1 Introduction

Recently, stochastic fractional differential equations and stochastic fractional partial differential equations have attracted more and more attention. It turns out that differential equations involving derivatives of non-integer orders have memory properties, which are called non-local properties. Because of the non-local property of the Caputo fractional derivatives in time, Caputo fractional differential equations are important to model and describe problems in many disciplines, such as engineering, physics, and chemistry. For more details, see [1–7].

Compared with the work on deterministic fractional differential equations, the study of stochastic fractional differential equations is still in its infancy. However, the majority of work is concerned about the existence and uniqueness of solutions; see [8–12]. Until quite recently, there were some authors who considered some types of Caputo fractional stochastic differential equations and Caputo fractional stochastic partial differential equations by different approaching. For example, in Ref. [13], the authors considered the existence of stable manifolds for a type of stochastic differential equations. The authors of paper [14] considered the averaging principle of a type of stochastic fractional differential under some conditions consistent with the stochastic differential equations. In [15], the existence of global forward attracting set for stochastic lattice systems with a Caputo fractional time derivative in the weak mean-square topology is established. In [16], the asymptotic distance between two distinct solutions is considered under a temporally weighted norm. Its worth mentioning that the Euler–Maruyama type approximate results for Caputo fractional stochastic differential equations have been established by [17]. For more related work, see [12, 18–22].

The Caratheodory approximation scheme was first considered by Caraheodory for ordinary differential equations, then Bell, Mohammad and Mao extended it to the stochastic differential equations case; see [23]. To the best of our knowledge, there is no work paying attention to the Caratheodory approximation for the Caputo fractional stochastic differential equation. In this paper, we will consider the Caratheodory approximation for the following type of Caputo fractional stochastic differential equation:

$$ \textstyle\begin{cases} D_{t}^{\alpha }X_{t}=f(t,X_{t})\,dt+g(t, X_{t})\,dB_{t},& t\geq 0, \\ X_{0}=x_{0} \in L^{2}(\Omega ,H), \end{cases} $$
(1.1)

where \(\alpha \in (\frac{1}{2}, 1)\). For more details see Sect. 2. The aim of this paper is to extend the Caratheodory approximate results for Eq. (1.1).

This article is organized as follows. In Sect. 2 we will give some assumptions and basic results that we need. The existence and uniqueness of solution will be discussed in Sect. 3. In the last section, we will consider the Caratheodory approximation for the Caputo fractional stochastic differential equations.

Throughout this paper, the letter C will denote positive constants whose value may change in different occasions. We will write the dependence of a constant on parameters explicitly if it is essential.

2 Preliminaries

We impose the following assumptions to guarantee the existence and uniqueness of solution, H denote a Hilbert space, its norm is denoted by \(|\cdot |\).

\(\mathbf{H1}\): Lipschitz condition: Let \(t\geq 0\) and constant \(k>0\), such that, for all \(x,y\in H\),

$$\begin{aligned} \bigl\vert f(t,x)-f(t,y) \bigr\vert ^{2}+ \bigl\vert g(t,x)-g(t,y) \bigr\vert ^{2}\leq k \vert x-y \vert ^{2}. \end{aligned}$$

\(\mathbf{H2}\): Growth condition: Let \(t\geq 0\) and constant \(k>0\), such that, for all \(x\in H\),

$$\begin{aligned} \bigl\vert f(t,x) \bigr\vert ^{2}+ \bigl\vert g(t,x) \bigr\vert ^{2}\leq k\bigl(1+ \vert x \vert ^{2}\bigr). \end{aligned}$$

The following generalization of Gronwall’s lemma for singular kernels is needed for us to establish our results; see [15, 24].

Lemma 2.1

Suppose \(b\geq 0\), \(\beta >0\) and \(a(t)\) is a nonnegative function locally integrable on \(0\leq t< T\) (some \(T\leq +\infty \)), and suppose \(u(t)\) is nonnegative and locally integrable on \(0\leq t< T\) with

$$ u(t)\leq a(t)+b \int _{0}^{t}(t-s)^{\beta -1}u(s)\,ds. $$

Then

$$ u(t)\leq a(t)+ \int _{0}^{t} \Biggl[\sum _{n=1}^{\infty } \frac{(b\Gamma (\beta ))^{n}}{\Gamma (n\beta )}(t-s)^{n\beta -1}a(s) \Biggr]\,ds, \quad 0\leq t < T, $$

where \(\Gamma (\cdot )\) is the Gamma function.

3 Well-posedness

In this section, we consider the existence and uniqueness of solution for the following equation under conditions \(\mathbf{H1}\) and \(\mathbf{H2}\):

$$ \textstyle\begin{cases} D_{t}^{\alpha }X_{t}=f(t,X_{t})\,dt+g(t, X_{t})\,dB_{t},& t\geq 0, \frac{1}{2}< \alpha < 1, \\ X_{0}=x_{0} \in L^{2}(\Omega ,H), \end{cases} $$
(3.1)

where \(B_{t}\) is a scalar Brownian motion, f and g are H-value functions.

Definition 3.1

An H-value \(\mathcal{F}_{t}\)-adapted stochastic process \(X_{t}\), \(t\in [0, T]\), is called a solution of the initial value problem (3.1), if \(X_{t}\in C([0,T];L^{2}(\Omega , H))\) and satisfies the following integral equation:

$$\begin{aligned} X_{t} =&x_{0}+\frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1}f(s,X_{s}) \,ds \\ &{}+\frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1}g(s,X_{s}) \,dB_{s}. \end{aligned}$$
(3.2)

The existence and uniqueness of solutions for Eq. (3.1) have been considered by our previous work [25]. Similar problem also considered by [16] under different framework. To make this paper self-contained, we just give the main part of the proof for the following theorem.

Theorem 3.1

([25])

Under conditions \(\mathbf{H1}\) and \(\mathbf{H2}\), for every \(x_{0}\in L^{2}(\Omega ,H)\), Eq. (3.1) has a unique mild solution \(X_{t}\in C([0,T];L^{2}(\Omega , H))\).

Proof

We prove the theorem by the contraction mapping principle. Using conditions \(\mathbf{H1}\) and \(\mathbf{H2}\), Lemma 2.1, we can derive that \(X_{t}\in C([0,T];L^{2}(\Omega , H))\).

Let

$$ S=\bigl\{ X_{t}|X_{t}\in C\bigl([0,T];L^{2}( \Omega , H)\bigr)\bigr\} $$

equipped with the norm

$$ \bigl\vert f(t) \bigr\vert _{\varsigma }=\sup_{0\leq t\leq T} E \bigl\vert f(t) \bigr\vert ^{2} $$

be the Banach space of all \(\mathcal{F}_{t}\)-adapted processes.

For any \(t\in [0,T]\) and \(X_{t}\in S\), define a mapping as follows:

$$ (\Phi X) (t)=x_{0}+\frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{ \alpha -1}f(s,X_{s}) \,ds+\frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{ \alpha -1}g(s,X_{s}) \,dB_{s}. $$

It is easy to verify that

$$ \Phi (\cdot ):C\bigl([0,T],L^{2}(\Omega ;H)\bigr)\rightarrow C \bigl([0,T],L^{2}( \Omega ;H)\bigr). $$

Let \(X_{t},Y_{t}\in S\), then

$$\begin{aligned} E \bigl\vert (\Phi X) (t)-(\Phi Y) (t) \bigr\vert ^{2} \leq & 2E \biggl\vert \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1} \bigl[f(s,X_{s})-f(s,Y_{s})\bigr]\,ds \biggr\vert ^{2} \\ &{}+ 2 E \biggl\vert \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1} \bigl[g(s,X_{s})-g(s,Y_{s})\bigr]\,dB_{s} \biggr\vert ^{2}. \end{aligned}$$

Denote \(\beta =2\alpha -1>0\), by the Cauchy–Schwartz inequality, Itô’s isometry formula and condition \(\mathbf{H1}\), we have

$$ E \bigl\vert (\Phi X) (t)-(\Phi Y) (t) \bigr\vert ^{2}\leq \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{\beta -1}E \vert X_{s}-Y_{s} \vert ^{2}\,ds. $$

Using mathematical induction methods, we can deduce the following fact:

$$ E \bigl\vert \bigl(\Phi ^{n} X\bigr) (t)-\bigl( \Phi ^{n} Y\bigr) (t) \bigr\vert ^{2}\leq \frac{1}{\beta }\biggl( \frac{2k(T+1)}{\Gamma (\alpha )^{2}}\biggr)^{n} \frac{\Gamma (\beta )^{n}}{\Gamma (n\beta )} t^{n\beta } \vert X_{t}-Y_{t} \vert _{ \varsigma }. $$
(3.3)

For \(n=1\), by simple calculation we get

$$ E \bigl\vert (\Phi X) (t)-(\Phi Y) (t) \bigr\vert ^{2}\leq \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \vert X_{t}-Y_{t} \vert _{\varsigma } \frac{t^{\beta }}{\beta }, $$

which satisfies Eq. (3.3) with \(n=1\).

Now, assuming that Eq. (3.3) is satisfied for \(n=j\), we claim that it is also correct for \(n=j+1\). We have

$$\begin{aligned} &E \bigl\vert \bigl(\Phi ^{j+1} X\bigr) (t)- \bigl(\Phi ^{j+1} Y\bigr) (t) \bigr\vert ^{2} \\ &\quad \leq \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{ \beta -1}E \bigl\vert \bigl(\Phi ^{j} X\bigr) (s)-\bigl(\Phi ^{j} Y \bigr) (s) \bigr\vert ^{2}\,ds \\ &\quad \leq \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{ \beta -1} \frac{1}{\beta }\biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}}\biggr)^{j} \frac{\Gamma (\beta )^{j}}{\Gamma (j\beta )} s^{j\beta } \vert X_{s}-Y_{s} \vert _{ \varsigma }\,ds \\ &\quad \leq \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}}\biggr)^{j+1} \frac{1}{\beta } \frac{\Gamma (\beta )^{j}}{\Gamma (j\beta )} \vert X_{t}-Y_{t} \vert _{\varsigma } \int _{0}^{t} (t-s)^{\beta -1}s^{j\beta } \,ds. \end{aligned}$$
(3.4)

To get the estimate for \(n=j+1\), we only need to consider the following integral:

$$ \int _{0}^{t} (t-s)^{\beta -1}s^{j\beta } \,ds. $$

Take \(s=tz\), then

$$\begin{aligned} \int _{0}^{t}(t-s)^{\beta -1}s^{j\beta } \,ds =& \int _{0}^{1}(1-z)^{ \beta -1}t^{\beta -1}t^{j\beta }z^{j\beta }t\,dz \\ =&t^{(j+1)\beta } \int _{0}^{1}(1-z)^{\beta -1}z^{j\beta }\,dz \\ =&t^{(j+1)\beta }B(j\beta +1,\beta ) \\ =&t^{(j+1)\beta } \frac{\Gamma (\beta )\Gamma (j\beta +1)}{\Gamma ((j+1)\beta +1)}, \end{aligned}$$

where \(B(\cdot ,\cdot )\) is the Beta function. Combining this result with Eq. (3.4) we have

$$\begin{aligned} &E \bigl\vert \bigl(\Phi ^{j+1} X\bigr) (t)- \bigl(\Phi ^{j+1} Y\bigr) (t) \bigr\vert ^{2} \\ &\quad \leq \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{j+1} \frac{1}{\beta }\frac{\Gamma (\beta )^{j}}{\Gamma (j\beta )} \vert X_{t}-Y_{t} \vert _{ \varsigma } t^{(j+1)\beta } \frac{\Gamma (\beta )\Gamma (j\beta +1)}{\Gamma ((j+1)\beta +1)} \\ &\quad = \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{j+1} \frac{1}{\beta } \Gamma (\beta )^{j+1} \frac{\Gamma (j\beta +1)}{\Gamma ((j+1)\beta +1)\Gamma (j\beta )}t^{(j+1) \beta } \vert X_{t}-Y_{t} \vert _{\varsigma } \\ &\quad = \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{j+1} \frac{1}{\beta } \Gamma (\beta )^{j+1} \frac{j\beta \Gamma (j\beta )}{(j+1)\beta \Gamma ((j+1)\beta )\Gamma (j\beta )}t^{(j+1) \beta } \vert X_{t}-Y_{t} \vert _{\varsigma } \\ &\quad \leq \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{j+1} \frac{1}{\beta }\Gamma (\beta )^{j+1} \frac{t^{(k+1)\beta }}{\Gamma ((j+1)\beta )} \vert X_{t}-Y_{t} \vert _{\varsigma }. \end{aligned}$$
(3.5)

Then we arrive at the following estimate for all n:

$$\begin{aligned} \bigl\vert \bigl(\Phi ^{n} X\bigr) (t)-\bigl(\Phi ^{n} Y\bigr) (t) \bigr\vert _{\varsigma }\leq \biggl( \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{n}\frac{1}{\beta }\Gamma ( \beta )^{n}\frac{T^{n\beta }}{\Gamma (n\beta )} \vert X_{t}-Y_{t} \vert _{ \varsigma }. \end{aligned}$$
(3.6)

If we can prove

$$\begin{aligned} \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{n} \frac{1}{\beta } \Gamma (\beta )^{n}\frac{T^{n\beta }}{\Gamma (n\beta )}< 1, \end{aligned}$$
(3.7)

for sufficient large n, then the theorem holds.

Consider the following series of positive terms:

$$ \sum_{n=1}^{\infty } \biggl( \frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{n} \frac{1}{\beta }\Gamma (\beta )^{n}\frac{T^{n\beta }}{\Gamma (n\beta )}. $$

We will show that

$$ \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)^{n}\frac{1}{\beta } \Gamma (\beta )^{n}\frac{T^{n\beta }}{\Gamma (n\beta )}\rightarrow 0, $$

as \(n\rightarrow +\infty \), which guarantees that Eq. (3.7) holds. Thanks to the d’Alembert discriminant method, we only need to justify

$$ \lim_{n\rightarrow \infty } \frac{(\frac{2k(T+1)}{\Gamma (\alpha )^{2}})\Gamma (\beta )T^{\beta }\Gamma (n\beta )}{\Gamma ((n+1)\beta ))}< 1. $$

Use the relationship of Gamma function and the Stirling formula, represented as follows:

$$ \Gamma (x)\sim \sqrt{2\pi }e^{-x}x^{x-\frac{1}{2}}, x\rightarrow \infty . $$

Then

$$\begin{aligned} &\lim_{n\rightarrow \infty } \frac{(\frac{2k(T+1)}{\Gamma (\alpha )^{2}})\Gamma (\beta )T^{\beta }\Gamma (n\beta )}{\Gamma ((n+1)\beta ))} \\ &\quad =\lim_{n\rightarrow \infty } \biggl(\frac{2k(T+1)}{\Gamma (\alpha )^{2}} \biggr)\Gamma (\beta )T^{\beta }e^{\beta } \sqrt{\frac{n+1}{n}} \biggl( \frac{n}{n+1} \biggr)^{n\beta } \frac{1}{(n\beta +\beta )^{\beta }}= 0, \end{aligned}$$

which shows that \(\Phi (\cdot )\) is a contraction mapping on \(C([0,T],L^{2}(\Omega ;H))\) for all \(T<\infty \). This completes the proof. □

4 Caratheodory’s approximate solutions

In this section, we consider the Caratheodory approximation for stochastic fractional differential equations. Similar to the stochastic differential equations approach, we try to give the definition of Caratheodory’s approximate solutions for stochastic fractional differential equations as follows.

For every integer \(n\geq 1\), define \(x_{n}(t)=x_{0}\) for \(-1\leq t\leq 0\) and

$$\begin{aligned} x_{n}(t) =&x_{0}+\frac{1}{\Gamma {(\alpha )}} \int _{0}^{t}(t-s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr)\,ds \\ &{}+ \frac{1}{\Gamma {(\alpha )}} \int _{0}^{t}(t-s)^{ \alpha -1}g \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr) \,dB_{s} \end{aligned}$$

for \(0< t\leq T\).

Note that, for \(0\leq t\leq \frac{1}{n}\), \(x_{n}(t)\) can be computed by

$$ x_{n}(t)=x_{0}+\frac{1}{\Gamma {(\alpha )}} \int _{0}^{t}(t-s)^{\alpha -1}f(s,x_{0}) \,ds +\frac{1}{\Gamma {(\alpha )}} \int _{0}^{t}(t-s)^{\alpha -1}g(s,x_{0}) \,dB_{s}, $$

then, for \(\frac{1}{n}< t\leq \frac{2}{n}\),

$$\begin{aligned} x_{n}(t) =&x_{n}\biggl(\frac{1}{n}\biggr)+ \frac{1}{\Gamma {(\alpha )}} \int _{ \frac{1}{n}}^{t}(t-s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \\ &{}+ \frac{1}{\Gamma {(\alpha )}} \int _{\frac{1}{n}}^{t}(t-s)^{\alpha -1}g \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr) \,dB_{s} \end{aligned}$$

and so on. By this approach, we can compute \(x_{n} (t)\) step by step on the intervals \([0,\frac{1}{n}], (\frac{1}{n}, \frac{2}{n}], \ldots \) .

Lemma 4.1

Under the condition \(\mathbf{H2}\), for all \(n\leq 1\), we have

$$ \sup_{0\leq t\leq T}E \bigl\vert x_{n}(t) \bigr\vert ^{2}\leq \Omega =: r_{1} \bigl(1+E_{2 \alpha -1,1} \bigl(r_{2}\Gamma (2\alpha -1)T^{2\alpha -1}\bigr) \bigr)< \infty , $$

where \(r_{1}=3E|x_{0}|^{2}+3 \frac{(kT^{(2\alpha -1)})(T+1)}{\Gamma {(\alpha )}^{2}(2\alpha -1)}\), \(r_{2}=3\frac{k(T+1)}{\Gamma {(\alpha )}^{2}}\) and \(E_{2\alpha -1, 1}(\cdot )\) is a two-parameter function of the Mittag-Leffler type (see [15]).

Proof

From the simple arithmetic inequality

$$ \vert a+b+c \vert ^{2}\leq 3\bigl( \vert a \vert ^{2}+ \vert b \vert ^{2}+ \vert c \vert ^{2}\bigr), $$

we have

$$\begin{aligned} E \bigl\vert x_{n}(t) \bigr\vert ^{2} \leq & 3E \vert x_{0} \vert ^{2}+3E \biggl\vert \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \biggr\vert ^{2} \\ &{}+3E \biggl\vert \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t-s)^{\alpha -1}g \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr) \,dB_{s} \biggr\vert ^{2} \\ :=&3I_{1}+3I_{2}+3I_{3}. \end{aligned}$$

By the Cauchy–Schwarz inequality and condition \(\mathbf{H2}\), we can estimate the term \(I_{2}\) as follows:

$$\begin{aligned} I_{2} \leq & \frac{Tk}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2( \alpha -1)}\biggl(1+E \biggl\vert x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\biggr)\,ds \\ \leq & \frac{Tk}{\Gamma (\alpha )^{2}} \biggl[ \frac{t^{2\alpha -1}}{2\alpha -1}+ \int _{0}^{t} (t-s)^{2(\alpha -1)}E \biggl\vert x_{n}\biggl(s- \frac{1}{n}\biggr) \biggr\vert ^{2}\,ds \biggr] \\ \leq &\frac{kT^{2\alpha }}{\Gamma (\alpha )^{2} (2\alpha -1)}+ \frac{Tk}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2(\alpha -1)} \sup _{0\leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds. \end{aligned}$$

Similarly, with Itô’s isometry formula and condition \(\mathbf{H2}\), we have an estimate for the stochastic integral term:

$$\begin{aligned} I_{3} \leq & \frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2( \alpha -1)}\biggl(1+E \biggl\vert x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\biggr)\,ds \\ \leq & \frac{kT^{2\alpha -1}}{\Gamma (\alpha )^{2} (2\alpha -1)}+ \frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2(\alpha -1)}\sup _{0 \leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds. \end{aligned}$$

Combining the estimate for \(I_{1}\), \(I_{2}\), \(I_{3}\), we arrive at

$$\begin{aligned} E \bigl\vert x_{n}(t) \bigr\vert ^{2}\leq r_{1}+r_{2} \int _{0}^{t} (t-s)^{(2\alpha -1)-1} \sup _{0\leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds, \end{aligned}$$
(4.1)

where we denote

$$ r_{1}=3E \vert x_{0} \vert ^{2}+3 \frac{(kT^{2\alpha -1})(T+1)}{\Gamma (\alpha )^{2} (2\alpha -1)} $$

and

$$ r_{2}=3\frac{k(T+1)}{\Gamma (\alpha )^{2}}. $$

Note that, for \(t_{1}\leq t_{2}\), we have

$$ \int _{0}^{t_{1}} (t_{1}-s)^{(2\alpha -1)-1} \sup_{0\leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds \leq \int _{0}^{t_{2}} (t_{2}-s)^{(2\alpha -1)-1} \sup_{0\leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds. $$

Then

$$ \sup_{0\leq r\leq t}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\leq r_{1}+r_{2} \int _{0}^{t} (t-s)^{(2 \alpha -1)-1}\sup _{0\leq r\leq s}E \bigl\vert x_{n}(r) \bigr\vert ^{2}\,ds. $$

Applying Lemma 2.1, we can directly obtain

$$\begin{aligned} \sup_{0\leq r\leq t}E \bigl\vert x_{n}(r) \bigr\vert ^{2} \leq & r_{1} \Biggl(1+ \int _{0}^{t} \sum_{n=1}^{\infty } \frac{(r_{2}\Gamma (2\alpha -1))^{n}}{\Gamma (2n\alpha -n)} (t-s)^{n(2 \alpha -1)-1}\,ds \Biggr) \\ \leq & r_{1} \Biggl(1+\sum_{n=1}^{\infty } \frac{(r_{2}\Gamma (2\alpha -1)T^{2\alpha -1})^{n}}{\Gamma (2n\alpha -n+1)} \Biggr) \\ =& r_{1} \bigl(1+E_{2\alpha -1,1}\bigl(r_{2}\Gamma (2\alpha -1)T^{2\alpha -1}\bigr) \bigr) < \infty , \end{aligned}$$

for all \(t\in [0,T]\), where \(E_{2\alpha -1,1}(\cdot )\) is a two-parameter function of the Mittag-Leffler type (see [15]). □

Lemma 4.2

Under the condition \(\mathbf{H2}\), for all \(n\geq 1\) and \(0\leq t_{0}< t\leq T\) with \(t-t_{0}\leq 1\), then

$$\begin{aligned} E \bigl\vert x_{n}(t)-x_{n}(t_{0}) \bigr\vert ^{2}\leq C(t-t_{0})^{2\alpha -1}. \end{aligned}$$

Proof

Taking \(0\leq t_{0}< t\leq T\), we note that

$$\begin{aligned}& E \bigl\vert x_{n}(t)-x_{n}(t_{0}) \bigr\vert ^{2} \\& \quad \leq 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1}f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds- \int _{0}^{t_{0}} (t_{0} -s)^{ \alpha -1}f\biggl(s,x_{n}\biggl(s-\frac{1}{n} \biggr)\biggr)\,ds \biggr\vert ^{2} \\& \qquad {}+ 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1} g \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr) \,dB_{s}\\& \qquad {}- \int _{0}^{t_{0}} (t_{0} -s)^{ \alpha -1}g\biggl(s,x_{n}\biggl(s-\frac{1}{n} \biggr)\biggr)\,dB_{s} \biggr\vert ^{2} \\& \quad =: 2(J_{1}+J_{2}). \end{aligned}$$

For \(J_{1}\), we have

$$\begin{aligned} J_{1} \leq & 2 E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{t_{0}}^{t} (t-s)^{ \alpha -1}f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \biggr\vert ^{2} \\ &{}+2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t_{0}} \bigl[(t-s)^{ \alpha -1}-(t_{0}-s)^{\alpha -1} \bigr]f\biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \biggr\vert ^{2} \\ =:& 2J_{11}+2J_{12}. \end{aligned}$$

Using the Cauchy–Schwartz inequality, \(t-t_{0}\leq 1\), we give an estimate for \(J_{11}\) as follows:

$$\begin{aligned} J_{11} \leq & \frac{1}{\Gamma (\alpha )^{2}} \int _{t_{0}}^{t} (t-s)^{2 \alpha -2}\,ds \int _{t_{0}}^{t} E \biggl\vert f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr) \biggr\vert ^{2}\,ds \\ \leq &\frac{k}{\Gamma (\alpha )^{2}(2\alpha -1)}(t-t_{0})^{2\alpha -1} \int _{t_{0}}^{t} \biggl[1+E \biggl\vert x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\biggr]\,ds \\ \leq & \frac{(\Omega +1) k}{\Gamma (\alpha )^{2}(2\alpha -1)}(t-t_{0})^{2 \alpha -1}, \end{aligned}$$

where

$$ \Omega =r_{1} \bigl(1+E_{2\alpha -1,1}\bigl(r_{2} \Gamma (2\alpha -1)T^{2 \alpha -1}\bigr) \bigr) $$

has been defined in Lemma 4.1.

For \(J_{12}\), we have the following result:

$$\begin{aligned} J_{12} =&E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t_{0}} \bigl[(t-s)^{ \alpha -1}-(t_{0}-s)^{\alpha -1} \bigr]f\biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \biggr\vert ^{2} \\ \leq & \frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t_{0}} \bigl[(t-s)^{ \alpha -1}-(t_{0}-s)^{\alpha -1} \bigr]^{2}\,ds \int _{0}^{t_{0}} \biggl[1+E \biggl\vert x_{n}\biggl(s- \frac{1}{n}\biggr) \biggr\vert ^{2}\biggr]\,ds \\ \leq & \frac{CTk}{\Gamma (\alpha )^{2}} \int _{0}^{t_{0}} \bigl[(t_{0}-s)^{2 \alpha -2}-(t-s)^{2\alpha -2} \bigr]\,ds \\ =& \frac{CTk}{\Gamma (\alpha )^{2}} \biggl[ \frac{(t-t_{0})^{2\alpha -1}}{2\alpha -1}+ \frac{t_{0}^{2\alpha -1}}{2\alpha -1}- \frac{t^{2\alpha -1}}{2\alpha -1} \biggr] \\ \leq & \frac{CTk}{\Gamma (\alpha )^{2}} \frac{(t-t_{0})^{2\alpha -1}}{2\alpha -1}. \end{aligned}$$

For \(J_{2}\), taking the Itô isometry formula and condition \(\mathbf{H2}\) into account, using similar estimate methods to \(J_{1}\), it can be shown that

$$\begin{aligned} J_{2}\leq \frac{Ck}{\Gamma (\alpha )^{2}} \frac{(t-t_{0})^{2\alpha -1}}{2\alpha -1}. \end{aligned}$$

Combining all the deduced estimates, we have

$$ E \bigl\vert x_{n}(t)-x_{n}(t_{0}) \bigr\vert ^{2}\leq C(t-t_{0})^{2\alpha -1}. $$

This completes the proof. □

Theorem 4.1

Under the conditions \(\mathbf{H1}\) and \(\mathbf{H2}\), let \(x(t)\) be the unique solution of equations (3.1). Then for \(n\geq 1\)

$$ \sup_{0\leq t\leq T}E \bigl\vert x(t)-x_{n}(t) \bigr\vert ^{2}\leq \frac{C}{n^{2\alpha -1}} $$
(4.2)

Proof

Note that

$$\begin{aligned} x(t)-x_{n}(t) =& \frac{1}{\Gamma (\alpha )} \biggl[ \int _{0}^{t} (t-s)^{ \alpha -1}f \bigl(s,x(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr)\,ds\biggr] \\ &{}+ \frac{1}{\Gamma (\alpha )}\biggl[ \int _{0}^{t} (t-s)^{\alpha -1} g \bigl(s,x(s)\bigr)\,dB_{s}- \int _{0}^{t} (t -s)^{\alpha -1}g \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr) \,dB_{s}\biggr]. \end{aligned}$$

Hence, employing a simple arithmetic inequality, we have

$$\begin{aligned}& E \bigl\vert x(t)-x_{n}(t) \bigr\vert ^{2} \\& \quad \leq 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1}f \bigl(s,x(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr)\,ds \biggr\vert ^{2} \\& \qquad {}+ 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1} g \bigl(s,x(s)\bigr)\,dB_{s}- \int _{0}^{t} (t -s)^{\alpha -1}g \biggl(s,x_{n}\biggl(s- \frac{1}{n}\biggr)\biggr) \,dB_{s} \biggr\vert ^{2} \\& \quad =: 2(I_{1}+I_{2}). \end{aligned}$$

For \(I_{1}\), we have

$$\begin{aligned} I_{1} \leq & 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1}f \bigl(s,x(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}f \bigl(s,x_{n}(s)\bigr)\,ds \biggr\vert ^{2} \\ &{}+ 2E\frac{1}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{\alpha -1}f \bigl(s,x_{n}(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}f \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr)\,ds \vert ^{2} \\ =:& 2(I_{11}+I_{12}). \end{aligned}$$

Using the Cauchy–Schwartz inequality and the condition \(\mathbf{H1}\), we have the following estimate for \(I_{11}\):

$$\begin{aligned} I_{11}\leq \frac{kT}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2 \alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2}\,ds. \end{aligned}$$

Similarly, for \(I_{12}\), we have

$$\begin{aligned} I_{12}\leq \frac{kT^{2\alpha -1}}{(2\alpha -1)\Gamma (\alpha )^{2}} \int _{0}^{t} E \biggl\vert x_{n}(s)-x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\,ds. \end{aligned}$$

Also, we can divide \(I_{2}\) into two parts as follows:

$$\begin{aligned} I_{2} \leq & 2E\frac{1}{\Gamma (\alpha )^{2}} \biggl\vert \int _{0}^{t} (t-s)^{ \alpha -1}g \bigl(s,x(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}g \bigl(s,x_{n}(s)\bigr)\,dB_{s} \biggr\vert ^{2} \\ &{}+ 2E\frac{1}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{\alpha -1}g \bigl(s,x_{n}(s)\bigr)\,ds- \int _{0}^{t} (t -s)^{\alpha -1}g \biggl(s,x_{n}\biggl(s-\frac{1}{n}\biggr)\biggr) \,dB_{s} \vert ^{2} \\ =:& 2(I_{21}+I_{21}). \end{aligned}$$

By the Itô isometry formula, we get

$$ I_{21}\leq \frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2 \alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2}\,ds $$

and

$$ I_{22}\leq \frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2 \alpha -2}E \biggl\vert x_{n}(s)-x_{n}\biggl(s-\frac{1}{n} \biggr) \biggr\vert ^{2}\,ds . $$

Combining with the estimate for \(I_{1}\) and \(I_{2}\), it is derived that

$$\begin{aligned} E \bigl\vert x(t)-x_{n}(t) \bigr\vert ^{2} \leq & \frac{k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2\alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2}\,ds \\ &{}+\frac{kT^{2\alpha -1}}{(2\alpha -1)\Gamma (\alpha )^{2}} \int _{0}^{t} E \biggl\vert x_{n}(s)-x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\,ds \\ &{}+\frac{k}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2\alpha -2}E \biggl\vert x_{n}(s)-x_{n}\biggl(s- \frac{1}{n} \biggr) \biggr\vert ^{2}\,ds, \end{aligned}$$
(4.3)

by Lemma 4.2, if \(s\geq \frac{1}{n}\), then

$$ E \biggl\vert x_{n}(s)-x_{n}\biggl(s- \frac{1}{n}\biggr) \biggr\vert ^{2}\leq \frac{C}{n^{2\alpha -1}}, $$

otherwise if \(0\leq s<\frac{1}{n}\),

$$ E \biggl\vert x_{n}(s)-x_{n}\biggl(s- \frac{1}{n}\biggr) \biggr\vert ^{2}=E \bigl\vert x_{n}(s)-x_{n}(0) \bigr\vert ^{2}\leq Cs^{2 \alpha -1}\leq \frac{C}{n^{2\alpha -1}}. $$

Following Eq. (4.3), we have

$$\begin{aligned} E \bigl\vert x(t)-x_{n}(t) \bigr\vert ^{2} \leq & \frac{k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2\alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2}\,ds \\ &{}+\frac{kT^{2\alpha -1}}{(2\alpha -1)\Gamma (\alpha )^{2}} \int _{0}^{t} E \biggl\vert x_{n}(s)-x_{n}\biggl(s-\frac{1}{n}\biggr) \biggr\vert ^{2}\,ds \\ &{}+\frac{T}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2\alpha -2}E \biggl\vert x_{n}(s)-x_{n}\biggl(s- \frac{1}{n} \biggr) \biggr\vert ^{2}\,ds \\ \leq & \frac{k(T+1)}{\Gamma (\alpha )^{2}} \int _{0}^{t} (t-s)^{2 \alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2}\,ds \\ &{}+\frac{kT^{2\alpha -1}(1+T)}{(2\alpha -1)\Gamma (\alpha )^{2}} \frac{1}{n^{2\alpha -1}} \\ =:&q_{1} \int _{0}^{t} (t-s)^{2\alpha -2}E \bigl\vert x(s)-x_{n}(s) \bigr\vert ^{2} \,ds+q_{2}. \end{aligned}$$

Applying Lemma 2.1, we obtain

$$ E \bigl\vert x(t)-x_{n}(t) \bigr\vert ^{2}\leq q_{2} \bigl(1+E_{2\alpha -1,1}\bigl(q_{1}\Gamma (2 \alpha -1)T^{2\alpha -1}\bigr) \bigr)=:\frac{C}{n^{2\alpha -1}}. $$

This completes the proof. □

Remark 4.1

When \(\alpha =1\), i.e. Eq. (1.1) becomes a stochastic differential equation, the convergent rate of the scheme in Theorem 4.1 coincides with the well-known convergent rate of the classical Caratheodory results; see [23].