Stochastic differential equations with piecewise constant arguments (SEPCA) can be regarded as generalizations of both stochastic delay differential equations (SDDEs) and differential equations with piecewise constant arguments (EPCA). The general form of SEPCA of retarded type is

$ dX(t)=f(t,X(t),X([t-p]))dt+g(t,X(t),X([t-p]))dW(t), t\geq 0 $

with the initial function $X(-1)=X_{-1}$ and $X(0)=X_0$. Where $[\cdot]$ signifies the greatest integer function, $p\in \mathbb{R}^+$. One-dimensional standard Wiener process $W(t)$ satisfies $\mathbb{E}(W(t))=0, \mathbb{E}(W(t)W(s))=\min\{t,s\}$ and the initial value $X_{-1},X_0$ are random variables. In addition, we assume that $f(t,0,0)=0$ and $g(t,0,0)=0$.

EPCA describe hybrid dynamical systems and combine properties of both differential and difference equations. They are appeared in modeling of various problems in real life such as biology, mechanics and electronics. Several important properties of the analytical solution of EPCA as well as numerical methods have been studied by many authors [1-4]. For more details of EPCA, the reader can see Wiener's book [5].

In many scientific and applied areas, including finance, ecology, computational biology and population dynamics, SDDEs are often used to model the corresponding systems. In recent years, there has been increasingly interesting in studying such equations, and their numerical treatments have also received considerable attention. Tudor and Tudor [6] and Tudor [7] first studied numerical solutions of SDDEs. Cao [8] derived some stability properties of Euler-Maruyama method for linear SDDEs. Buckwar and Shardlow [9] considered weak approximation methods for SDDEs. Rathinasamy and Balachandran [10] analyzed the mean-square stability of the semi-implicit Euler method for linear stochastic differential equations with multiple delays and Markovian switching. Hu and Huang [11] studied mean-square stability of stochastic $\theta$-methods for stochastic delay integro-differential equations. Xiao et al. [12] analyzed convergence and stability of semi-implicit Euler methods for a linear stochastic pantograph equations. Recently, Wang and Chen [13] have given results of mean-square stability of semi-implicit Euler methods for nonlinear neutral SDDEs. We note that most of the above numerical stability results are focused on the mean-square stability, few results have been found in the references that involve T (Trajectory)-stability of numerical method for SDDEs. The definition of T-stability of numerical schemes for stochastic differential equations was introduced by Saito and Mitsui [14]. Burrage et al. [15] extended this concept from weak approximation to strong approximation. Burrage and Tian [16] discussed the T-stability of the composite Euler method for stochastic ordinary differential equations (SODEs). Cao [17] studied the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. For linear stochastic delay integro-differential equations, Rathinasamy and Balachandran [18] considered the T-stability of the split-step $\theta$-methods. Motivated by the work of Cao [17] and Rathinasamy and Balachandran [18], the present paper will focus on both the mean-square stability and the T-stability of the Euler-Maruyama method for SEPCA.

In this paper, we consider the following scalar SEPCA of retarded type

$\begin{equation}\label{e:101} \left\{ \begin{aligned} & dX(t)=(a_1X(t)+a_2X([t-1]))dt+(a_3X(t)+a_4X([t-1]))dW(t), t\geq 0,\\ & X(-1)=X_{-1},X(0)=X_0, \end{aligned}\right. \end{equation}$

(1.1)

where $a_1,a_2,a_3,a_4 \in \mathbb{R}$. The major objective of this paper is to illustrate that the Euler-Maruyama method applied to (1.1) is both mean-square stable and T-stable under the condition which guarantees the stability of the analytical solution.

The structure of this paper is organized as follows. In Section 2 we will introduce some necessary notations and hypotheses of (1.1) and discuss the stability properties of its analytical solution. In Section 3, the Euler-Maruyama method will be used to produce the numerical solutions. Moreover, our main results will be shown and proved in this section. Conclusion is provided in Section 4.

Let $(\Omega,\mathcal{F},P)$ be a complete probability space with a filtration $\mathcal{F}_{t\geq 0}$ satisfying the usual conditions. The Wiener process $W(t)$ in (1.1) be $\mathcal{F}_t$-adapted and independent of $\mathcal{F}_0$, $L^1([0,\infty),\mathbb{R})$ and $L^2([0,\infty),\mathbb{R})$ denote the family of all real valued measurable $\mathcal{F}_t$-adapted process $f(t)_{t\geq 0}$ such that for every $T>0, \displaystyle\int_0^T|f(t)|dt<\infty$ w.p.1 and $\displaystyle\int_0^T|f(t)|^2dt<\infty$ w.p.1, respectively. Moreover, assume that the initial value $X_{-1}$ and $X_0$ are $\mathcal{F}_0$-measurable and $\mathbb{E}(X_0)^2<\infty$.

Definition 2.1 A stochastic process $X(t)$ is called a solution of (1.1) on $[0, \infty)$ if it has the following properties:

(ⅰ) $X(t)$ is continuous and $\mathcal{F}_{t\geq 0}$-adapted;

(ⅱ) $f(t,X(t),X([t-1]))_{t\geq 0}\in L^1([0,\infty),\mathbb{R})$ and $g(t,X(t),X([t-1]))_{t\geq 0}\in L^2([0,\infty),\mathbb{R})$;

(ⅲ) (1.1) is satisfied on every interval $[n, n+1)\subset [0, \infty)$ with integral end-points almost surely.

Definition 2.2 If any solution $X(t)$ of (1.1) satisfies

$ \lim\limits_{t \rightarrow \infty}\mathbb{E}(X(t))^2=0, $

then the zero solution of the (1.1) is called mean-square stable.

Lemma 2.3 [19] The solution $x=0$ of equation

$\begin{equation}\label{e:102} x'(t)=ax(t)+a_0x([t])+a_1x([t-1]) \end{equation}$

(2.1)

is asymptotically stable (the solution $x(t)\to 0$ as $t\to \infty$) if and only if the inequalities

$\begin{equation}\label{e:103} \left\{ \begin{aligned} &|a_1|<\frac{a}{e^a-1},\\ &a_1-\frac{a(e^a+1)}{e^a-1}<a_0<-a-a_1 \end{aligned}\right. \end{equation}$

(2.2)

hold.

From Lemma 2.3, we can easily obtain the following result.

Corollary 2.4 The solution $x=0$ of equation

$ x'(t)=ax(t)+a_1x([t-1]) $

is asymptotically stable if and only if the inequality

$ -\frac{a}{e^a-1}<a_1<-a $

holds.

According to Corollary 2.4 and Lemma 2.1 in [20], we have the following Result.

Theorem 2.5 If $A_2 \geq 0$ and $A_1+A_2<0$, then any continuous and positive solution $x(t)$ of system

$\begin{equation}\label{e:104} \left\{ \begin{aligned} &x'(t)\leq A_1x(t)+A_2x([t-1]),\\ &x(-1)=x_{-1},x(0)=x_0 \end{aligned}\right. \end{equation}$

(2.3)

is asymptotically stable.

Hence, the result on the stability of analytical solution is obtained.

Theorem 2.6 If condition

$\begin{equation}\label{e:105} 2a_1+a_3^2+a_4^2+2|a_2+a_3a_4|<0 \end{equation}$

(2.4)

holds, then the solution of (1.1) is mean-square stable.

Proof By Itô formula, we have

$\begin{equation}\label{e:106} \begin{aligned} dX^2(t) =&(2X(t)(a_1X(t)+a_2X([t-1])+(a_3X(t)+a_4X([t-1]))^2)dt\\ & +2X(t)(a_3X(t)+a_4X([t-1]))dW(t)\\ \leq& ((2a_1+a_3^2+|a_2+a_3a_4|)X^2(t)+(a_4^2+|a_2+a_3a_4|)X^2([t-1]))dt\\ & +2X(t)(a_3X(t)+a_4X([t-1]))dW(t), \end{aligned} \end{equation}$

(2.5)

let

$ Y(t)=\mathbb{E}(X^2(t)), $

then

$ dY(t) \leq \{(2a_1+a_3^2+|a_2+a_3a_4|)Y(t)+(a_4^2+|a_2+a_3a_4|)Y([t-1])\}dt, $

that is

$ Y'(t)\leq (2a_1+a_3^2+|a_2+a_3a_4|)Y(t)+(a_4^2+|a_2+a_3a_4|)Y([t-1]). $

By using Theorem 2.5 and condition (2.4), we have

$ \lim\limits_{t\rightarrow \infty}\mathbb{E}(X(t))^2=\lim\limits_{t\rightarrow \infty}Y(t)=0, $

which completes the proof.

Let $h=1/m$ be a given stepsize with integer $m \geq 1$, and the gridpoints $t_n$ be defined by $t_n=nh$. Let $n=km+l$ $(l=0,1,\cdots,m-1),$ applying the Euler-Maruyama method to (1.1), we have

$\begin{equation}\label{e:107} X_{n+1}=X_n+(a_1X_n+a_2X_{(k-1)m})h+(a_3X_n+a_4X_{(k-1)m})\triangle W_n, \end{equation}$

(3.1)

where $X_n=X(t_n)$, the increments $\triangle W_n:=W(t_{n+1})-W(t_n)$ are independent $N(0,h)-$distributed Gaussian random variables. We assume $X_n$ to be $\mathcal{F}_{t_n}$-measurable at the mesh-points $t_n$. For the convergence of the Euler-Maruyama method, we refer the interested reader to [21-23]. In the next two subsections, we will focus on its stability property.

3.1 Mean-square Stability 

Definition 3.1 Under the condition (2.4), a numerical method applied to (1.1) is said to be mean-square stable, if there exists a $h_0(a_1,a_2,a_3,a_4)>0$, such that the numerical solution sequence ${X_n}$ produced by this numerical scheme satisfies

$\begin{equation}\label{e:108} \lim\limits_{n \rightarrow \infty}\mathbb{E}(X_n)^2=0 \end{equation}$

(3.2)

for every stepsize $h \in (0,h_0(a_1,a_2,a_3,a_4))$ with $h=1/m$ for an integer $m$.

Let

$ $\begin{aligned} &P=(1+a_1h)^2+a_3^2h+|(1+a_1h)a_2h+a_3a_4h|,\\ &Q=a_2^2h^2+a_4^2h+|(1+a_1h)a_2h+a_3a_4h|,\\ &h_0(a_1,a_2,a_3,a_4)=-\frac{a_3^2+a_4^2+2a_1+2|a_2+a_3a_4|}{(|a_1|+|a_2|)^2}, \end{aligned}$ $

we have the following Lemma.

Lemma 3.2 If condition (2.4) is satisfied, then inequality $P+Q<1$ holds for any $h \in (0,h_0(a_1,a_2,a_3,a_4))$.

Proof It is obvious that $h_0(a_1,a_2,a_3,a_4)>0$ from condition (2.4), and

$\begin{equation}\label{e:109} \begin{aligned} P+Q &=(1+a_1h)^2+a_3^2h+a_2^2h^2+a_4^2h+2|(1+a_1h)a_2h+a_3a_4h|\\ &\leq (a_1^2+a_2^2)h^2+(a_3^2+2a_1+a_4^2)h+1+2h|a_2+a_3a_4|+2h^2|a_1a_2|\\ &=(|a_1|+|a_2|)^2h^2+(a_3^2+a_4^2+2a_1+2|a_2+a_3a_4|)h+1, \end{aligned} \end{equation}$

(3.3)

therefore $P+Q<1$ if and only if

$ 0<h<-\frac{a_3^2+a_4^2+2a_1+2|a_2+a_3a_4|}{(|a_1|+|a_2|)^2}=h_0(a_1,a_2,a_3,a_4), $

the proof is completed.

Then the first main Theorem of this paper is obtained.

Theorem 3.3 Assume the condition (2.4) holds, then the Euler-Maruyama method applied to (1.1) is mean-square stable with $h \in (0,h_0(a_1,a_2,a_3,a_4)))$.

Proof It follows from (3.1) that

$\begin{equation}\label{e:110} X_{n+1}=(1+a_1h+a_3 \triangle W_n)X_n+(a_2h+a_4 \triangle W_n)X_{(k-1)m}. \end{equation}$

(3.4)

Squaring both sides of the above equality, yields

$\begin{equation}\label{e:111} \begin{aligned} X_{n+1}^2= &(1+a_1h+a_3 \triangle W_n)^2 X_n^2+(a_2h+a_4 \triangle W_n)^2X_{(k-1)m}^2\\ &+2(1+a_1h+a_3 \triangle W_n)(a_2h+a_4 \triangle W_n)X_nX_{(k-1)m}. \end{aligned} \end{equation}$

(3.5)

Using the elementary inequality $2xyab \leq |xy|(a^2+b^2)$ we have

$\begin{equation}\label{e:112} \begin{aligned} X_{n+1}^2 \leq &(1+a_1h+a_3 \triangle W_n)^2 X_n^2+(a_2h+a_4 \triangle W_n)^2X_{(k-1)m}^2\\ &+|(1+a_1h+a_3 \triangle W_n)(a_2h+a_4 \triangle W_n)|(X_n^2+X_{(k-1)m}^2), \end{aligned} \end{equation}$

(3.6)

that is

$\begin{equation}\label{e:113} \begin{aligned} X_{n+1}^2 \leq &\{(1+a_1h+a_3 \triangle W_n)^2+|(1+a_1h+a_3 \triangle W_n)(a_2h+a_4 \triangle W_n)|\} X_n^2\\ &+\{(a_2h+a_4 \triangle W_n)^2+|(1+a_1h+a_3 \triangle W_n)(a_2h+a_4 \triangle W_n)|\}X_{(k-1)m}^2. \end{aligned} \end{equation}$

(3.7)

Denote

$ Y_n=\mathbb{E}(X_n^2). $

Note that $\mathbb{E}(\triangle W_n)=0$ and $\mathbb{E}(\triangle W_n)^2=h $, the inequality (3.7) reduces to

$\begin{equation}\label{e:114} \begin{aligned} Y_{n+1} \leq &[(1+a_1h)^2+a_3^2h+ |(1+a_1h)a_2h+a_3a_4h|]Y_n\\ &+[a_2^2h^2+a_4^2h+|(1+a_1h)a_2h+a_3a_4h|]Y_{(k-1)m}, \end{aligned} \end{equation}$

(3.8)

which is equivalent to

$\begin{equation}\label{e:115} Y_{n+1} \leq PY_n+QY_{(k-1)m}. \end{equation}$

(3.9)

Namely

$\begin{equation}\label{e:116} Y_{n+1}\leq (P+Q)\max\left\{Y_n, Y_{(k-1)m} \right\}. \end{equation}$

(3.10)

By virtue of Lemma 3.2, the iteration of inequality (3.10) implies

$ \lim\limits_{n\rightarrow \infty}Y_n=\lim\limits_{n\rightarrow \infty}\mathbb{E}(X_n)^2=0. $

It is shown that the method is mean-square stable.

3.2 T-stability 

The mean-square stability may still cause difficulty from the point of view of computer implementation. In order to learn more about the nature of numerical scheme, we often need to evaluate the value of the expectation $\mathbb{E}|X_n|^2$ where $X_n$ is an approximating sequence of the solution sample path. In a certain probability, $X_n$ may happen to overflow in computer simulations. This actually interferes with the evaluation of $X_n$. To overcome this difficulty, T-stability is introduced with respect to the approximate sequence of sample path (trajectory) by Saito and Mitsui [14].

Definition 3.4 [15] Under the condition (2.4), the numerical scheme equipped with a specified driving process is said to be T-stable if

$\begin{equation}\label{e:117} \lim\limits_{n \rightarrow \infty}|X_n|=0 \end{equation}$

(3.11)

holding for the driving process.

The so called specified driving process is to approximate $\triangle W_n$ by random variable with specified distribution. In this paper, we treat the Euler-Maruyama scheme with two-point random variables. The wiener increment $\triangle W_n$ is taken as $U_n \sqrt{h}$ whose probability distribution is given by $P(U_n=\pm 1)=1/2$, where $P$ indicates probability.

By formula (3.4) and $\triangle W_n=U_n \sqrt{h}$, we have

$\begin{equation}\label{e:118} \begin{aligned} |X_{n+1}| &\leq \left(\left|1+a_1h+a_3 \triangle W_n\right|+\left|a_2h+a_4 \triangle W_n\right|\right)\max\{|X_n|,|X_{(k-1)m}|\}\\ &=\left(\left|1+a_1h+a_3 \sqrt{h} U_n\right|+\left|a_2h+a_4 \sqrt{h} U_n\right|\right)\max\{|X_n|,|X_{(k-1)m}|\}. \end{aligned} \end{equation}$

(3.12)

Considering the average of $n+1$ steps by recursiving (3.4), we call it average stability function with

$\begin{equation}\label{e:119} \begin{aligned} & R^{(2)}(h;a_1,a_2,a_3,a_4)\\ =&\left(\left|1+a_1h+a_3 \sqrt{h}\right|+\left|a_2h+a_4 \sqrt{h}\right|\right)\left(\left|1+a_1h-a_3 \sqrt{h}\right|+\left|a_2h-a_4 \sqrt{h}\right|\right)\\ =&\left|(1+a_1h)^2-a_3^2h\right|+\left|a_2^2h^2-a_4^2h\right|\\ & +\left|[(1+a_1h)a_2h-a_3a_4h]+[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ & +\left|[(1+a_1h)a_2h-a_3a_4h]-[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|, \end{aligned} \end{equation}$

(3.13)

if $R^{(2)}(h;a_1,a_2,a_3,a_4)<1$, then $\lim\limits_{n \rightarrow \infty}|X_n|=0$, so the numerical method is T-stable. Therefore, the second main theorem of this paper is given as follows.

Theorem 3.5 Assume condition (2.4) is satisfied, denote

$\begin{eqnarray*} && h_1=\min\left\{\frac{\displaystyle 1}{\displaystyle |a_1|},-\frac{\displaystyle 2(a_1+|a_2|)-(|a_3|-|a_4|)^2}{\displaystyle (a_1+|a_2|)^2}\right\},\\ && h_2=\min\left\{\frac{\displaystyle 1}{\displaystyle |a_1|},\frac{\displaystyle -k_0+\sqrt{k_0^2-8(a_1^2-a_2^2)|a_4|}}{\displaystyle 2(a_1^2-a_2^2)}\right\},\\ && h_3=\frac{\displaystyle -k_0-\sqrt{k_0^2-8(a_1^2-a_2^2)|a_4|}}{\displaystyle 2(a_1^2-a_2^2)},\\ && h_4=\min\left\{\frac{\displaystyle 1}{\displaystyle |a_1|},\frac{\displaystyle -k_1-\sqrt{k_1^2-8(a_1-|a_2|)^2}}{\displaystyle 2(a_1-|a_2|)^2}\right\},\\ && h_5=\frac{\displaystyle -k_1+\sqrt{k_1^2-8(a_1-|a_2|)^2}}{\displaystyle 2(a_1-|a_2|)^2},\\ && h_6=\min\left\{\frac{\displaystyle 1}{\displaystyle |a_1|},\frac{\displaystyle -k_2-\sqrt{k_2^2-8(a_1^2-a_2^2)(1-|a_4|)}}{\displaystyle 2(a_1^2-a_2^2)}\right\},\\ && h_7=\frac{\displaystyle -k_2+\sqrt{k_2^2-8(a_1^2-a_2^2)(1-|a_4|)}}{\displaystyle 2(a_1^2-a_2^2)}, \\ && S_1=\left\{h \left| h \geq \frac{a_4^2}{a_2^2} \right. \right\}\\ && \bigcap \left\{h \left| h \geq \frac{a_3^2-2a_1+\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2}, h \leq \frac{a_3^2-2a_1-\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2} \right. \right\}, \\&& S_2=\left\{h \left| h < \frac{a_4^2}{a_2^2} \right. \right\}\\ && \bigcap \left\{h \left| h \geq \frac{a_3^2-2a_1+\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2}, h \leq \frac{a_3^2-2a_1-\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2} \right. \right\}, \\&& S_3= \left\{h \left| h < \frac{a_4^2}{a_2^2} \right. \right\}\\ && \bigcap \left\{h \left| \frac{a_3^2-2a_1-\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2}< h< \frac{a_3^2-2a_1+\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2} \right. \right\}, \\&& S_4=\left\{h \left| h \geq \frac{a_4^2}{a_2^2} \right. \right\}\\ && \bigcap \left\{h \left| \frac{a_3^2-2a_1-\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2}< h< \frac{a_3^2-2a_1+\sqrt{a_3^2-4a_1}|a_3|}{2a_1^2} \right. \right\}, \end{eqnarray*}$

where $k_0=2a_1+a_4^2-a_3^2+2|a_2a_3|+2a_1|a_4|, k_1=2a_1-2|a_2|-(|a_3|+|a_4|)^2, k_2=2a_1-2|a_2a_3|-2a_1|a_4|-a_3^2+a_4^2$, then

(ⅰ) If $h \in S_1$ and satisfies $h<h_1$, the Euler-Maruyama method for (1.1) is T-stable;

(ⅱ) If $h \in S_2$ and satisfies $h_3<h<h_2$, the Euler-Maruyama method for (1.1) is T-stable;

(ⅲ) If $h \in S_3$ and satisfies $h<h_4$ or $h>h_5$, the Euler-Maruyama method for (1.1) is T-stable;

(ⅳ) If $h \in S_4$ and satisfies $h<h_6$ or $h>h_7$, the Euler-Maruyama method for (1.1) is T-stable.

Proof By virtue of (3.13), if the inequality

$\begin{equation}\label{e:120} \begin{aligned} & \left|(1+a_1h)^2-a_3^2h\right|+\left|a_2^2h^2-a_4^2h\right|\\ &+\left|[(1+a_1h)a_2h-a_3a_4h]+[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ &+\left|[(1+a_1h)a_2h-a_3a_4h]-[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right| <1 \end{aligned} \end{equation}$

(3.14)

holds for $h<1/|a_1|$, then $R^{(2)}(h;a_1,a_2,a_3,a_4)<1$ means that $|X_n| \rightarrow 0 $ $(n \rightarrow \infty)$. Hence, the Euler-Maruyama method is T-stable. In addition, we notice that $h_i>0, i=1,2,\cdots,7$ and $k_j<0,j=0,1,2$ under the condition (2.4) and Remark 3.6. The proof will be considered in four cases as follows.

(ⅰ) If $h \in S_1$, then $a_2^2h^2-a_4^2h \geq 0, (1+a_1h)^2-a_3^2h \geq 0$. So we have

$\begin{equation}\label{e:121} \begin{aligned} & \left|[(1+a_1h)a_2h-a_3a_4h]+[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ &+\left|[(1+a_1h)a_2h-a_3a_4h]-[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ =&2\left|(1+a_1h)a_2h-a_3a_4h\right|, \end{aligned} \end{equation}$

(3.15)

therefore, by inequality (3.14), we can obtain

$\begin{equation}\label{e:122} \begin{aligned} & (1+a_1h)^2-a_3^2h+a_2^2h^2-a_4^2h+2|(1+a_1h)a_2h-a_3a_4h|\\ \leq& (1+a_1h)^2-a_3^2h+a_2^2h^2-a_4^2h+2(1+a_1h)|a_2|h+2|a_3a_4|h\\ =&1+[2(a_1+|a_2|)-(|a_3|-|a_4|)^2]h+(a_1+|a_2|)^2h^2 \\ <&1, \end{aligned} \end{equation}$

(3.16)

namely

$ [2(a_1+|a_2|)-(|a_3|-|a_4|)^2]h+(a_1+|a_2|)^2h^2<0. $

It is easy to find that the inequality (3.14) holds when $h<h_1.$

(ⅱ) If $h \in S_2$, then $a_2^2h^2-a_4^2h < 0, (1+a_1h)^2-a_3^2h \geq 0$. So we have

$\begin{equation}\label{e:123} \begin{aligned} & \left|[(1+a_1h)a_2h-a_3a_4h]+[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ &+\left|[(1+a_1h)a_2h-a_3a_4h]-[a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}]\right|\\ =&2\left|a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}\right|. \end{aligned} \end{equation}$

(3.17)

Hence, the inequality (3.14) can be written by

$\begin{equation}\label{e:124} \begin{aligned} & (1+a_1h)^2-a_3^2h+a_4^2h-a_2^2h^2+2\left|a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}\right|\\ \leq& (1+a_1h)^2-a_3^2h+a_4^2h-a_2^2h^2+2|a_2a_3|h\sqrt{h}+2(1+a_1h)|a_4|\sqrt{h}\\ \leq& (1+a_1h)^2-a_3^2h+a_4^2h-a_2^2h^2+2|a_2a_3|h+2(1+a_1h)|a_4|\\ =&1+2|a_4|+[2a_1+a_4^2-a_3^2+2|a_2a_3|+2a_1|a_4|]h+(a_1^2-a_2^2)h^2 \\ <&1, \end{aligned} \end{equation}$

(3.18)

consequently

$ (a_1^2-a_2^2)h^2+k_0h+2|a_4|<0. $

We can easily derive that $a_1^2-a_2^2>0, k_0^2-8(a_1^2-a_2^2)|a_4|>0$. So the inequality (3.14) holds when $h_3<h<h_2.$

(ⅲ) If $h \in S_3$, then $a_2^2h^2-a_4^2h < 0, (1+a_1h)^2-a_3^2h <0$. Hence (3.15) holds, (3.14) can be written as

$\begin{equation}\label{e:125} \begin{aligned} & a_3^2h-(1+a_1h)^2+a_4^2h-a_2^2h^2+2|(1+a_1h)a_2h-a_3a_4h|\\ \leq& a_3^2h-(1+a_1h)^2+a_4^2h-a_2^2h^2+2(1+a_1h)|a_2|h+2|a_3a_4|h\\ =&-1-(a_1-|a_2|)^2h^2+[(|a_3|+|a_4|)^2-2a_1+2|a_2|]h\\ <&1, \end{aligned} \end{equation}$

that is

$ (a_1-|a_2|)^2h^2+k_1h+2>0, $

so inequality (3.14) holds when $h<h_4$ or $h>h_5$.

(ⅳ) If $h \in S_4$, then $a_2^2h^2-a_4^2h \geq 0, (1+a_1h)^2-a_3^2h <0$. Hence (3.17) holds, (3.14) can be written as

$\begin{equation}\label{e:126} \begin{aligned} & a_3^2h-(1+a_1h)^2+a_2^2h^2-a_4^2h+2\left|a_2a_3h\sqrt{h}-(1+a_1h)a_4\sqrt{h}\right|\\ \leq& a_3^2h-(1+a_1h)^2+a_2^2h^2-a_4^2h+2|a_2a_3|h+2(1+a_1h)|a_4|\\ =&-1+(a_2^2-a_1^2)^2h^2+[a_3^2-a_4^2-2a_1+2|a_2a_3|+2a_1|a_4|]h+2|a_4|\\ <&1, \end{aligned} \end{equation}$

that is

$ (a_1^2-a_2^2)^2h^2+k_2h+2-2|a_4|>0, $

so inequality (3.14) holds when $h<h_6$ or $h>h_7$.

Therefore, the proof is completed.

Remark 3.6 In Theorem 3.5, if anyone of the following inequalities holds

$ k_1^2-8(a_1-|a_2|)^2 < 0,k_2^2-8(a_1^2-a_2^2)(1-|a_4|) < 0, $

we can let the corresponding stepsize $h$ equals 1.

In this paper, we discuss the mean-square stability and T-stability of Euler-Maruyama method for linear SEPCA of retarded type. It may be worthwhile to remark that the paper makes a meaningful exploratory for T-stability of numerical method. We believe that this topic will be gained more and more attention by scientists and engineers. T-stability of numerical methods for SEPCA of advanced type will be considered in the further work.

Acknowledgments 

The authors would like to thank the reviewer and Professor Mingzhu Liu for their constructive suggestions and helpful assistance.