∆,H T.
(22)
then
E|x∆(t)|m≤P1+P2t
0
(E|x∆(s)|m+E|ˆx∆(s)|m)ds
≤P1+P2t
0sup
0≤r≤s
E|x∆(r)|mds.
If the right side is not decrease with t, the above for-
mula holds for all t∈[0, T ].
We are able to notice
sup
0≤r≤t
E|x∆(r)|m≤P1+P2t
0sup
0≤r≤s
E|x∆(r)|mds.
Theorem 2.2 bring that
sup
0≤r≤t
E|x∆(r)|m≤P.
Here ∆∈(0,∆∗]and P is independent of ∆, then
(19) can eb detect.
3.3 Strong convergence
Lemma 3.5. We set Assumptions 3.1 and 3.2 are ture.
Let Z > |x0|is a number with real value. We have
the following conclusion:
1.Define
θ=inf{t≥0 : |x(t)| ≥ Z},
where inf ∅=∞.θis a stopping time. We obtained
that
P(θZ≤T)≤C
Z2.
2.For any ∆∈(0,∆∗), Defined ν∆,Z as
ν∆,Z =inf{t≥0 : |x∆(t)| ≥ Z}.
then
P(ν∆,Z ≤T)≤C
Z2.
3.For any n∈(2, m],
lim
∆→0
E|x∆(T)−x(T)|n= 0,lim
∆→0
E|¯x∆(T)−x(T)|n= 0.
The proof of these lemmas is similar to the [24]
(section 3.2), so we will not go into much detail here.
4 Simulation
Consider the Gilpin-Ayala model driven by frac-
tion Brownian Motion,
dNt=Nt1−Nt
Kθ(rdt +βdBH
t).(23)
Let f(Nt) = rNt1−Nt
Kθand g(Nt) =
Nt1−Nt
Kθ. We can prove that there is a unique
continuous solution Nt,0< Nt< K. Obvi-
ously, the Assumption 3.1 and 3.2 are satisfied. Set
θ= 1 and choose η(s) = s2, futhermore, (4) is
hold. We also can select a strictly decreasing func-
tion k(∆) = ∆−1
2.
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There is a truncated EM numerical solution N∆(t)
of Eq.(23). The form is
N(tk+1)
∆=N(tk)
∆+f∆(N(tk)
∆)∆ + g∆(N(tk)
∆)∆BH
k,
where the truncated functions are f∆(Nt) =
f(Nt), g∆(Nt) = g(Nt).
The image of the orbit for this equation is shown
as Figure 1.
Fig.1: Sample orbit for Equation (23)
And by Lemma 3.5, we can claim that N∆(t)is
strongly convergent to N(t).
We simulate the convergence rate of equation (23).
And we set N0= 1,T= 1,H= 0.6,θ= 1,β=1
2,
r= 1, and simulate 1000 sample trajectories.
Next we will focus on the error at the endpoint
t=T, and compute the average error δ=|N∆(T)−
N(T)|.N(T)represents real solution of equation
(23). The result is shown in the Figure 2.
Fig.2: The average error δ
In this graph, the solid blue line connected by as-
terisks represents the approximation to δagainst ∆t
on log-log scale and that’s implies the numerical so-
lution of the equation (23) is convergent. And as ∆t
decreases, δ=|N∆(T)−N(T)|also decreases ac-
cordingly.
5 Conclusion
This paper extend the method in [24] to SDEs
driven by fractional noise. Regarding the research in
this paper, we can draw the following conclusions:
1.This method will be applied to the nonlinear
stochastic differential equations of fractional Brow-
nian motion without linear growth condition.
2.The moment boundedness of the solution is
guaranteed by using the stopping time and its strong
convergence is proved.
3.We use the Gilpin-Ayala equation as an exam-
ple to simulate the convergence rate of the numerical
solution, and verify the error of this method.
In future works, this numerical method can be ap-
plied to some stochastic models of fractional Brow-
nian motion. But it was limited to satisfying the lo-
cal Lipschitz condition. On the basis of this research,
we can continue to study some numerical methods of
equations without Lipschitz condition. In addition,
there will be some inspiration for the study of some
other noise-driven stochastic differential equations.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Suxin Wang, instructed and checked the rea-
sonableness and correctness of the article.
Lei Yang is responsible for the derivation of calcu-
lations, simulation design and the writing of articles.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Supported by the Fundamental Research Funds for
the Central Universities(Grant No. 3122023032),
Ministry of Education of the People’s Re-
public of China Humanities and Sciences
Youth Foundation(Grant No. 22YJC630055).
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Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International , CC BY 4.0)
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