Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel

Abstract

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis.

1. Introduction

Mathematical models of natural and physical phenomena have always played a central and critical role in science since the times of Newton and Euler. The fractional derivatives have been crucial in applied mathematics and mathematical sciences since the time of Leibniz. Note that the foundation of fractional derivatives was pioneered by Liouville. These fractional systems are often of evolutionary type, time-dependent, linear or nonlinear cases; within their range of important applications we find aircraft simulation, fractional Newtonian fluids, flow through porous media, weather prediction, image restoration, image processing, and inverse image processing, just to name a few. Among others, these fractional systems can be used to describe behaviors of the most fascinating and challenging issues in the mathematical community. Despite much progress, the field is still open, and it is often quiet challenging to obtain regular and global solutions to the fractional Navier–Stokes and hydrodynamics models of turbulence for just the case of a deterministic setting. For the fractional stochastic counterparts, it is even much more harder in certain cases, for example, the existence of invariant measure in any dimension.

Over the last few decades, stochastic fractional partial differential equations (SFPDEs) have been investigated by several authors, and many results were obtained on its theory and also applied to various real world phenomena. We refer, for example, to [1,2,3,4,5,6]. The analysis of systems with a Caputo derivative of variable order without a singular kernel presents a cutting edge and a mathematical framework for modeling complex systems influenced by both randomness and non-local interactions. Instead of approaches to calculus, these equations make use of the adaptable nature of variable-order Caputo derivatives, which adjust to varying conditions and capture subtle memory effects. By incorporating stochastic terms to accommodate randomness in system dynamics often represented by Wiener or Gaussian processes, SFPDEs with variable-order Caputo derivatives provide a tool for analyzing intricate phenomena across various fields, such as some branches of physics (nematic liquid crystal), mechanics (stochastic electro-rheological fluids), and climate, atmospheric, ocean dynamics (shallow water systems), turbulent flow in gas pipelines, and porous media problems.

The application of the Caputo derivative with variable order goes beyond standard partial differential equations, allowing for the exploration of anomalous diffusion, non-local interactions and spatio-temporal correlations in diverse systems. The lack of kernels in the variable-order Caputo derivative improves stability and enables efficient computational methods for solving SFPDEs. This progress creates opportunities to study emerging behaviors, multi-fractal patterns, and complex dynamics in disciplines such as physics, biology, finance, and engineering, to name just a few.

Currently, notable works from several researchers delve into the characteristics and uses of variable-order Caputo derivatives in calculus. These investigations offer perspectives on the underpinnings and real world significance of utilizing variable-order Caputo derivatives to model and analyze intricate systems influenced by stochastic fractional partial differential equations. In this direction, several authors extended many results in order to incorporate the definition of the Caputo derivative to model useful problems and also complex phenomena either through the temporal variable or via a spacial component or as space–time fractional derivatives. For instance, we refer to a more recent manuscript in this direction; see [7] and the references therein for attenuation of the Caputo–Djrbashian fractional time derivative in modeling some inverse problems for wave equations. For the solution of stochastic evolution equation using the Marchaud fractional derivative, we refer to the work in Zhou et al. [8], and more recent results in this direction can be obtained in [9,10,11,12,13,14] and the references therein. For the exact solution using the conformable derivative, we refer to papers [12] and Abu-Shady and Mohammed in [15,16,17], and further extension can be found in the references [18,19,20,21]. For the the Wick-type fractional derivative, we refer to, for instance, the manuscript 0f Kim H. et al. [22]. For the fractional PDEs and their numerical solutions, we refer to, e.g., the work in [23,24], where further references can also obtained. For the random attractor in the fractional sense, see, for example, the manuscript [25]. For the fraction derivatives in the sense of Jumarie’s modified Riemann–Liouvile type, we refer the reader to [26], where further references are given therein. Most recent results concerning the various extensions of the Riemann–Liouville fractional derivatives can be found in the work of Jacky and Anna in [27].

Stochastic processes in many fields and applications can be described by classical stochastic differential equations (SDEs) of the form

$$\begin{array}{c}\hfill du=f(u(t))dt+b(u(t))dW.\end{array}$$

(1)

Let us mention a few, such as engineering, finance, and physics, because fractional stochastic differential equations are becoming a powerful tool in modeling problems in many disciplines. Many stochastic processes can be described (or modeled) by the system given in (1), for instance, a particle in the viscoelastic medium has random-movement Brownian motion with its resistance of a memory effect that leads to a fractional stochastic differential equation (FSDE); see [28,29,30,31,32,33] for more details. Despite knowing that multiple time-scale processes are not easy to model by using FSDE, however, we can emphasize the modeling of the accuracy of numerical approximation of non-physical initial weak singularity problems through FSDE; see instance, [24,34] where further references can be found therein. Henceforth, it is clearly challenging to obtain the probabilistic strong solutions for SFPDEs. Therefore, various numerical approximations of the solutions to both the deterministic FPDEs and SFPDEs have been proposed to deal with the issues of strong convergence and error estimates based on the Euler–Maruyama approximating techniques. See, for instance, the examples of a range of numerical approximations in the recent results in [32,35,36,37,38,39,40,41,42,43,44]. For the convergence of relative entropy for the Euler–Maruyama approximation, we refer to the most recent work of Yu in [43]; the split-step truncated EM approach was observed in [45]. Further exposition can be found in [46,47].

For the first time, Zhiwei et al. [33] analyzed, discretized, and obtained a solution of a two-time-scale variable-order time-FSDE using the Riemann–Liouville fractional differential operator with the non-Lipschitz weakly singular kernel. Similarly, Hong et al. [48] obtained the solution of the variable-order time-fractional diffusion equations and a variable-order FSDE of two-time scales with a Riemann–Liouville fractional differential operator presented by Xiangcheng et al. [49]. More detailed exposition about the Caputo derivatives can be found in [50] in this direction. For more engineering applications, we refer to [51]. On the other hand, for the solvability of time-fractional diffusion equations of constant order, we refer to Martin [24]. However, all the above papers dealt with the numerical approximations of FSPDEs with kernels. Therefore, the asymptotic and ergodic environment of the type of probabilistic solutions in this work have not yet been studied in the literature. Thus, so far, the asymptotic behavior of the underlying problem can still be treated as an open problem.

To the best of our knowledge, the strong convergence of the EM scheme for the nonlinear fractional stochastic differential equations without singular kernels has not yet been investigated in the existing literature. Having said that, therefore, the present work is the first one to consider proving the existence and uniqueness of a probabilistic strong solution to fractional stochastic differential equations with non-singular kernels subjected to nonlinear perturbation. Hence, this motivates us to state and prove a range of results on a priori estimates, error estimates, and the strong convergence of the EM approximation method for the problem under consideration. In fact, the underlying problem describes the more complex random phenomena with non-local effect. In addition, we develop and prove the strong convergence of the Euler-type scheme. Finally, as a matter of fact, we present our numerical experiments in order to substantiate the analysis of the problem under consideration. The current results from the numerical experiments are then compared with the results in the existing literature using both MATLAB and Python programmings. The main difference between the results carried out in this paper and those in [33] is highlighted by the existence and uniqueness carried out not only without the Lipschitz conditions as used in [33] but also the fact that we do not use the kernel, as in [33]. Instead of adapting the Riemann–Liouville fractional differential operator, as in [33], we use the new Caputo fractional derivative with variable order and study numerical solutions of FSDE without a singular kernel.

In this paper we consider the new Caputo–Fabrizio fractional derivative with variable order and study numerical solutions of FSDE without a singular kernel in the form

$$du=(-\lambda {}_0{}^{C}D_t^{\alpha }u+f(t,u))dt+b(t,u)dW,t\in [0,T],u(0)=u_0,\lambda \ge 0,$$

(2)

where f and b are nonlinear external terms representing the drift and the intensity of the noise, respectively. Here, W is a Wiener process and ${}_0{}^{C}D_t^{\alpha }u$ is the new Caputo–Fabrizio fractional derivative of order $\alpha$; see the subsection below for more details and properties of this operator. Note the assumptions on both function f and b will be subsequently specified.

1.1. Important Properties of ${}_0{}^{C}D_t^{\alpha }$

We recall from [50] that the Caputo fractional derivative of order $\alpha$ is defined by

$$\begin{array}{c}\hfill D_t^{\alpha }u(t):={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\int }_a^t\mathcal{K}(t,s)u^{\prime }(s)ds\end{array}$$

where $\alpha \in [0,1]$, $a\in [-\infty ,t)$, and $u\in H^1(a,b)$ for $a<b$ and $a,b\in \mathbf{R}$. Here, $\mathcal{K}(t,s)$ is the kernel, defined as

$$\mathcal{K}(t,s):={(t-s)}^{-\alpha }\mathrm{for}s\le t.$$

We can list some of the properties and drawbacks of this operator in the remark below.

Remark 1.

Note that, for $s=t$, the kernel in the above definition becomes singular, that is, $\mathcal{K}(t,t)=0$. This singularity is disadvantageous in modeling real-world phenomena. This fact can reduce the application of this operator to more complex problems. Furthermore, we also note in the definition of the operator $D_t^{\alpha }$, the presence of the function $u^{\prime }(t)$, meaning that the derivative of u must exist in the ordinary sense before we can define or compute any expression with the above defined operator.

Now set

$${\mathcal{K}}_0:=e^{-{\displaystyle \frac{\alpha }{1-\alpha }}(t-s)}$$

and change the constant ${\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}$ by ${\displaystyle \frac{1}{\sqrt{2\pi (1-{\alpha }^2)}}}$ in the definition of the operator $D_t^{\alpha }$. Then, according to [52], the new Caputo–Fabrizio derivative of order $\alpha \in (0,1)$, ${}_0{}^{C}D_t^{\alpha }u$ can be defined as FD without a singular kernel as follows (after normalizing):

$$\begin{array}{c}\hfill {}_0{}^{C}D_t^{\alpha }u={\displaystyle \frac{1}{(1-\alpha )}}{\int }_a^t\dot{u}(\tau )e^{{\displaystyle \frac{-\alpha (t-\tau )}{(1-\alpha )}}}d\tau .\end{array}$$

Remark 2.

At first glance, we notice that, for $t=s$, the new kernel ${\mathcal{K}}_0(t,t)=1\ne 0$. In contrast to $\mathcal{K}$, the new kernel ${\mathcal{K}}_0$ is not singular and it is, in addition, very much a regular kernel. Moreover, as for the operator $D_t^{\alpha }$, we also observe that ${}_0{}^{C}D_t^{\alpha }$ requires the function u to admit a first derivative in the ordinary sense. However, this can be addressed below.

Arguing similarly as in [50], one can easily define the operator ${}_0{}^{C}D_t^{\alpha }$ for function $u\in L^1(a,b)$ and $\alpha \in [0,1]$ by setting

$${}_0{}^{C}D_t^{\alpha }u(t)={\displaystyle \frac{\alpha M(\alpha )}{1-\alpha }}{\int }_a^t[u(t)-u(s)]{\mathcal{K}}_0(t,s)ds$$

where $M(\alpha )$ is a normalized constant depending on $\alpha$.

From this definition, we remark that the system modeled by the operator ${}_0{}^{C}D_t^{\alpha }u$ can describe non-local systems, which are able to model fluctuations that cannot be captured by the usual operator $D_t^{\alpha }$.

1.2. Motivational Background

Exploring systems that display unpredictability and distant connections is fundamental in contemporary science and engineering practices. The mathematical representation of these systems frequently necessitates instruments that surpass traditional calculus by integrating components capable of managing both randomness and memory impacts. Fractional stochastic partial differential equations have recently been among the most popular areas in describing and understanding mathematically complex behaviors of robust systems such as the Navier–Stokes and hydrodynamics models of turbulence fluids. It is worth mentioning that the presence of stochastic forces in a model (fractional type) can lead to new phenomena that are very difficult to describe using their deterministic fractional counterparts. The generalization of Navier–Stokes equations by fractional Navier–Stokes systems can be used to better describe and analyze in practice the compressible and incompressible fluids dynamics. Due to the challenging task in computing and analyzing the behavior of some complex phenomena, several researchers have recently proposed different classes of models. Among those models, we mention the shell fractional system that consists of infinitely many differential equations that can be used to model behaviors similar to the fractional Navier–Stokes, and many other similar properties of the classical Navier–Stokes can be studied. Moreover, other things that can be explored are the attempt at relaxing some assumptions and homogeneity, studying boundary conditions and boundary layers, and treating other physical phenomena.

Utilizing adaptable derivatives in these equations allows for a more accurate representation of real-world events that helps capture nuances that are often missed by the conventional models.

The enhanced capabilities of the order Caputo derivative without singular kernels represent a major step forward in fractional calculus by enhancing stability and effectiveness in computational techniques across various industries, like physics with prevalent anomalous diffusion phenomena; biology with intricate spatio-temporal patterns; and finance, with market dynamics displaying fractal characteristics. By removing kernels from the models structure, it not only enhances its resilience but also makes it more viable computationally, thus paving the way for exploring fresh avenues in simulating and comprehending systems that are impacted by both deterministic and stochastic elements.

Innovations in the field have revealed that SFPDEs incorporating Caputo derivatives of varying orders offer perspectives on complex system behaviors that conventional models find challenging to explain fully. For example, intricate phenomena characterized by scale processes, non-local interactions, and intricate random dynamics present in viscoelastic materials or financial markets stand to benefit from this method. These progressions not only enrich our theoretical comprehension but also hold practical significance in refining algorithms and enhancing numerical techniques applied in simulations across diverse scientific domains.

The task within this manuscript is to initiate the study of some interesting classes of nonlinear FSPDEs driven by white noise and also to investigate several results: first the issues of well-posedness (existence and uniqueness) of these systems, rate of convergence, strong convergence, a priori estimates, and higher integrability on the solutions, error estimates, and the approximate solutions.

In our research project, we are seeking to extend our knowledge in this particular field even further by investigating a novel approach not previously explored. We examine how a robust strong probabilistic solution can exist for fractional stochastic differential equations characterized by non-singular kernels and impacted by nonlinear perturbations for the very first time ever. This study opens up pathways for delving deeper into intricate stochastic systems and enhancing computational methodologies for better outcomes overall. Our findings underscore distinctions and advancements compared to current techniques in this area of study and lay down a strong groundwork that will shape future investigations in stochastic processes and fractional calculus. The discoveries have the potential to result in advancements across various sectors, such as material science and financial engineering, by addressing uncertainties and memory influences that are essential in these fields.

To the best of our knowledge, the strong convergence of the EM scheme for nonlinear fractional stochastic differential equations without singular kernels has not yet been studied in the corresponding literature. Therefore, this work is the first one to consider proving the existence and uniqueness of a probabilistic strong solution to the fractional stochastic differential equations with non-singular kernel subjected to nonlinear white noise. Hence, this clearly motivates us to establish new results on a priori bounds, higher integrability, error estimates, and the strong convergence of the EM approximation method for the problem under consideration.

1.3. Organization of the Paper

The remaining sections of this paper are structured as follows. In Section 2, we prove that the convergence of the Euler–Maruyama approximation and the associated numerical solution has a bounded pth moment for a some $p\in [2,\infty )$. In Section 3, we impose further assumptions on the nonlinear external forces f and g to ensure that $u(t)$ has bounded moments. In Section 4, we focus on establishing an optimal rate of convergence of the method established in Section 3. This section is augmented to a more widely used implicit variant of Euler–Maruyama by relating the notion introduced in Section 2 of implicit methods with the result obtained in Section 4. In Section 5, we draw concluding remarks. The major tools are listed in Appendix A.

2. Convergence, Continuity, and Uniqueness of the Solution for Equation (2)

In this section, we prove the existence of numerical solutions of FSDE (2). We start by reformulating FSDE (2) similar to Yang et al. [33].

Let $(\mathsf{\Omega },\mathcal{F},\mathcal{P},\mathbb{F})$ be a filtered probability space equipped with a filtration satisfying the usual conditions. Let W be a Wiener process defined on this probability space, $(\mathsf{\Omega },\mathcal{F},\mathcal{P})$ associated with the filtration, and let $u_0$ be a random variable that is independent of the Wiener process W. We consider $W(·)$ to be a Brownian motion with respect to the filtration

$$\begin{array}{c}\hfill {\mathcal{F}}_t=\mathcal{U}\{W(s):s\in [0,t],u_0\},\end{array}$$

where ${\mathcal{F}}_t$ is the natural $\sigma$-algebra generated by $u_0$, which is a second-order variable, and ${\mathcal{F}}_t$ is independent of Brownian motion $W(·)$ with a history up to time t. Here and subsequently, we assume that the filtration $\mathbb{F}$ is complete and right-continuous. Indeed, hereafter, we assume that the nonlinear external terms $f:\mathbb{R}⟶{\mathbb{R}}^m\mathrm{and}b:\mathbb{R}⟶{\mathbb{R}}^{d\times m}$ are continuous with respect to the first variable t and Lipschitz with respect to the second variable u. We denote the Euclidean vector norm by the symbol $|·|$.

In order to obtain the existence and uniqueness of solution (strong) to the problem under consideration, we need to provide some assumptions on a the nonlinear terms, the drift f, and the diffusion coefficients b that will be needed subsequently.

Assumptions:

(i)

Suppose that $f\mathrm{and}b$ are both one-sided Lipschitz nonlinear continuous functions and, in addition, the function b is Lipschitz nonlinear continuous function with respect to u, respectively, i.e., for $R\in (0,\infty )$, then there exists a constant $C_R>0$ depending only on R, such that $\forall u,v\in {\mathbb{R}}^m$; it holds that

$$\begin{array}{c}\hfill \langle f(u)-f(v),u-v\rangle \le C_R{|u-v|}^2,\end{array}$$

(3)

$$\begin{array}{c}\hfill \langle b(u)-b(v),u-v\rangle \le C_R{|u-v|}^2,\end{array}$$

(4)

$$\begin{array}{c}\hfill {|b(u)-b(v)|}^2\le C_R{|u-v|}^2.\end{array}$$

(5)

(ii)

We assume that the nonlinear term b is measurable a.e., t, and that there exists $L_C\in (0,\infty )$ such that

$$\begin{array}{cc}\hfill & \left|f(u)\right|\le L_C(1+\left|u\right|),\forall u\in {\mathbb{R}}^m,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \left|b(u)\right|\le L_C(1+\left|u\right|),\forall u\in {\mathbb{R}}^m.\hfill \end{array}$$

(7)

(iii)

Let $\alpha \in C^1([0,T])$, the space of functions that are differentiable and continuous on the interval $[0,T]$ such that there is ${\alpha }^{*}\in (0,1)$ such that

$$\begin{array}{c}\hfill 0\le \alpha (y)\le {\alpha }^{*}.\end{array}$$

(8)

To represent Equation (2) in stochastic process form, first we integrate the fractional derivative part of (2) from $0\mathrm{to}t$ by applying the same technique in Xiangcheng et al. [49]. We first have

$$\begin{array}{cc}\hfill {\int }_0^t-\lambda {}_0{}^{C}D_y^{\alpha (y)}udy=& {\int }_0^t{\int }_0^y{\displaystyle \frac{-\lambda }{(1-\alpha (y))}}\dot{u}(s)e^{{\displaystyle \frac{-\alpha (y)(t-y)}{(1-\alpha (y))}}}dsdy\hfill \\ \hfill =& {\int }_0^t{\int }_s^t{\displaystyle \frac{-\lambda }{(1-\alpha (y))}}\dot{u}(s)e^{{\displaystyle \frac{-\alpha (y)(y-s)}{(1-\alpha (y))}}}dyds\hfill \\ \hfill =& {\int }_0^{t}u(s){\int }_s^t{\displaystyle \frac{-\lambda }{(1-\alpha (y))}}{e}^{{\displaystyle \frac{-\alpha (y)(y-s)}{(1-\alpha (y))}}}dyds\hfill \\ \hfill -& {\int }_0^{t}u(s){\displaystyle \frac{d}{ds}}\left[{\int }_s^t{\displaystyle \frac{\lambda }{(1-\alpha (y))}}{e}^{{\displaystyle \frac{-\alpha (y)(y-s)}{(1-\alpha (y))}}}dy\right]ds\hfill \\ \hfill =& -u(0){\int }_0^t{\displaystyle \frac{\lambda }{(1-\alpha (y))}}{e}^{{\displaystyle \frac{-\alpha (y)(y)}{(1-\alpha (y))}}}dy-{\int }_0^t\mathcal{K}(t,s)u(s)ds,\hfill \end{array}$$

(9)

where

$$\begin{array}{c}\hfill \mathcal{K}(t,s)={\displaystyle \frac{d}{ds}}\left[{\int }_s^t{\displaystyle \frac{\lambda }{(1-\alpha (y))}}{e}^{{\displaystyle \frac{-\alpha (y)(y-s)}{(1-\alpha (y))}}}dy\right].\end{array}$$

(10)

Our main objective is to find a stochastic process u that will satisfy the stochastic integral below. For this purpose, modulo a stopping time argument, we can insert Equation (10) in (2), with $t\in [0,T]$, to generate a stochastic process $u(·)$, which is progressively measurable on $\mathcal{F}(·)$ such that the following stochastic integral holds:

$$\begin{array}{cc}\hfill u(t)=& u(0)+u(0){\int }_0^t{\displaystyle \frac{\lambda }{(1-\alpha (s))}}{e}^{{\displaystyle \frac{-\alpha (s)(s)}{(1-\alpha (s))}}}ds+{\int }_0^t\mathcal{K}(t,s)u(s)ds\hfill \\ \hfill & +{\int }_0^{t}f(u(s))ds+{\int }_0^{t}b(u(s))dW(s)a.s..\hfill \end{array}$$

(11)

Our first and primary goals are to investigate the strong convergence of numerical approximation in the scenario, where f and b are both one-sided Lipschitz with b in addition being locally Lipschitz. Subsequently, $Q_i$ denote some generic positive constants that may change from time to time throughout the paper.

Theorem 1.

If Assumptions (3) to (8) hold, then SDE (2) has a unique solution u such that

$$\begin{array}{cc}\hfill & \mathbb{E}[\underset{0\le t\le T}{sup}{\left|u(t)\right|}^2]\le \mathsf{\Theta }{E}_0{\rm Y}<\infty ,t\in [0,T].\hfill \\ \hfill \mathit{where} & \mathsf{\Theta }=3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}{T}^2\mathbb{E}({\left|u_0\right|}^2)+6L^2(T^2+Q_{1}T)(1+\mathbb{E}{(\left|u_0\right|)}^2),\hfill \\ & \mathsf{\Lambda }=3({\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+C_R(T+Q))and\hfill \end{array}$$

(12)

$$\begin{array}{cc}& E_0(\mathsf{\Lambda }T)=\sum _{n=0}^{\infty }{{\rm Y}}^n=\sum _{n=0}^{\infty }{\displaystyle \frac{{\mathsf{{\rm Y}}}^n}{\mathsf{\Gamma }(0\times n+1)}},\mathsf{{\rm Y}}=\mathsf{\Lambda }T\ge 0.\hfill \end{array}$$

(13)

In order to proceed with the proof of this result in Theorem 1, we rely on the Picard iteration approach. For this purpose, we define a general term of the Picard sequence ${\left\{Z_n\right\}}_{n=0}^{\infty }$ by setting

$$\begin{array}{cc}\hfill Z_{n+1}(t)=& Z_0+u_0{\int }_0^t{\displaystyle \frac{\lambda }{(1-\alpha (s))}}{e}^{{\displaystyle \frac{-\alpha (s)(s)}{(1-\alpha (s))}}}ds+{\int }_0^t\mathcal{K}(t,s)Z_n(s)ds\hfill \\ \hfill +& {\int }_0^{t}f(Z_n(s))ds+{\int }_0^{t}b(Z_n(s))dW(s)\mathbf{for}n\ge 1,\hfill \end{array}$$

(14)

with $Z_0:=u(0)=u_0$.

This above Picard sequence can be generated using the Picard iteration method. Therefore, we follow the arguments of the celebrated work of Etienne Pardoux in [53] to deal with the terms in the sequence. Before we delve into the proof of Theorem 1, we first introduce the following result.

Lemma 1.

Under Assumptions (3), (4), (6), and (7), there is a constant $C>0$ such that for any $n\ge 1$ and $t\in [0,T]$, we have the estimate

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_n(x)\right|}^2\le C.\end{array}$$

(15)

Proof. 

Applying Ito’s formula to the function ${\left|x\right|}^2$, we have

$$\begin{array}{cc}\hfill {\left|Z_{n+1}(s)\right|}^2=& |Z_0{|}^2+2u_0{\int }_0^x\langle \mathcal{K}(x,s),Z_n(s)\rangle ds+2{\int }_0^x\langle Z_n(s),f(Z_n(s))\rangle ds\hfill \\ \hfill +& 2{\int }_0^x\langle Z_n(s),b(Z_n(s))\rangle dW(s)+{\int }_0^x{\left|b(Z_n(s))\right|}^{2}ds\hfill \\ \hfill =& |Z_0{|}^2+K_1(x)+K_2(x)+K_3(x)+K_4(x).\hfill \end{array}$$

(16)

In order to estimate every term in (16), we rely on Assumptions (3), (4), (6), and (7), the Burkholder–Davis–Gundy (BDG) inequality, Young’s inequality, and the Cauchy–Swarz inequality to deal with each term in (16). For this purpose, we deal with the integrals in (16) separately. Therefore, we take mathematical expectation of both sides of (16) then apply the supremum to both sides to obtain the corresponding relation from which we estimate the remaining terms as below. We have

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}\left|K_1(x)\right|& =\mathbb{E}\underset{0\le x\le t}{sup}\left|{\int }_0^x\langle \mathcal{K}(x,s),Z_n(s)\rangle ds\right|\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^x{\left|\mathcal{K}(x,s)\right|}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\int }_0^x{\left|Z_n(s)\right|}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C_{\epsilon }{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+\epsilon \mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(r)\right|}^{2}ds.\hfill \end{array}$$

(17)

Using the assumption in item (i) along with Fubini’s theorem, we infer from the Young and Cauchy inequalities that there exists a constant $L_C$ such that

$$\begin{array}{cc}\mathbb{E}\underset{0\le x\le t}{sup}\left|K_2(x)\right|=\hfill & \mathbb{E}\underset{0\le x\le t}{sup}\left|{\int }_0^x\langle Z_n,f(Z_n(s))\rangle ds\right|\hfill \\ \hfill & \le L_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x\left|Z_n(s)\right|(1+|Z_n(s)|)ds\hfill \\ \hfill & \le L_C\mathbb{E}\underset{0\le x\le t}{sup}\left[\epsilon {\int }_0^x{\left|Z_n(s)\right|}^{2}ds+C_{\epsilon }{\int }_0^x{1}^{2}ds+{\int }_0^x{\left|Z_n(s)\right|}^{2}ds\right]\hfill \\ \hfill & \le C_{\epsilon }+(1+\epsilon )\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds.\hfill \end{array}$$

(18)

As for the term in $K_3$, we first apply the BDG inequality and then use Assumption (7) and Young’s inequality to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}\left|{\int }_0^x\langle Z_n(s),b(Z_n(s))\rangle dW(s)\right|\hfill \\ \hfill & \le C\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x{\left|Z_n(s)\right|}^2{\left|b(Z_n(s))\right|}^{2}ds\right|}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C\mathbb{E}\left[{\left[\underset{0\le x\le t}{sup}{\left|Z_n(s)\right|}^2\right]}^{{\displaystyle \frac{1}{2}}}\underset{0\le x\le t}{sup}{\left|{\int }_0^x(1+{\left|Z_n(s)\right|}^2)ds\right|}^{{\displaystyle \frac{1}{2}}}\right]\hfill \\ \hfill & \le \eta \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_n(s)\right|}^2+t{C}_{\eta }+C_{\eta }\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds.\hfill \end{array}$$

(19)

Similarly, in the case for the bound of $K_3(x)$, we can also estimate the term $K_4(t)$ using Assumption (7) to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|b(Z_n(s))\right|}^{2}ds\hfill \\ \hfill & \le L_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left(1+\left|Z_n(r)\right|\right)}^{2}ds\hfill \\ \hfill & \le T{L}_C+L_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(r)\right|}^{2}ds.\hfill \end{array}$$

(20)

Combining all the estimates (17)–(20), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_{n+1}(x)\right|}^2\hfill \\ \hfill & \le |Z_0{|}^2+C_{\epsilon }+(1+\epsilon )\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds\hfill \\ \hfill & +\eta \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_n(s)\right|}^2+t{C}_{\eta }+C_{\eta }\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds\hfill \\ \hfill & +C_{\epsilon }+(1+\epsilon )\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds+T{L}_C+L_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(r)\right|}^{2}ds\hfill \\ \hfill & \le C+\eta \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_n(s)\right|}^2+2(1+\epsilon +L_C+C_{\eta })\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(r)\right|}^{2}ds.\hfill \end{array}$$

(21)

On the other hand, raising the expression in (14) to the power 2 and the taking mathematical expectation and the supremum and using Jensen’s inequality, we obtain that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_{n+1}(x)\right|}^2\hfill \\ \hfill & \le 5\mathbb{E}|Z_0{|}^2+\mathbb{E}\underset{0\le x\le t}{sup}{u}_0{\int }_0^x{\displaystyle \frac{\lambda }{(1-\alpha (s))}}{e}^{{\displaystyle \frac{-\alpha (s)(s)}{(1-\alpha (s))}}}ds+10\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x\mathcal{K}(x,s)ds\right|}^2\hfill \\ \hfill & +10\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}f(Z_n(s))ds\right|}^2+10\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}b(Z_n(s))dW(s)\right|}^2\hfill \\ \hfill & =5|Z_0{|}^2+M_1(x)+M_2(x)+M_3(x)+M_4(x).\hfill \end{array}$$

(22)

Similarly as in the estimates for $K_1(t)-K_4(t)$, we rely on the same techniques to deal with the estimates for $M_1(t)-M_4(t)$. Therefore, we use Assumptions (3)–(8) along with the BDG, Young, Cauchy, and Jensen inequalities to obtain the corresponding relations needed for the estimates. We hence deduce the estimate for $M_1(t)$ and $M_2(t)$ as follows:

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{M}_1(x)+\mathbb{E}\underset{0\le x\le t}{sup}{M}_2(x)\le 2{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T.\end{array}$$

(23)

Due to Assumption (6) along with the Young and Cauchy inequalities, we infer that

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{M}_3(x)& \le \mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}f(Z_n(s))ds\right|}^2\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^x\left|f(Z_n(s))\right|ds\right]}^2\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^x(1+\left|Z_n(s)\right|)ds\right]}^2\hfill \\ \hfill & \le T^2+\mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^x\left|Z_n(s)\right|ds\right]}^2\hfill \\ \hfill & \le T^2+\mathbb{E}\underset{0\le x\le t}{sup}{\left[{\left({\int }_0^x{1}^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_0^x{\left|Z_n(s)\right|}^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}\right]}^2\hfill \\ \hfill & \le T^2+T{C}_{\kappa }+\kappa \mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds.\hfill \end{array}$$

(24)

As for the stochastic term, we use Assumption (7) and apply Burkholder–Davis–Gundy’s inequality along with the Young and Cauchy inequalities to obtain

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{M}_4(x)& \le \mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}b(Z_n(s))dW(s)\right|}^2\hfill \\ \hfill & \le C\mathbb{E}{\int }_0^t{\left|b(Z_n(s))\right|}^{2}ds\hfill \\ \hfill & \le C{L}_C\mathbb{E}{\int }_0^t(1+{\left|Z_n(s)\right|}^2)ds\hfill \\ \hfill & \le TC{L}_C+C{L}_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds.\hfill \end{array}$$

(25)

Combining all estimates (23)–(25), we obtain

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_{n+1}(x)\right|}^2\le C(\lambda ,{\alpha }^{*},\kappa ,T)+C(T,\kappa )\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds.\end{array}$$

(26)

In order to carry on with the remaining parts of the proof, we define the function $H(t)$ by setting

$$H_n(t):=\mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_{n+1}(x)\right|}^2,\mathbf{for}n\in \mathbb{N}.$$

We remark by definition of $H_n(t)$ that

$$\begin{array}{c}\hfill H_n(t)\le C(\lambda ,{\alpha }^{*},\kappa ,T)+C(T,\kappa )\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_n(s)\right|}^{2}ds\end{array}$$

(27)

from which we infer that

$$\begin{array}{c}\hfill H_n(t)\le Q_T{\int }_0^x{\left|H_{n-1}(s)\right|}^{2}ds,\end{array}$$

(28)

where Q is a constant depending only on $\lambda ,{\alpha }^{*},T$, and $\kappa$.

Arguing similarly as in [33], we heavily rely on the principle of mathematical induction to estimate the function $H_n(t)$. For this purpose, we first address the case of $n=0$ and deal with the corresponding relation as follows:

We infer from relation (16) and BDG inequality (25) with $n=0$ that

$$\begin{array}{cc}\hfill H_0(t)& =\mathbb{E}\underset{0\le x\le t}{sup}{\left|Z_1(x)\right|}^2\hfill \\ \hfill =& |Z_0{|}^2+2u_0{\int }_0^x\langle \mathcal{K}(x,s),Z_0(s)\rangle ds+2{\int }_0^x\langle Z_0(s),f(Z_0(s))\rangle ds\hfill \\ \hfill +& 2{\int }_0^x\langle Z_0(s),b(Z_0(s))\rangle dW(s)+{\displaystyle \frac{1}{2}}{\int }_0^x{\left|b(Z_0(s))\right|}^{2}ds\hfill \\ \hfill & \le \mathbb{E}|Z_0{|}^2+2\mathbb{E}{u}_0\underset{0\le x\le t}{sup}{\int }_0^x\langle \mathcal{K}(x,s),Z_0(s)\rangle ds+2\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x\langle Z_0(s),f(Z_0(s))\rangle ds\hfill \\ \hfill +& 2\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x\langle Z_0(s),b(Z_0(s))\rangle dW(s)+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|b(Z_0(s))\right|}^{2}ds\hfill \\ \hfill & \le \mathbb{E}|Z_0{|}^2+{\displaystyle \frac{{\lambda }^2}{1-{\alpha }^{*}}}\mathbb{E}|u_0{|}^2+T{L}_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|Z_0\right|}^{2}ds\hfill \\ \hfill & \le C\mathbb{E}|u_0{|}^2=:\mathsf{\Gamma }<\infty ,\hfill \end{array}$$

(29)

where $\mathsf{\Gamma }$ is a constant depending only on ${\alpha }^{*},\lambda$, $\mathbb{E}|u_0{|}^2$, and T. It implies the estimate of the first term $H_0(t)$. This holds true.

For $n=1$, we have

$$\begin{array}{ll}{H}_1(t)& \le Q_T{\int }_0^x{H}_0(s)ds\\ & =Q_T{\int }_0^x\mathsf{\Gamma }ds\\ & =Q_T{\int }_0^t\mathsf{\Gamma }ds\\ & =Q_T\mathsf{\Gamma }t<\infty .\end{array}$$

(30)

This holds true as well.

Similarly, for $n=2$ we have

$$\begin{array}{ll}\hfill H_2(t)& \le Q_T{\int }_0^x{H}_1(s)ds\\ \hfill & =Q_T{\int }_0^x{\mathsf{\Gamma }}^{2}tds\\ & =Q_T{\int }_0^t{\mathsf{\Gamma }}^{2}tds\\ & =Q_T{\mathsf{\Gamma }}^2{t}^2<\infty .\end{array}$$

(31)

This is true.

Now, we assume that the induction step and the inductive hypothesis both hold true; for $n=k+1$ we have that

$$\begin{array}{ll}\hfill H_{k+1}(t)& \le Q_T{\int }_0^x{H}_k(s)ds\\ & =Q_T{\int }_0^x{\mathsf{\Gamma }}^k{t}^{k-1}ds\\ & =Q_T{\int }_0^t{\mathsf{\Gamma }}^k{t}^{k-1}ds\\ & =Q_T{\mathsf{\Gamma }}^k{t}^k<\infty ,\end{array}$$

(32)

from which we deduce that the series (32) can be written as

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{Q}_T{(\mathsf{\Gamma }t)}^n=Q_T\sum _{n=0}^{\infty }{(\mathsf{\Gamma }t)}^n{Q}_T{E}_0(\mathsf{\Gamma }t)<\infty \mathbf{for}\mathbf{any}t\in [0,T].\end{array}$$

(33)

Here, $E_0$ stands for the Mittag-Leffler function and the above series duly converges. Therefore,

$$\sum _{n=0}^{\infty }\mathbb{E}\underset{0\le t\le T}{sup}{\left|Z_{n+1}(t)\right|}^2<\infty \mathbf{a}.\mathbf{s}.$$

from which, along with Chebyshev’s inequality, we infer that

$$\begin{array}{cc}\hfill \mathbb{P}\left(\underset{0\le t\le T}{sup}{\left|Z_{n+1}(t)\right|}^2\ge 2^{-n}\right)& \le 4^n\mathbb{E}\underset{0\le t\le T}{sup}{\left|Z_{n+1}(t)\right|}^2\hfill \\ \hfill & \le 4^n{(\mathsf{\Gamma }t)}^n{Q}_T\hfill \\ \hfill & \le 4^n{\mathsf{\Gamma }}^n{T}^n{Q}_T.\hfill \end{array}$$

(34)

Hence, we use the Borel–Cantelli lemma to deduce that the sequence ${\left\{Z_n\right\}}_{n=0}^{\infty }$ converges uniformly to the process $u(t)$ on $[0,T]$. □

We are now ready to proceed with the Proof of Theorem 1.

Proof. 

In order to obtain the integrability in Theorem 1, we need to estimate every term in the right-hand side of (14). For this purpose, we first rely on the result of Lemma 1 and use the same sequence ${\left\{Z_n\right\}}_{n\ge 1}$ as defined above. We therefore notice that the difference $Z_{n+1}(t)-Z_n(t)$ is given by

$$\begin{array}{cc}\hfill Z_{n+1}(t)-Z_n(t)=& {\int }_0^t\mathcal{K}(t,s)(Z_{n+1}(s)-Z_n(s))ds\hfill \\ \hfill +& {\int }_0^t(f(Z_{n+1}(s))-f(Z_n(s))ds+{\int }_0^t[b(Z_{n+1}(s))-b(Z_n(s))]dW(s).\hfill \end{array}$$

In order to obtain the best estimates, we first apply Ito’s formula to the functions ${\left|x\right|}^2$ and $d(Z_{n+1}-Z_n)$. Second, we estimate the terms in the expression of $|Z_{n+1}-Z_n{|}^2$ directly without using the expressions coming from Ito’s formula and then combine the terms coming from both relations to get the best estimates required for the proof of this theorem.

For this purpose, on one hand, applying Ito’s formula to the functions ${\left|x\right|}^2$ and $d(Z_{n+1}-Z_n)$, we obtain

$$\begin{array}{cc}\hfill {|Z_{n+1}(s)-Z_n(s)|}^2=& 2{\int }_0^t\langle \mathcal{K}(t,s),Z_n(s)-Z_{n-1}(s)\rangle ds\hfill \\ \hfill +& 2{\int }_0^t\langle Z_n(s)-Z_{n-1}(s),f(Z_n(s))-f(Z_{n-1}(s))\rangle ds\hfill \\ \hfill +& 2{\int }_0^t\langle Z_n(s)-Z_{n-1}(s),b(Z_n(s))-b(Z_{n-1}(s))\rangle dW(s)\hfill \\ \hfill +& {\displaystyle \frac{1}{2}}{\int }_0^t{|b(Z_n(s))-b(Z_{n-1}(s))|}^{2}ds\hfill \\ \hfill =& J_1(t)+J_2(t)+J_3(t)+J_4(t),\hfill \end{array}$$

from which we infer that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{|Z_{n+1}(x)-Z_n(x)|}^2\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}\left|J_1(x)\right|+\mathbb{E}\underset{0\le x\le t}{sup}\left|J_2(x)\right|+\mathbb{E}[\underset{0\le x\le t}{sup}\left|J_3(x)\right|+\mathbb{E}\underset{0\le x\le t}{sup}\left|J_4(x)\right|.\hfill \end{array}$$

(35)

Our next task is to separately estimate every term on the right side of Equation (35). To bound the first term of (35), we can apply Lemma A1, the Cauchy–Schwarz inequality, and Assumptions (3) to (8). We also apply the Young’s inequality modulo a stopping time to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}\left|J_1(x)\right|\hfill \\ \hfill & =\mathbb{E}\underset{0\le x\le t}{sup}|{\int }_0^x\left|\mathcal{K}(x,s)\right||Z_n(s)-Z_{n-1}(s)\left|ds\right|\hfill \\ \hfill & \le C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|\mathcal{K}(x,s)\right|}^{2}ds+C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \hfill & \le C{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds.\hfill \end{array}$$

(36)

We can proceed to estimate the term $J_2(x)$. Note that, by virtue of triangle and Jensen inequalities and due to the assumption in item (i) along with Fubini’s Theorem, we have

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}\left|J_2(x)\right|=& \mathbb{E}\underset{0\le x\le t}{sup}\left|{\int }_0^x\langle Z_n-Z_{n-1},f(Z_n(s))-f(Z_{n-1}(s))\rangle ds\right|\hfill \\ \hfill & \le C_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds.\hfill \end{array}$$

(37)

In order to proceed with the estimate of the term $J_3(x)$, we first use the one-sided Lipschitz assumption given in item (i), secondly the BDG inequality, and thirdly Young’s inequality to assert that there exists a constant $C_T>0$ such that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}\left|J_3(x)\right|\hfill \\ \hfill & \le \mathbb{E}{\left[{\int }_0^t|Z_n(s)-Z_{n-1}{(s)|}^2{|b(Z_n(s))-b(Z_{n-1}(s))|}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le \mathbb{E}\left({\left[\underset{0\le x\le t}{sup}{|Z_n(x)-Z_{n-1}(x)|}^2\right]}^{{\displaystyle \frac{1}{2}}}{\left[\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}d(s)\right]}^{{\displaystyle \frac{1}{2}}}\right)\hfill \\ \hfill & \le C\mathbb{E}\underset{0\le x\le t}{sup}|Z_n(x)-Z_{n-1}{(x)|}^2+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}d(s)\hfill \\ \hfill & \le C_T\mathbb{E}\underset{0\le x\le t}{sup}{|Z_n(x)-Z_{n-1}(x)|}^2.\hfill \end{array}$$

(38)

Finally, by applying Ito’s isometry, Lemma A2 (Appendix A), and Assumptions (3) to (8), we can bound the third term of (35) and obtain using a stopping time argument that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x\left[b(Z_n(s))-b(Z_{n-1}(s))\right]ds\right|}^2\hfill \\ \hfill & \le c_1\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|b(Z_n(s))-b(Z_{n-1}(s))|}^{2}ds\hfill \\ \hfill & \le Q_3\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds,\hfill \end{array}$$

(39)

where $Q_3$ is constant depending on R only. This gives us the estimate on the last term $J_4(x)$.

Now we combine all the estimates from (35) to (39) and (35) as well as applying Young’s inequality to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}|Z_{n+1}(x)-Z_n(x)){|}^2\hfill \\ \hfill & \le C({\alpha }^{*},\lambda ,T)+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+\hfill \\ \hfill & 3C_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+3Q_1{C}_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \hfill & \le C({\alpha }^{*},\lambda ,T)+{\mathsf{\Psi }}_1\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds,\hfill \end{array}$$

(40)

where ${\mathsf{\Psi }}_1=C_R(T+Q_1)$.

On the other hand, we notice that

$$\begin{array}{c}\hfill |Z_{n+1}(x)-Z_n{(x)|}^2={|I_1(x)+I_2(x)+I_3(x)|}^2,\end{array}$$

(41)

where

$$\begin{array}{cc}\hfill I_1(x)& ={\int }_0^x\mathcal{K}(x,s)\left[Z_n(s)-Z_{n-1}(s)\right]ds,\hfill \\ \hfill I_2(x)& ={\int }_0^x\left[f(Z_n(s))-f(Z_{n-1}(s))\right]ds,\hfill \\ \hfill I_3(x)& ={\int }_0^x\left[b(Z_n(s))-b(Z_{n-1}(s))\right]dW(s).\hfill \end{array}$$

This indicates to us that we must also estimate every term in (41). Then, by the Jensen inequality, we have

$$\begin{array}{cc}\hfill {|Z_{n+1}(x)-Z_n(x)|}^2\le & 3\left[{\left|I_1(x)\right|}^2+{\left|I_2(x)\right|}^2+{\left|I_3(x)\right|}^2\right],\hfill \end{array}$$

from which we infer that

$$\begin{array}{ccc}\hfill \mathbb{E}[\underset{0\le x\le t}{sup}{|Z_{n+1}(x)-Z_n(x)|}^2]& \le & 3\mathbb{E}[\underset{0\le x\le t}{sup}[{\left|I_1(x)\right|}^2]+3\mathbb{E}[\underset{0\le x\le t}{sup}[{\left|I_2(x)\right|}^2]\hfill \\ & & +3\mathbb{E}[\underset{0\le x\le t}{sup}[{\left|I_3(x)\right|}^2].\hfill \end{array}$$

(42)

Our next task is to separately bound every term on the right side of Equation (42). To bound the first term of (42), we can apply Lemma A1, the Cauchy–Schwarz inequality, and Assumptions (3) to (8). We also apply the Young’s inequality modulo a stopping time to obtain

$$\begin{array}{c}\mathbb{E}\underset{0\le x\le t}{sup}{\left|I_1(x)\right|}^2\hfill \\ =\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x\mathcal{K}(x,s)(Z_n(s)-Z_{n-1}(s))ds\right|}^2\hfill \\ \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^t{\left|\mathcal{K}(x,s)\right|}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \le \mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{\left|\mathcal{K}(x,s)\right|}^{2}ds+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \le {\displaystyle \frac{{\lambda }^2}{{(1-\alpha *)}^2}}t+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \le {\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds.\hfill \end{array}$$

(43)

Next, by applying Cauchy and Young’s inequalities and Assumptions (3) to (8), we can bound the second term of (35) as follows:

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{\left|I_2(x)\right|}^2=& \mathbb{E}\left[\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}f(Z_n(s))-f(Z_{n-1}(s))ds\right|}^2\right]\hfill \\ \hfill & \le \mathbb{E}\left(\left[\underset{0\le x\le t}{sup}{\left[{\int }_0^{x}1ds\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\int }_0^x{|f(Z_n)(s)-f(Z_{n-1}(s))ds|}^2\right]}^{{\displaystyle \frac{1}{2}}}\right]\right)\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}\left[{\int }_0^{x}1ds+{\int }_0^x{|f(Z_n)(s)-f(Z_{n-1}(s))ds|}^2\right]\hfill \\ \hfill & \le T+C_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \hfill & \le T+C_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds.\hfill \end{array}$$

(44)

Finally, by applying Ito isometry, Lemma A2, Burkh$\ddot{\mathrm{o}}$lder–Davis–Gundy, and Assumptions (3) to (8), we can bound the third term of (42) and obtain using a stopping time $\tau$ that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{\left[I_3(x)\right]}^2\hfill \\ \hfill & =\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x\left[b(Z_n(s))-b(Z_{n-1}(s))\right]dW(s)\right|}^2\hfill \\ \hfill & =\mathbb{E}{\int }_0^t{|b(Z_n(x))-b(Z_{n-1}(x))|}^{2}dx\hfill \\ \hfill & \le Q_1{C}_R{\int }_0^t{|Z_n(x)-Z_{n-1}(x)|}^{2}dx,\hfill \end{array}$$

(45)

where $Q_1$ is constant as the one given by Lemma A2. Now we combine all inequalities (36)–(45) with relation (35) and apply Young’s inequality to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{|Z_{n+1}(x)-Z_n(x)|}^2\hfill \\ \hfill & \le 3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+\hfill \\ \hfill & +3T{C}_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+3Q_1{C}_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \hfill & \le 3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+{\mathsf{\Psi }}_2\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds,\hfill \end{array}$$

(46)

where ${\mathsf{\Psi }}_1=C_R(T+Q_1)$.

Combining estimates (40) and (46), we infer that there exists a constant $\mathsf{\Psi }$ such that

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{|Z_{n+1}(x)-Z_n(x)|}^2\hfill \\ \hfill & \le 3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+\hfill \\ \hfill & +3T{C}_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds+3Q_1{C}_R\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds\hfill \\ \hfill & \le 3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}T+{\mathsf{\Psi }}_2\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|Z_n(s)-Z_{n-1}(s)|}^{2}ds,\hfill \end{array}$$

(47)

where $\mathsf{\Psi }={\displaystyle \frac{1}{2}}({\mathsf{\Psi }}_1+{\mathsf{\Psi }}_2)$.

Next, we use mathematical induction to find the estimate for the terms in relation (47). For this purpose, for $n\ge 1$, we reformulate and estimate the terms in the right-hand side of Equation (42) as follows. Set

$$\begin{array}{c}\hfill h_n(t):=\mathbb{E}\underset{0\le x\le t}{sup}{|Z_{n+1}(x)-Z_n(x)|}^2\le 3C({\alpha }^{*},T)+\mathsf{\Psi }{\int }_0^x{h}_{n-1}(s)ds,\end{array}$$

where

$$h_{n-1}(t):=\mathbb{E}\underset{0\le r\le t}{sup}{|Z_n(r)-Z_{n-1}(r)|}^2\mathrm{for}n\ge 1.$$

Using Equation (35) with $n=0$, applying the Jensen inequality, and using Assumptions (3) to (8), we obtain that

$$\begin{array}{cc}\hfill h_0(t)& =\mathbb{E}\underset{0\le x\le t}{sup}{\left|(Z_1(x)-Z_0(x))\right|}^2\hfill \\ \hfill & \le 3\mathbb{E}\underset{0\le x\le t}{sup}\left({\left|{\int }_0^x\mathcal{K}(x,s)u_{0}ds\right|}^2+{\left|{\int }_0^{x}f(u_0)ds\right|}^2+{\left|{\int }_0^{x}b(u_0(s))dW(s)\right|}^2\right)\hfill \\ \hfill & \le 3\mathbb{E}\underset{0\le x\le t}{sup}\left(u_0^2{\left({\int }_0^x\left|\mathcal{K}(x,s)\right|ds\right)}^2+{\left({\int }_0^x\left|f(u_0)\right|ds\right)}^2+Q_1{\int }_0^x{\left|b(u_0)\right|}^{2}ds\right)\hfill \\ \hfill & \le 3\mathbb{E}\left(\underset{0\le x\le t}{sup}{\left({\int }_0^{x}L(1+|u_0)|ds\right)}^2+Q_1{\int }_0^x{L}^2{(1+\left|u_O\right|)}^{2}ds\right)+3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}{T}^2\mathbb{E}({\left|u_0\right|}^2)\hfill \\ \hfill & \le 3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}{T}^2\mathbb{E}{\left|u_0\right|}^2+3L^2{T}^2\mathbb{E}{(1+\left|u_0\right|)}^2+3Q_1{L}^{2}T\mathbb{E}{(1+\left|u_0\right|)}^2=\mathsf{\Theta }<\infty ,\hfill \end{array}$$

(48)

where $\mathsf{\Theta }=3{\displaystyle \frac{{\lambda }^2}{{(1-{\alpha }^{*})}^2}}{T}^2\mathbb{E}{\left|u_0\right|}^2+6L^2(T^2+Q_{1}T)(1+\mathbb{E}{(\left|u_0\right|)}^2)$. Therefore, $h_0(t)$ is bounded.

Our next task is to estimate $h_n(t)$ for $n\ge 1$. Again, we easily can prove this by mathematical induction as in the aforementioned technique.

For $n=1$, using (48), we have

$$\begin{array}{c}\hfill h_1(t)=:\mathbb{E}\underset{0\le x\le t}{sup}{|Z_2(s)-Z_1(s)|}^2\le \mathsf{\Psi }{\int }_0^t\mathsf{\Theta }ds=\mathsf{\Psi }t\mathsf{\Theta }<\infty .\end{array}$$

(49)

Similarly for $n=2$, we have

$$\begin{array}{cc}\hfill h_2(t):=& \mathbb{E}\underset{0\le x\le t}{sup}{|Z_3(s)-Z_2(s)|}^2\le {(\mathsf{\Psi }t)}^2\mathsf{\Theta }<\infty .\hfill \end{array}$$

Now, suppose that the relation holds for $n=m$, i.e.,

$$\begin{array}{c}\hfill h_m(t):=\mathbb{E}\underset{0\le x\le t}{sup}{|Z_m(s)-Z_{m-1}(s)|}^2\le \mathsf{\Psi }{\int }_0^t{h}_{m-1}(s)ds\le {(\mathsf{\Psi }t)}^m\mathsf{\Theta }<\infty .\end{array}$$

(50)

We claim that the assumption also holds true for $n=m+1$:

$$\begin{array}{c}\hfill h_{m+1}(t)\le \mathsf{\Psi }{\int }_0^t{h}_m(s)ds\le {(\mathsf{\Psi }t)}^{m+1}\mathsf{\Theta }<\infty \end{array}$$

from which we therefore infer that

$$\begin{array}{c}\hfill h_n(t)=\mathbb{E}\underset{0\le x\le t}{sup}{|Z_{n+1}(s)-Z_n(s)|}^2\le {(\mathsf{\Psi }t)}^n\mathsf{\Theta }<\infty ,\forall n\ge 0.\end{array}$$

(51)

Moreover, the series defined by the right-hand side of relation (51) converges to the Mittag-Leffler function

$$\begin{array}{cc}\hfill \sum _{m=0}^{\infty }{(\mathsf{\Psi }t)}^n\mathsf{\Theta }=\sum _{m=0}^{\infty }{(\mathsf{\Psi })}^n{t}^n\mathsf{\Theta }=\mathsf{\Theta }\sum _{m=0}^{\infty }{\displaystyle \frac{{(\mathsf{\Psi })}^n{t}^n}{\mathsf{\Gamma }(0.n+1)}}=& \mathsf{\Theta }{E}_0(\mathsf{\Psi }T)<\infty ,t\in [0,T]\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }\mathbb{E}\underset{0\le t\le T}{sup}{|Z_{n+1}(t)-Z_n(t)|}^2<\infty ,a.s..\end{array}$$

(52)

Due to Chebyshev’s inequality, we infer from (51) and (52) that

$$\begin{array}{cc}\hfill \mathcal{P}{(\underset{t\in [0,T]}{sup}|Z_{n+1}(t)-Z_n(t))|}^2\ge 2^{-n})\le & 4^n\mathbb{E}{[\underset{t\in [0,T]}{sup}|Z_{n+1}(t)-Z_n(t))|}^2]\hfill \\ & \le 4^n{(\mathsf{\Psi }t)}^n\mathsf{\Theta }\le 4^n{(\mathsf{\Psi }T)}^n\mathsf{\Theta }.\hfill \end{array}$$

Using the Borel–Cantelli lemma, the sequence

$$\begin{array}{c}\hfill Z_n(t)=u_0+\sum _{m=1}^n[Z_m(t)-Z_{m-1}(t)]\end{array}$$

uniformly converges to limit u on $[0,T]$ and solves the underlying problem (11) a.s. We establish the continuity of u based on the continuity of ${\left\{Z_n\right\}}_{n=0}^{\infty }$ and the above result. From this we can therefore assert that the solution of FSDE (2) is the continuous function u.

Let $\overline{u}(t)$ be the solution of (2), different from $u(t)$. By following similar procedures as applied in (46), we have

$$\begin{array}{c}\hfill \mathbb{E}[\underset{0\le x\le t}{sup}{|u(x)-\overline{u}(x)|}^2]\le \mathsf{\Psi }{\int }_0^t\mathbb{E}[\underset{0\le r\le s}{sup}{|u(r)-\overline{u}(r)|}^2]ds,\forall t\in [0,T].\end{array}$$

(53)

By applying Lemma A3 (Generalized Gronwall inequality) on (53), we obtain that

$$\begin{array}{c}\hfill \mathbb{E}[\underset{0\le x\le t}{sup}{|u(x)-\overline{u}(x)|}^2]=0.\end{array}$$

This implies

$$\begin{array}{c}\hfill u(t)=\overline{u}(t)\mathrm{a}.\mathrm{s}..\end{array}$$

We conclude that the problem (2) admits a unique solution $u(t)$. □

In this paper, it is crucial to note that we are mainly interested in the strong convergence of the Euler–Maruyama scheme. Therefore, it is clear that we need to focus on the approximation of the sample paths of the fractional stochastic partial differential equations. Our main task is to state and prove results similar to those in [33] while showing the convergence.

The results in the above theorem can be generalized to compute the expected error $\mathbb{E}[\underset{0\le s\le t}{sup}{\left|u(s)\right|}^p]$ for $2\le p<\infty$, if the assumption $\mathbb{E}[\underset{0\le s\le t}{sup}{\left|u_0\right|}^p]<\infty$ is satisfied. This fact is stated and proved in the result below.

Theorem 2.

If Assumptions (3) to (8) hold, then FSDE (2) has a unique solution u such that

$$\begin{array}{cc}\hfill & \mathbb{E}[\underset{0\le x\le t}{sup}{\left|u(x)\right|}^p]<\infty ,t\in [0,T].\hfill \end{array}$$

(54)

Proof. 

Recall from (11) that

$$\begin{array}{cc}\hfill u(t)=& u(0)+u(0){\int }_0^t{\displaystyle \frac{\lambda }{(1-\alpha (s))}}{e}^{{\displaystyle \frac{-s\alpha (s)}{(1-\alpha (s))}}}ds+{\int }_0^t\mathcal{K}(t,s)u(s)ds\hfill \\ \hfill & +{\int }_0^{t}f(u(s))ds+{\int }_0^{t}b(u(s))dW(s)a.s..\hfill \end{array}$$

(55)

We raise (55) to the power p and, since $p>2$, we can apply Jensen’s inequality to the corresponding relation to obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\underset{0\le x\le t}{sup}{|u(t)|}^p\hfill \\ \hfill & \le 5^{p-1}\mathbb{E}|u_0{|}^p+5^{p-1}\mathbb{E}{|u_0|}^p\underset{0\le x\le t}{sup}{\left|{\int }_0^x{\displaystyle \frac{\lambda }{1-\alpha (s)}}{e}^{{\displaystyle \frac{-s\alpha (s)}{1-\alpha (s)}}}ds\right|}^p+5^{p-1}\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^x\mathcal{K}(x,s)u(s)ds\right|}^p+\hfill \\ \hfill & +5^{p-1}\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}f(u(s))ds\right|}^p+5^{p-1}\mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}b(u(s))dW(s)\right|}^p\hfill \\ \hfill & =5^{p-1}\mathbb{E}{|u_0|}^p+5^{p-1}{\mathcal{I}}_1(x)+5^{p-1}{\mathcal{I}}_2(x)+5^{p-1}{\mathcal{I}}_3(x)+5^{p-1}{\mathcal{I}}_4(x).\hfill \end{array}$$

(56)

Similarly as in the proof of Theorem 1, we can estimate the term ${\mathcal{I}}_1(x)$ as follows:

$$\begin{array}{c}\hfill {\mathcal{I}}_1(x)\le {\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}T.\end{array}$$

(57)

To estimate the term ${\mathcal{I}}_2(x)$, we apply Holder’s inequality to get

$$\begin{array}{cc}\hfill {\mathcal{I}}_2(x)& \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\int }_0^x{|\mathcal{K}(x,s)|}^{p-1}ds\right]}^{{\displaystyle \frac{p}{p-1}}}{\int }_0^x{|u(s)|}^{p}ds\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}{x}^{{\displaystyle \frac{p}{p-1}}}{\left[{\displaystyle \frac{{\lambda }^{p-1}}{{(1-{\alpha }^{*})}^{p-1}}}\right]}^{{\displaystyle \frac{p}{p-1}}}{\int }_0^x{|u(s)|}^{p}ds\hfill \\ \hfill & \le T^{{\displaystyle \frac{p}{p-1}}}{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds.\hfill \end{array}$$

(58)

In order to estimate the term ${\mathcal{I}}_3(x)$, we deploy Assumption (6) in item (ii) along with Holder’s inequality to obtain that

$$\begin{array}{cc}\hfill {\mathcal{I}}_3(x)& \le \mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}f(u(s))ds\right|}^p\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}{\left[{\left({\int }_0^x{1}^{(p-1)}ds\right)}^{{\displaystyle \frac{1}{p-1}}}{\left({\int }_0^x{|f(u(s))|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\right]}^p\hfill \\ \hfill & \le \mathbb{E}\underset{0\le x\le t}{sup}\left[x^p{\int }_0^x{|f(u(s))|}^{p}ds\right]\hfill \\ \hfill & \le T^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|f(u(s))|}^{p}ds\hfill \\ \hfill & \le T^p{L}_C\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{(1+|u(s)|)}^{p}ds\hfill \\ \hfill & \le T^{p}C{L}_C^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{(1+|u(s)|}^p)ds\hfill \\ \hfill & \le T^{p+1}C{L}_C^p+T^{p}C{L}_C^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds.\hfill \end{array}$$

(59)

As for the term ${\mathcal{I}}_4(x)$, we use Assumption (7) and we apply Holder and Bukholder–Davis–Gundy inequalities to assert that there exists a constant $C>0$ such that

$$\begin{array}{cc}\hfill {\mathcal{I}}_4(x)\le & \mathbb{E}\underset{0\le x\le t}{sup}{\left|{\int }_0^{x}b(u(s))dW(s)\right|}^p\hfill \\ \hfill & \le C\mathbb{E}{\left|{\int }_0^t{|b(u(s))|}^{2}ds\right|}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C\mathbb{E}{\left[{\left({\int }_0^t{1}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_0^t{|b(u(s))|}^{p}ds\right)}^{{\displaystyle \frac{2}{p}}}\right]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C\mathbb{E}\left[t^{{\displaystyle \frac{p-2}{2}}}{\int }_0^t{|b(u(s))|}^{p}ds\right]\hfill \\ \hfill & \le T^{{\displaystyle \frac{p-2}{2}}}C\mathbb{E}{L}_C^p{\int }_0^t{(1+|u(s)|}^p)ds\hfill \\ \hfill & \le T^{{\displaystyle \frac{p}{2}}}C{L}_C^p+T^{{\displaystyle \frac{p-2}{2}}}C{L}_C^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds,\hfill \end{array}$$

(60)

where q is the conjugate of ${\displaystyle \frac{p}{2}}$ and it is given by $q={\displaystyle \frac{p}{p-2}}$.

Combining all the estimates (57)–(60), we infer from (56) that

$$\begin{array}{cc}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{|u(t)|}^p\le & 5^{p-1}\mathbb{E}|u_0{|}^p+5^{p-1}\mathbb{E}{|u_0|}^p{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}T\hfill \\ \hfill & +5^{p-1}{T}^{{\displaystyle \frac{2p-1}{p-1}}}{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds++5^{p-1}{T}^{p+1}C{L}_C^p\hfill \\ \hfill & +5^{p-1}{T}^{p+1}C{L}_C^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds+5^{p-1}{T}^{{\displaystyle \frac{p}{2}}}C{L}_C^p+\hfill \\ \hfill & +5^{p-1}{T}^{{\displaystyle \frac{p}{2}}}C{L}_C^p\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds\hfill \end{array}$$

(61)

from which we infer that

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{|u(x)|}^p\le {\kappa }_1+{\kappa }_2\mathbb{E}\underset{0\le x\le t}{sup}{\int }_0^x{|u(s)|}^{p}ds,\end{array}$$

(62)

where ${\kappa }_1$ and ${\kappa }_2$ are constants depending on $p,T$, and $\lambda$.

By virtue of Gronwall’s inequality applied to (62), we deduce that there exists a constant $\kappa$ such that

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le x\le t}{sup}{|u(t)|}^p\le \kappa .\end{array}$$

(63)

Arguing similarly as in the proof of Theorem 1, it is sufficient to assert that there exists a constant ${\kappa }_3$ depending only on $p,T,\lambda$ such that

$$\begin{array}{c}\hfill \mathbb{E}\underset{0\le t\le T}{sup}{|u(t)|}^p\le {\kappa }_3.\end{array}$$

(64)

This concludes the desired proof of the theorem. □

3. Strong Convergence of the Euler–Maruyama Scheme

In this section, we develop the Euler–Maruyama scheme of (2) and conveniently investigate its strong convergence under some more weaker assumptions than those in the literature. Mainly, we will study the strong convergence of the fractional SPDEs with one-sided Lipschitz coefficients. Before stating and proving results similar to those above, we first construct the Euler–Maruyama approximation scheme.

3.1. Derivation of the Scheme

Let us first fix $N\in \mathbb{N}$ and suppose $n\in [0,N]\mathrm{and}\tau :={\displaystyle \frac{T}{N}}$ and set $t_n:=n\tau$ and $\mathsf{\Delta }{W}_n=W({\tau }_{n+1})-W({\tau }_n)$. Using the Euler quadrature at $t_n$ for $n\in [1,N]$, we can discretize the right-hand side of Equation (11).

First, let us discretize the second term using

$$\begin{array}{cc}\hfill u(0){\int }_0^{t_n}{\displaystyle \frac{\lambda }{(1-\alpha (t_n))}}{e}^{{\displaystyle \frac{-s\alpha (t_n)}{(1-\alpha (t_n))}}}ds=& u(0)\sum _{l=0}^{n-1}{\int }_{t_l}^{t_{l+1}}{\displaystyle \frac{\lambda }{(1-\alpha (t_n))}}{e}^{{\displaystyle \frac{-s\alpha (t_n)}{(1-\alpha (t_n))}}}ds\hfill \\ \hfill =& u(0){\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-t_n\alpha (t_n)}{(1-\alpha (t_n))}}}).\hfill \end{array}$$

(65)

Second, we discretize the third term. Instead of classical SDEs or constant-order FSDEs, its kernel is non-local without a convolution structure. It is difficult to discretize, but to simplify, we first bound the kernel using (A3), then discretize the bounded equation.

$$\begin{array}{cc}\hfill {\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds\le & {\displaystyle \frac{\lambda }{1-{\alpha }^{*}}}{\int }_0^{t_n}u(s)ds={\displaystyle \frac{\lambda }{1-{\alpha }^{*}}}\sum _{l=0}^{n-1}{\int }_{t_l}^{t_{l+1}}u(s)ds\hfill \\ & \approx {\displaystyle \frac{\lambda }{1-{\alpha }^{*}}}{\int }_{t_l}^{t_{l+1}}u(t_l)ds=\frac{\lambda \mathsf{\Delta }t}{1-{\alpha }^{*}}\sum _{l=0}^{n-1}u(t_l)\hfill \end{array}$$

$$\begin{array}{c}\hfill ={\displaystyle \frac{\lambda \tau }{1-{\alpha }^{*}}}\sum _{l=0}^{n-1}u(t_l).\end{array}$$

(66)

Next, let us discretize the fourth term:

$$\begin{array}{cc}\hfill {\int }_0^{t}f(u(s))ds=& \sum _{l=0}^{n-1}{\int }_{t_l}^{t_{l+1}}f(u(s))ds\hfill \\ & \approx {\int }_{t_l}^{t_{l+1}}f(u(t_l))ds=\tau \sum _{l=0}^{n-1}f(u(t_l)).\hfill \end{array}$$

(67)

Finally, let us discretize the fifth term:

$$\begin{array}{cc}\hfill {\int }_0^{t}b(u(s))dW=& \sum _{l=0}^{n-1}{\int }_{t_l}^{t_{l+1}}b(u(s))dW(s)\hfill \\ & \approx \sum _{l=0}^{n-1}{\int }_{t_l}^{t_{l+1}}b(u(t_l))dW(s)=\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l,\hfill \end{array}$$

(68)

where $\mathsf{\Delta }{W}_l\sim N(0,\tau )$ and $\sim N(0,\tau )$ is a Gaussian random variable with zero mean and variance $\tau$.

The Euler–Maruyama scheme of (2) is built by incorporating discretizations (65) through (68) and (11), from which we define recursively $v_n$, for $n\in \{1,2,\dots ,N\}$ and the initial value $v_0:=u_0$, by setting

$$\begin{array}{cc}\hfill v_n=& u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-t_n\alpha (t_n)}{(1-\alpha (t_n))}}})+{\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l\hfill \\ \hfill & \approx u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-\alpha (t_n)(t_n)}{(1-\alpha (t_n))}}})+{\displaystyle \frac{\lambda \tau }{1-{\alpha }^{*}}}\sum _{l=0}^{n-1}u(t_l)\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l.\hfill \end{array}$$

(69)

The Euler–Maruyama approximation (EMA) gives us the closest values for the solution of the underlying problem, but this is done only at the grid points, hence it makes the EMA $v_n$ for $n\in \{1,2,\dots ,N\}$ much easier to implement in the numerical aspect of the problem. We are now ready to introduce the following result.

Theorem 3.

Under Assumptions (3)–(7), for each $n\in \{1,2,3,\dots ,N\}$, the solution of (69), $v_n$, satisfies the estimate

$$\begin{array}{c}\hfill \mathbb{E}|v_n^2|\le Q_1(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))),\end{array}$$

(70)

where the constants $Q_1$ and $Q_2$ are given by

$$\begin{array}{c}\hfill Q_1=5\mathbb{E}[u_0^2](1+{({\displaystyle \frac{\lambda }{{\alpha }^{*}}})}^2)+10L^{2}T(T+1),\\ \hfill Q_2=10L^{2}TN(T+1)+5N{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2.\end{array}$$

(71)

Proof. 

Using Jensen’s inequality with $m=5$ along with the Cauchy inequality, we can bound the solution of (69), $v_n$, as follows:

$$\begin{array}{cc}\hfill \mathbb{E}|v_n^2|\le & 5\mathbb{E}|u_0^2|+5\mathbb{E}{(u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-\alpha (t_n)(t_n)}{(1-\alpha (t_n))}}}))}^2+5\mathbb{E}{({\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds)}^2\hfill \\ & +5{\tau }^2\mathbb{E}(\sum _{l=0}^{n-1}f{(u(t_l))}^2+5\mathbb{E}{(\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l)}^2.\hfill \end{array}$$

(72)

Let us bound the second term of (72), so we have

$$\begin{array}{cc}\hfill 5\mathbb{E}[(u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}{(1-e^{{\displaystyle \frac{-\alpha (t_n)(t_n)}{(1-\alpha (t_n))}}})}^2]& \le 5\mathbb{E}[{(u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}})}^2]\hfill \\ & \le 5{({\displaystyle \frac{\lambda }{{\alpha }^{*}}})}^2\mathbb{E}[{(u_0)}^2].\hfill \end{array}$$

(73)

Next, let us bound the fourth term of (72) in a similar fashion as in [33]. We apply Cauchy’s inequality and use Assumptions (3) through (8) to get

$$\begin{array}{cc}\hfill 5{\tau }^2\mathbb{E}[{(\sum _{l=0}^{n-1}f(v_l))}^2]\le & 10L^2{T}^2\tau \sum _{l=0}^{n-1}\mathbb{E}[1+v_l^2]\hfill \\ & \le 10L^2{T}^2+10L^{2}T\tau \sum _{l=0}^{n-1}\mathbb{E}[v_l^2]\hfill \\ & \le 10L^2{T}^2+10L^2{T}^2\sum _{l=0}^{n-1}\mathbb{E}[v_l^2].\hfill \end{array}$$

(74)

Now we estimate the fifth term of (72) in a similar fashion as in [33]. We use Ito’s isometry and Assumptions (3) to (8) to obtain the desired bound as follows:

$$\begin{array}{cc}\hfill 5\mathbb{E}[{(\sum _{l=0}^{n-1}b(v_l)\mathsf{\Delta }{w}_l)}^2]=& 5\tau \sum _{l=0}^{n-1}\mathbb{E}[b{(v_l)}^2]\le 10L^2\tau \sum _{l=0}^{n-1}\mathbb{E}[1+v_l^2]\hfill \\ & \le 10L^{2}T+10L^2\tau \sum _{l=0}^{n-1}\mathbb{E}[v_l^2]\hfill \\ & \le 10L^{2}T+10L^{2}T\sum _{l=0}^{n-1}\mathbb{E}[v_l^2].\hfill \end{array}$$

(75)

We will now examine the estimate of the third component of (72), which is unfamiliar for numerical approximations of conventional SDEs. Using Assumptions (3) to (8) and relation (66), we obtain the following:

$$\begin{array}{cc}\hfill 5\mathbb{E}[{({\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds)}^2]\le & 5\mathbb{E}({({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2{(\sum _{l=0}^{n-1}u(t_l))}^2)\hfill \\ & =5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2\sum _{l=0}^{n-1}\mathbb{E}{(u(t_l))}^2\hfill \\ & \le 5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2\sum _{l=0}^{n-1}\mathbb{E}[v_l^2].\hfill \end{array}$$

(76)

Combining approximations (73), (74), (75), and (76) into (72) and applying the inequality ${(n-l)}^{1-{\alpha }^{*}}\le N^{1-{\alpha }^{*}}$ to infer that

$$\begin{array}{cc}\hfill \mathbb{E}[v_n^2]& \le 5\mathbb{E}[u_0^2]+5{({\displaystyle \frac{\lambda }{{\alpha }^{*}}})}^2\mathbb{E}[{(u_0)}^2]+10L^2{T}^2+10L^2{T}^2\sum _{l=0}^{n-1}\mathbb{E}[v_l^2]+10L^{2}T\hfill \\ \hfill & +10L^{2}T\sum _{l=0}^{n-1}\mathbb{E}[v_l^2]+5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2\sum _{l=0}^{n-1}\mathbb{E}[v_l^2]\hfill \\ \hfill & =Q_1+(10L^{2}T(T+1)+5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2){\displaystyle \frac{N^{{\alpha }^{*}}}{N^{{\alpha }^{*}}}}\sum _{l=0}^{n-1}{\displaystyle \frac{\mathbb{E}[v_l^2]{(n-l)}^{1-{\alpha }^{*}}}{{(n-l)}^{1-{\alpha }^{*}}}}\hfill \\ \hfill & \le Q_1+(10L^{2}T(T+1)+5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2){\displaystyle \frac{N^{{\alpha }^{*}}}{N^{{\alpha }^{*}}}}\sum _{l=0}^{n-1}{\displaystyle \frac{\mathbb{E}[v_l^2]{(N)}^{1-{\alpha }^{*}}}{{(n-l)}^{1-{\alpha }^{*}}}}\hfill \\ \hfill & =Q_1+(10L^{2}T(T+1)+5{({\displaystyle \frac{\lambda T}{1-{\alpha }^{*}}})}^2){\displaystyle \frac{N}{N^{{\alpha }^{*}}}}\sum _{l=0}^{n-1}{\displaystyle \frac{\mathbb{E}[v_l^2]}{{(n-l)}^{1-{\alpha }^{*}}}}\hfill \\ \hfill & =Q_1+{\displaystyle \frac{Q_2}{N^{{\alpha }^{*}}}}\sum _{l=0}^{n-1}{\displaystyle \frac{\mathbb{E}[v_l^2]}{{(n-l)}^{1-{\alpha }^{*}}}},\hfill \end{array}$$

(77)

where $Q_1$ and $Q_2$ are given as in (71).

By applying the result from Lemma A4 and using (77), we deduce that

$$\begin{array}{c}\hfill \mathbb{E}[v_n^2]\le Q_1(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))).\end{array}$$

Hence, this concludes the proof of the desired result stated in this theorem. □

For us to study the strong convergence, we first fix $(\mathsf{\Omega },\mathcal{F},\mathbb{F},\mathbb{P})$, the filtered probability space with the complete filtration satisfying the usual conditions, $\mathbb{F}$. Subsequently, we assume Assumptions (3)–(8) hold.

3.2. Auxiliary Equation and Its Error Estimates

In order to study the strong convergence of the Euler–Maruyama scheme, we construct an equivalent form of the extension of the Euler–Maruyama approximation, which is a new continuous-time stochastic process $v(t)$ on $[0,T]$ based on the step function $\widehat{s}=\widehat{s}(s)$. Suppose $\widehat{s}=t_n$ for $t_n\le s\le t_{n+1}$ and $n\in \{0,1,2,\dots ,N-1\}$, hence the equivalent form is given by

$$\begin{array}{cc}\hfill v(t)=& u(0)+u(0){\int }_0^t{\displaystyle \frac{\lambda }{(1-\alpha (s))}}{e}^{{\displaystyle \frac{-\alpha (s)\widehat{s}}{(1-\alpha (s))}}}ds+{\int }_0^t\mathcal{K}(t,s)v(\widehat{s})ds\hfill \end{array}$$

$$\begin{array}{c}\hfill +{\int }_0^{t}f(v(\widehat{s}))ds+{\int }_0^{t}b(v(\widehat{s}))dW\mathrm{for}\mathrm{all}t\in [t_n,t_{n+1})\mathrm{a}.\mathrm{s}.\end{array}$$

(78)

Lemma 2.

Let ${\left\{v_n\right\}}_{n=0}^N$ be the solution of the Euler–Maruyama scheme and v be the continuous-time stochastic process defined by (78), then $\left\{v(t_n)\right\}=v_n$ for $0\le n\le N$.

Proof. 

The proof of this result is a simple follow up of the result in Lemma 2 of Zhiwei et al. [33], where a detailed description of the proof is given. □

The aim of this section is to prove the convergence. We start by stating that the p-moment of the Euler–Maruyama approximation can also be estimated using the same techniques similar to the previous result. For this purpose, let $N\ge 1$ be a fixed natural integer and set $\mathsf{\Delta }={\displaystyle \frac{T}{N}}$ and $0<t_1<t_2<\cdots <t_N:=T$, where ${\tau }_n:=n\mathsf{\Delta }$ for $n\in \{1,2,\dots ,N\}$. Now we can state the following result.

Lemma 3.

For $n\in \{0,1,2,\dots ,N\}$ and N fixed, let $v_n$ be the Euler–Maruyama approximation at time $t_n$ given in

$$\begin{array}{cc}\hfill v_n=& u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-t_n\alpha (t_n)}{(1-\alpha (t_n))}}})+{\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l\hfill \\ \hfill & \approx u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-\alpha (t_n)(t_n)}{(1-\alpha (t_n))}}})+{\displaystyle \frac{\lambda \tau }{1-{\alpha }^{*}}}\sum _{l=0}^{n-1}u(t_l)\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l.\hfill \end{array}$$

(79)

Then, there exists a finite constant $C>0$ such that

$$\begin{array}{c}\hfill \mathbb{E}(|v_n{|}^p)\le C\mathit{for}\mathit{all}n\ge 1\mathit{and}p\ge 1.\end{array}$$

(80)

Proof. 

For $n=0$, we have $v_0:=u_0$ and, since $\mathbb{E}(|u_0{|}^p)<\infty$ for any $p\ge 1$ by hypothesis, we can directly assert and verify that $\mathbb{E}(|v_0{|}^p)<\infty$. It is enough to show using mathematical induction techniques that

$$\mathbb{E}(|v_n{|}^p)<\infty$$

which will follow from

$$\mathbb{E}(|v_{n+1}{|}^p)<\infty .$$

For this purpose, we recall the Euler–Maruyama approximation at time $t_n$ given in

$$\begin{array}{cc}\hfill v_n=& u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-t_n\alpha (t_n)}{(1-\alpha (t_n))}}})+{\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l\hfill \\ \hfill & \approx u_0+u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-\alpha (t_n)(t_n)}{(1-\alpha (t_n))}}})+{\displaystyle \frac{\lambda \tau }{1-{\alpha }^{*}}}\sum _{l=0}^{n-1}u(t_l)\hfill \\ \hfill & +\tau \sum _{l=0}^{n-1}f(u(t_l))+\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l.\hfill \end{array}$$

(81)

since $v(t_n)=v_n$.

Therefore, we first fix $n\ge 1$ and we assume that

$$\mathbb{E}(|v_n{|}^p)<\infty .$$

Having said that, to carry on with the required proof, it is sufficient to raise relation (81) to the power p and using the assumptions on f and g along with applying Jensen’s inequality with $m=5$; we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}(|v_{n+1}{|}^p)\hfill \\ \hfill & \le 5^{p-1}\mathbb{E}{|v_0|}^p+5^{p-1}\mathbb{E}{\left|u_0{\displaystyle \frac{\lambda }{\alpha (t_n)}}(1-e^{{\displaystyle \frac{-t_n\alpha (t_n)}{(1-\alpha (t_n))}}})\right|}^p+5^{p-1}\mathbb{E}{\left|{\int }_0^{t_n}\mathcal{K}(t,s)u(s)ds\right|}^p\hfill \\ \hfill & +5^{p-1}\mathbb{E}{\left|\tau \sum _{l=0}^{n-1}f(u(t_l))\right|}^p+5^{p-1}\mathbb{E}{\left|\sum _{l=0}^{n-1}b(u(t_l))\mathsf{\Delta }{W}_l\right|}^p\hfill \\ \hfill & \le 5^{p-1}\mathbb{E}|v_0{|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(\alpha (t_n))}^p}}{\displaystyle \frac{1}{{(1-{\alpha }^{*})}^p}}\mathbb{E}{|v_0|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}{\mathsf{\Delta }}^p\mathbb{E}{\left|u(t_n)\right|}^p\hfill \\ \hfill & +5^{p-1}{\mathsf{\Delta }}^p{\tau }^p\mathbb{E}{\left|f(u(t_n))\right|}^p+5^{p-1}{\mathsf{\Delta }}^p\mathbb{E}{\left|b(u(t_n)\right|}^p{|\mathsf{\Delta }{W}_l|}^p\hfill \\ \hfill & \le 5^{p-1}\mathbb{E}|v_0{|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(\alpha (t_n))}^p}}{\displaystyle \frac{1}{{(1-{\alpha }^{*})}^p}}\mathbb{E}{|v_0|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}{\mathsf{\Delta }}^p\mathbb{E}{\left|u(t_n)\right|}^p\hfill \\ \hfill & +5^{p-1}{\mathsf{\Delta }}^p{\tau }^p{C}_2\mathbb{E}\left[1+{\left|u(t_n)\right|}^p\right]+5^{p-1}{\mathsf{\Delta }}^p{C}_3\mathbb{E}\left[1+{\left|u(t_n)\right|}^p\right]\mathbb{E}{\left[|\mathsf{\Delta }{W}_l{|}^{2p}\right]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(82)

By virtue of BDG’s inequality, we assert that there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \mathbb{E}|\mathsf{\Delta }{W}_l{|}^{2p}\le c_p{\mathsf{\Delta }}^p\le c_p{T}^p,\end{array}$$

(83)

from which we infer that

$$\begin{array}{cc}\hfill \mathbb{E}(|v_{n+1}{|}^p)& \le 5^{p-1}\mathbb{E}|v_0{|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(\alpha (t_n))}^p}}{\displaystyle \frac{1}{{(1-{\alpha }^{*})}^p}}\mathbb{E}{|v_0|}^p+5^{p-1}{\displaystyle \frac{{\lambda }^p}{{(1-{\alpha }^{*})}^p}}{\mathsf{\Delta }}^p\mathbb{E}{\left|v_n\right|}^p\hfill \\ \hfill & +5^{p-1}{\mathsf{\Delta }}^p{\tau }^p{C}_2\mathbb{E}\left[1+{\left|v_n\right|}^p\right]+5^{p-1}{\mathsf{\Delta }}^p{C}_3\mathbb{E}\left[1+{\left|v_n\right|}^p\right]C{T}^{{\displaystyle \frac{p}{2}}}.\hfill \end{array}$$

(84)

Therefore, we deduce that

$$\begin{array}{c}\hfill \mathbb{E}(|v_{n+1}{|}^p)<\infty \mathrm{for}\mathrm{any}n\ge 1\mathrm{and}p\ge 1.\end{array}$$

(85)

This finally proves the desired result. □

We are now ready to state and prove the following theorem concerning the convergence of the Euler–Maruyama approximating scheme.

Theorem 4.

The following estimate holds for the continuous time stochastic process v defined by (78) for any $t\in [t_n,t_{n+1})$ and $n\in \{0,1,2,\dots ,N-1\}$. Then

$$\begin{array}{c}\hfill \mathbb{E}[{(v(t)-v(t_n))}^2]\le M_1+M_2.\\ \hfill \mathit{Here} \mathit{the} \mathit{constants} M_1\mathit{and} M_2 \mathit{are} \mathit{given} \mathit{by} \\ \hfill M_1=6L^2\tau Q_1(1+\tau ) \mathit{and}\\ \hfill M_2=6\tau Q_1[L^2(1+\tau )+5{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2][1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))].\end{array}$$

(86)

Proof. 

Due to relations (78) and (11) at time $t_n$, for $t\in [t_n,t_{n+1})$ and $n\in [0,N-1]$, we implement Jensen’s inequality at $m=3$ to obtain

$$\begin{array}{cc}\hfill & {\left({\int }_a^{c}fds-{\int }_a^{b}gds\right)}^2={\left({\int }_a^b(f-g)ds+{\int }_b^{c}fds\right)}^2,\mathrm{for}a<b<c.\hfill \\ \hfill \mathrm{Hence}\\ \hfill & {\left({\int }_a^b(f-g)ds+{\int }_b^{c}fds\right)}^2\le 2{\left({\int }_a^b(f-g)ds\right)}^2+2{\left({\int }_b^{c}fds\right)}^2\hfill \end{array}$$

to obtain

$$\begin{array}{cc}\hfill \mathbb{E}[{(v(t)-v(t_n))}^2]\le & 3\mathbb{E}{\left({\int }_0^t\mathcal{K}(t,s)v(\widehat{s})ds-{\int }_0^{t_n}\mathcal{K}(t_n,s)v(\widehat{s})ds\right)}^2\hfill \\ \hfill & +3\mathbb{E}{\left({\int }_0^{t}f(v(\widehat{s}))ds-{\int }_0^{t_n}f(v(\widehat{s}))ds\right)}^2\hfill \\ \hfill & +3\mathbb{E}{\left({\int }_0^{t}b(v(\widehat{s}))dW(s)-{\int }_0^{t_n}b(v(\widehat{s}))dW(s))\right)}^2\hfill \\ \hfill & \le 6\mathbb{E}{\left({\int }_0^{t_n}(\mathcal{K}(t,s)-\mathcal{K}(t_n,s))v(\widehat{s})ds\right)}^2+6\mathbb{E}{\left({\int }_{t_n}^t\mathcal{K}(t,s)v(\widehat{s})ds\right)}^2\hfill \\ \hfill & +3\mathbb{E}{\left({\int }_{t_n}^{t}f(v(\widehat{s}))ds\right)}^2+3\mathbb{E}{\left({\int }_{t_n}^{t}b(v(\widehat{s}))dW(s)\right)}^2.\hfill \end{array}$$

(87)

Next, we bound the third term of (87) using Theorem 3 and Assumptions (3) to (8) to get

$$\begin{array}{cc}\hfill 3\mathbb{E}{\left({\int }_{t_n}^{t}f(v(\widehat{s}))ds\right)}^2& \le 3\tau {\int }_{t_n}^t\mathbb{E}[f{(v(\widehat{s}))}^2]ds\le 6L^2\tau {\int }_{t_n}^t(1+\mathbb{E}[|v_n{|}^2]ds\hfill \\ \hfill & =6L^2{\tau }^2(1+\mathbb{E}[|v_n{|}^2])\le 6L^2{\tau }^2{Q}_1[1+(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*})))].\hfill \end{array}$$

(88)

We employ It$\widehat{o}$ isometry, Theorem 3, and Assumptions (3) to (8) to bound the last term of the right-hand side of (87).

$$\begin{array}{cc}\hfill 3\mathbb{E}{\left({\int }_{t_n}^{t}b(v(\widehat{s}))dW(s)\right)}^2& =3\mathbb{E}{\int }_{t_n}^{t}b{(v(\widehat{s}))}^{2}ds\le 6L^2\tau (1+\mathbb{E}[|v_n{|}^2])\hfill \\ \hfill & \le 6L^2\tau [1+Q_1(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*})))].\hfill \end{array}$$

(89)

The remaining terms of (87) have neither local nor convolutional structures, which are unfamiliar in conventional SDEs and constant-order FSDEs. Let us estimate them one by one. Using the Cauchy inequality and (A3), we will link the second term (87).

$$\begin{array}{cc}\hfill 6\mathbb{E}{\left({\int }_{t_n}^t\mathcal{K}(t_n,s)v(\widehat{s})ds\right)}^2& \le 6{\int }_{t_n}^t{|\mathcal{K}(t_n,s)|}^2\mathbb{E}{[v(\widehat{s})]}^{2}ds\hfill \\ \hfill & \le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2{\int }_{t_n}^t\mathbb{E}{[v(\widehat{s})]}^{2}ds\le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau \mathbb{E}{[v_n]}^2\hfill \\ \hfill & \le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau Q_1[1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))].\hfill \end{array}$$

(90)

Next, we use the Cauchy–Schwarz inequality to bound the first term of (87) from the inequality below:

$$\begin{array}{c}6\mathbb{E}{\left({\int }_0^{t_n}(\mathcal{K}(t,s)-\mathcal{K}(t_n,s))v(\widehat{s})ds\right)}^2\hfill \\ \le 6{\int }_0^{t_n}{|\mathcal{K}(t,s)-\mathcal{K}(t_n,s)|}^2\mathbb{E}{[v_n]}^{2}ds\hfill \\ \le 12{\int }_0^{t_n}({|\mathcal{K}(t,s)|}^2+{|\mathcal{K}(t_n,s)|}^2)\mathbb{E}{[v_n]}^{2}ds\hfill \\ \le 24{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau \mathbb{E}{[v_n]}^2\le 24{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau Q_1[1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))].\hfill \end{array}$$

(91)

Inserting the outcomes of (88) through (91) into (87), we get

$$\begin{array}{cc}\hfill & \mathbb{E}[{(v(t)-v(t_n))}^2]\hfill \\ \hfill & \le 6L^2{\tau }^2{Q}_1[1+(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))]+6L^2\tau [1+Q_1(1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*})))]\hfill \\ & +6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau Q_1[1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))]+24{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2\tau Q_1[1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))]\hfill \\ & =6L^2\tau (Q_1\tau +1)+6\tau Q_1[L^2(1+\tau )+5{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2][1+E_{{\alpha }^{*}}(Q_2\mathsf{\Gamma }({\alpha }^{*}))]\hfill \\ & =M_1+M_2,\hfill \end{array}$$

where $M_1$ and $M_2$ are given in (86). Hence the proof is complete. □

3.3. Error Estimate of the Euler–Maruyama Scheme

In this part, we can estimate the error of the strong convergence of the Euler–Maruyama scheme (69), which is the article’s main contribution. In order to do this, we start by estimating the error of the process $u(t)-v(t)$.

Theorem 5.

Let $uandv$ be the solutions to the variable-order FSDE (2) and the auxiliary (78), respectively. Then, the following estimate holds

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{max}\mathbb{E}[{|u(t)-v(t)|}^2]\le Q_3{E}_{{\alpha }^{*}}[Q_4\mathsf{\Gamma }({\alpha }^{*})t^{{\alpha }^{*}}],\\ \hfill \mathit{where} \mathit{the} \mathit{constants} Q_3\mathit{and} Q_4\mathit{are} \mathit{given} \mathit{by} \\ \hfill Q_3=6T[M_1+M_2]({({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2+L^2(T+1))\mathit{and} \\ \hfill Q_4=6({({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2+L^2(T+1)){\int }_0^t\mathbb{E}{[u(s)-v(s)]}^{2}ds.\end{array}$$

(92)

Proof. 

For any $0\le t\le T$, let $t\in [t_n,t_{n+1})$ for some $n\in [0,N-1)$. Subtract Equation (78) from Equation (11) to obtain

$$\begin{array}{c}\hfill \mathbb{E}[{(u(t)-v(t))}^2]=J_1+J_2+J_3,\end{array}$$

(93)

where

$$\begin{array}{c}\hfill J_1=3\mathbb{E}[{({\int }_0^t\mathcal{K}(t,s)(u(s)-v(\widehat{s}))ds)}^2],\\ \hfill J_2=3\mathbb{E}[({\int }_0^t(f{(u(s)-f(v(\widehat{s}))ds)}^2],\\ \hfill J_3=3\mathbb{E}[{({\int }_0^t(b(u(s))-b(u(\widehat{s}))dW))}^2].\end{array}$$

Let us bound $J_1$ by Lemma A1 and Theorem 4 to obtain

$$\begin{array}{cc}\hfill J_1& =3\mathbb{E}[{({\int }_0^t\mathcal{K}(t,s)(u(s)-v(\widehat{s}))ds)}^2]\hfill \\ & \le 6\mathbb{E}[{({\int }_0^t\mathcal{K}(t,s)(u(s)-v(s))ds)}^2]+6\mathbb{E}[{({\int }_0^t\mathcal{K}(t,s)(v(s)-v(\widehat{s}))ds)}^2]\hfill \\ & \le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2{\int }_0^t\mathbb{E}{[(u(s)-v(s))]}^{2}ds+6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2{\int }_0^t\mathbb{E}[{(v(s)-v(\widehat{s}))}^2]ds\hfill \\ & \le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2{\int }_0^t\mathbb{E}{[(u(s)-v(s))]}^{2}ds+6T{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2[M_1+M_2].\hfill \end{array}$$

(94)

Apply Assumptions (3) to (8), the Cauchy inequality, and Theorem 4 to estimate $J_2$ by the following inequality:

$$\begin{array}{cc}\hfill J_2=& 3\mathbb{E}[({\int }_0^t(f{(u(s)-f(v(\widehat{s}))ds)}^2]\hfill \\ & \le 3T{L}^2[({\int }_0^t\mathbb{E}({(u(s)-(v(\widehat{s}))ds)}^2]\hfill \\ & \le 6T{L}^2({\int }_0^t(\mathbb{E}[{|u(s)-v(s)|}^2]+\mathbb{E}[{|v(s)-v(\widehat{s})|}^2])ds\hfill \\ & =6T{L}^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6T{L}^2{\int }_0^t\mathbb{E}[{|v(s)-v(\widehat{s})|}^2]ds\hfill \\ & \le 6T{L}^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6T^2{L}^2[M_1+M_2].\hfill \end{array}$$

(95)

We use It$\widehat{o}$ isometry, Assumptions (3) to (8), and Theorem 4 and split

$$\begin{array}{c}\hfill u(s)-v(\widehat{s})=(u(s)-v(s))+(v(s)-v(\widehat{s}))\end{array}$$

to bound $J_3$ by

$$\begin{array}{cc}\hfill J_3& =3\mathbb{E}[{({\int }_0^t(b(u(s))-b(u(\widehat{s}))dW))}^2]=3[({\int }_0^t\mathbb{E}(b(u(s))-b{(u(\widehat{s}))}^2{(dW)}^2]\hfill \\ & \le 3L^2{\int }_0^t(\mathbb{E}[{|u(s)-u(\widehat{s})|}^2]ds\le 6L^2{\int }_0^t(\mathbb{E}[{|u(s)-v(s)|}^2]+\mathbb{E}[{|v(s)-u(\widehat{s})|}^2])ds\hfill \\ & =6L^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6L^2{\int }_0^t\mathbb{E}[{|v(s)-u(\widehat{s})|}^2]ds\hfill \\ & \le 6L^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6T{L}^2[M_1+M_2].\hfill \end{array}$$

(96)

Substitute estimates (78), (95), and (96) into (93) to obtain

$$\begin{array}{cc}\hfill \mathbb{E}[{(u(t)-v(t))}^2]& \le 6{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2{\int }_0^t\mathbb{E}{[u(s)-v(s)]}^{2}ds+6T{({\displaystyle \frac{\lambda }{1-{\alpha }^{*}}})}^2[M_1+M_2]\hfill \\ & +6T{L}^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6T^2{L}^2[M_1+M_2]\hfill \\ & +6L^2{\int }_0^t\mathbb{E}[{|u(s)-v(s)|}^2]ds+6T{L}^2[M_1+M_2]\hfill \\ & \le Q_3+Q_4{\int }_0^t\mathbb{E}{[u(s)-v(s)]}^{2}ds\hfill \\ & \le Q_3+Q_4{\int }_0^t{\displaystyle \frac{\mathbb{E}{[u(s)-v(s)]}^2}{{(t-s)}^{1-{\alpha }^{*}}}}ds.\hfill \end{array}$$

(97)

Apply the results of Lemma A3 to complete the proof of the results in Theorem 5 from (92),

$$\begin{array}{c}\hfill \mathbb{E}[{(u(t)-v(t))}^2]\le Q_3{E}_{{\alpha }^{*}}[Q_4\mathsf{\Gamma }({\alpha }^{*})t^{{\alpha }^{*}}].\end{array}$$

Hence, the proof of this theorem is complete. □

4. Numerical Results and Simulations

In this section, we focus on the approximation and numerical methods for the problem under consideration. In this study, we conduct numerical experiments by employing the Euler–Maruyama scheme (69) to assess its efficacy and performance.

4.1. Strong Convergence of the Euler–Maruyama Scheme

In this subsection, we conduct a numerical test that shows the strong convergence of the Euler–Maruyama scheme (69) to (2), along with the calculation of the corresponding errors. For more detailed exposition on the Euler–Maruyama topic, we refer to the work in [33,54]. First, we deal with the numerical part of the noise by setting

$$\begin{array}{c}\hfill \mathsf{\Delta }{\omega }_i=Z_i\sqrt{\mathsf{\Delta }{t}_i}\end{array}$$

(98)

where $Z_i\in N(0,1)$ is a random discrete-time Gaussian process and normally distributed. Note that each set of $\{{\omega }_0,{\omega }_1,\dots \dots ,{\omega }_n\}$ produced by the Euler–Maruyama method is an approximate solution of stochastic process $u(t)$. Moreover, each ${\omega }_t$ is a stochastic process that has an approximated solution that is not unique.

Let $u(t_n,{\omega }_j)$ be the jth independent sample path of the FSDE (2) evaluated at $t_n$ with the numerical approximation $v_n({\omega }_j)$ defined using the Euler–Maruyama scheme (69) for $n=0,1,2,·,N$ and $j=1,2,...,M$. Compute and estimate the mean-square distance (error) between numerical solution $u(t_n,{\omega }_j)$ and reference solution $v_n({\omega }_j)$. The sample mean of the error $e_{\tau }$ and the convergence rate k are defined by

$$\begin{array}{c}\hfill e_{\tau }=\underset{0\le n\le N}{max}{[{\displaystyle \frac{1}{M}}\sum _{j=1}^M{|u(t_n,{\omega }_j)-v_n({\omega }_j)|}^2]}^{{\displaystyle \frac{1}{2}}}\le Q{N}^{-k},\end{array}$$

(99)

where

$$\begin{array}{c}\hfill k={log}_2{\displaystyle \frac{e_{\tau }}{e_{\frac{\tau }{2}}}}.\end{array}$$

(100)

In addition, the variable order $\alpha (t)$ is chosen by

$$\begin{array}{c}\hfill \alpha (t)=\alpha (1)+[\alpha (0)-\alpha (1)][1-t-{\displaystyle \frac{sin(2\pi (1-t))}{2\pi }}].\end{array}$$

(101)

For more details, we refer to [33,55,56].

Next, by illustration, we demonstrate the convergence and stability properties of the methods presented in this paper.

Taking into account all of the preceding formulas, we can now define the structure algorithm:

• Find the reference solution at $N_{ref}=2^{10}=1024$ because the analytical solution is unknown a priori;
• In the numerical experiment, the inputs for the simulation are:
  • the time interval $[0,T]=[0,1]$;
  • $\lambda =1$;
  • $M=2^{10}=1024$;
  • with the initial condition $u(0)=u_0,\mathbb{E}[u^2(0)]=\mathbb{E}[u_0^2]<\infty$;
• Define the corresponding grid $({\omega }_j,\mathsf{\Delta }\omega ,\mathsf{\Delta }t,t_n)$ and notice that
  • $\mathsf{\Delta }t={\displaystyle \frac{T}{N}}\mathrm{and}{t}_n=n\mathsf{\Delta }t,n=0,1,2,\dots \dots ,N$;
  • $\mathsf{\Delta }{\omega }_i=Z_i\sqrt{\mathsf{\Delta }{t}_i},Z_i\in N(0,1)$;
• Compute the variable order $\alpha$ using

  $$\begin{array}{c}\hfill \alpha (t)=\alpha (1)+[\alpha (0)-\alpha (1)][1-t-{\displaystyle \frac{sin(2\pi (1-t))}{2\pi }}];\end{array}$$
• Compute the sample mean of the error $e_{\tau }$ as follows:

  $$\begin{array}{c}\hfill e_{\tau }=\underset{0\le n\le N}{max}{[{\displaystyle \frac{1}{1024}}\sum _{j=1}^{1024}{|u(t_n,{\omega }_j)-v_n({\omega }_j)|}^2]}^{{\displaystyle \frac{1}{2}}}\le Q{N}^{-k};\end{array}$$
• Compute the convergence rate k using (100) and set

  $$\begin{array}{c}\hfill k={log}_2[{\displaystyle \frac{e_{\tau }}{e_{\frac{\tau }{2}}}}].\end{array}$$

Example 1.

For a linear variable-order FSDE, we choose $f(u)=b(u)=-uand{u}_0=0.1$ in the variable-order FSDE (2). We present the error $e_{\tau }$ for different mesh size N and different choices of $\alpha (0)and\alpha (1)$ in (101). We observe that the Euler–Maruyama scheme (69) demonstrates the convergence rates and we note that both are in agreement with the theoretical results obtained in Theorem 5.

In the next examples, we include a one-sided Lipschitz case of nonlinear function f and b in order to illustrate the numerical examples and verify the efficiency of the Euler–Maruyama scheme and the validity of Theorems 1–3.

Example 2.

For a linear variable-order FSDE, we consider a nonlinear function $f(u)=b(u)=sinuand{u}_0=0.1$ in the variable-order FSDE (2). The error $e_{\tau }$ for different mesh size N and difference choices of $\alpha (0)and\alpha (1)$ is presented in (101). We observe that the Euler–Maruyama scheme (69) demonstrates convergence rates and both are in agreement with the theoretical results obtained in Theorem 5.

We present the numerical results in

Table 1. Convergence of the scheme for the linear FSDE in Example 1.

$(\mathit{\alpha }(0),\mathit{\alpha }(1))$	(0.2, 0.1)		(0.6, 0.3)		(0.8, 0.5)
N	$e_{\tau }$	k	$e_{\tau }$	k	$e_{\tau }$	k
8	4.40 $e^{-3}$	0.00	4.73 $e^{-3}$	0	8.77 $e^{-}3$	0
16	4.38 $e^{-3}$	0.0070642	5.04 $e^{-3}$	0.0924739	9.49 $e^{-3}$	0.1148251
32	4.33 $e^{-3}$	0.0141644	5.12 $e^{-3}$	0.0222736	9.72 $e^{-3}$	0.0334769
64	4.25 $e^{-3}$	0.0284441	4.96 $e^{-3}$	0.0445478	9.39 $e^{-3}$	0.0494342
128	4.08 $e^{-3}$	0.0570356	4.47 $e^{-3}$	0.1524107	8.22 $e^{-3}$	0.1926919
256	3.78 $e^{-3}$	0.1109219	3.97 $e^{-3}$	0.1714552	8.22 $e^{-3}$	0.00
$min(1-{\alpha }^{*},0.5)$		0.5		0.4		0.2

and

Table 2. Convergence of scheme for the nonlinear FSDE in Example 2.

$(\mathit{\alpha }(0),\mathit{\alpha }(1))$	(0.2, 0.1)		(0.6, 0.3)		(0.8, 0.5)
N	$e_{\tau }$	k	$e_{\tau }$	k	$e_{\tau }$	k
16	4.38 $e^{-2}$	0.009004	5.04 $e^{-2}$	0.08139	9.49 $e^{-2}$	0.1234129
32	4.33 $e^{-4}$	0.00900	5.12 $e^{-3}$	0.02227360	9.72 $e^{-4}$	0.0231000
64	6.25 $e^{-2}$	0.03411000	4.96 $e^{-3}$	0.0445478	9.39 $e^{-1}$	0.0494342
128	4.08 $e^{-2}$	0.32949910	4.47 $e^{-3}$	0.1524107	8.22 $e^{-3}$	0.1926919
$min(1-{\alpha }^{*},0.5)$		0.5		0.4		0.2

to see the observations as in Example 1 and Example 2, respectively.

Since the above results for the nonlinear are not closely related, we can compute the sample paths many times with higher sample size to conclude the outcomes for this specific exam. For the following two examples, we assert the given functions satisfy the assumptions in the above theorems, therefore the result holds for these cases.

Example 3.

In this example, we investigate the strong convergence and the convergence rate for the SFPDEs on the interval $[0,1]$. Along with the theorems mentioned above, we also verify the one-sided Lipschitz conditions on both nonlinear external terms, f and g. For a nonlinear variable-order FSDE, we choose $f(u)=e^{-u}$ and $b(u)=u+e^{-u}$ and $u_0=0.1$ in the variable-order FSDE (2).

$$du=(-{\lambda }_0^C{D}_t^{\alpha }u+e^{-u}dt+(u+e^{-u})dW,t\in [0,T],u(0)=u_0.$$

We present the error $e_{\tau }$ for different mesh size N and different choices of $\alpha (0)and\alpha (1)$ in (101). Again, we observe that the Euler–Maruyama scheme (69) demonstrates the convergence rates that are in agreement with the theoretical analysis in Theorem 5.

Another example we can include is the following case.

Example 4.

In this example, we investigate the strong convergence and the convergence rate for the SFPDEs on the interval $[0,1]$. Along with the theorems mentioned above, we also verify the one-sided Lipschitz conditions on both nonlinear external terms, f and g. For a nonlinear variable-order FSDE, consider $f(u)={\displaystyle \frac{{\gamma }_1}{u}}-{\gamma }_{2}u$, where ${\gamma }_1$ and ${\gamma }_2$ are Lambert constants that can be determined, and $b(u)=u+\sqrt{u}$ and $u_0=0.1$ in the variable-order FSDE (2).

$$du=-\lambda {}_0{}^{C}D_t^{\alpha }u+({\displaystyle \frac{{\gamma }_1}{u}}-{\gamma }_{2}u)dt+(u+\sqrt{u})dW,t\in [0,T],u(0)=u_0.$$

We present the error $e_{\tau }$ for different mesh size N and different choices of $\alpha (0)and\alpha (1)$ in (101). Again, we observe that the Euler–Maruyama scheme (69) demonstrates the convergence rates and we note that both are in agreement with the theoretical analysis in the result obtained in Theorem 5.

4.2. Performance of the Variable-Order FSDE

We compare the performance of the variable-order FSDE (2), for $\lambda =0$, with the standard SDE. In this section of the numerical experiment, we used the same time interval and other variables as in the previous section. For computing various combinations of solutions of ($\alpha (0),\alpha (1)$), we use the Euler–Maruyama scheme (69) with a mesh size $N=2^{10}$.

In Figure 1 and Figure 2 below, we show the results for $(\alpha (0),\alpha (1))=(0.2,0.1)\mathrm{and}(\alpha (0),\alpha (1))=(0.8,0.5)$, respectively. Each figure shows the plots with values between $0.1\mathrm{and}1$. We note that the FSDE solution (2) and the standard SDE solution are nearly identical for $\lambda <1$ (here $\lambda =0.1$ in this case). In contrast to $\lambda =1$, the FSDE solution (2) exhibits a considerable difference from the SDE solution.

5. Concluding Remarks

In this article, we investigate a non-Lipschitz non-singular kernel variable order fractional stochastic differential equation subjected to a nonlinear multiplicative white noise that loses the convolution structure due to the variable order fractional differential operator. In addition to the well-posedness and moment estimates of the problem, we developed and proved a generalized Euler–Maruyama scheme and showed the strong convergence of the EM approximation scheme.

In particular, the strong convergence results obtained in this paper had the same result as the classical stochastic differential equations when taking $\alpha =0$ and $\alpha =1$ in the variable-order FSDE; we obtained the results related to the classical version of the problem under consideration.

To support the theoretical findings, we conducted numerical experiments. In particular, when the variable-order FSDE is $\alpha =0$ and $\alpha =1$, the behavior of the obtained convergence resembles the well-known convergence rates of the Euler–Maruyama scheme for conventional SDEs.

There are future perspectives and research potential directions which are mainly inspired from the need of sound mathematical analysis to extend the results obtained in this paper. We will mainly be interested in the question of global regularity and asymptotic expansion of the solutions for these SFPDEs. Additionally, many other open problems in this direction, such as the invariant measure and the ergodicity of the associated solutions, are crucial for future perspective work. The long-time behavior of a fractional stochastic system is of great relevance in long-term prediction of weather/climate or natural phenomena. Investigating the well-posedness of the the SFPDEs driven by Gaussian jumps and Levy processes and with non-regular intensity is still an open problem that can be addressed in future work.