The Existence and Averaging Principle for a Class of Fractional Hadamard Itô–Doob Stochastic Integral Equations

Abstract

In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. To accomplish this, we harness powerful mathematical tools, including the Hölder and Gronwall inequalities.

1. Introduction

The theory of fractional calculus is an ancient subject that appeared in the same century as classical calculus. The question of fractional derivatives was raised as early as 1695 by Leibniz in a letter to l’Hospital, but when Leibniz asked what the derivative of a function means when the order is $\frac{1}{2}$, Leibniz replied that this leads to a paradox. This letter to l’Hospital was considered the first incident of fractional calculus theory. Later, Euler’s functions made the definition of fractional operators possible. However, a systematic development of the subject did not occur until the nineteenth century with Riemann, Liouville, Grünwald, and Letnikov (see [1,2,3]).

Fractional calculus has practical applications across a range of disciplines, including biology, machine learning, and physics. A notable example is found in the study of anomalous diffusion phenomena, where fractional variants of the diffusion equation serve as indispensable tools for investigation (as highlighted in references such as [4,5,6]). These applications underscore the versatility and potency of fractional calculus in unraveling intricate phenomena that conventional calculus may struggle to address effectively.

Caputo–Hadamard and Hadamard derivatives are crucial classes of fractional derivatives. The scientific community has extensively studied Hadamard fractional integral equations (HFIE); for further details, readers can consult [7,8,9]. The FHIDSIE represents a significant category within HFIE, drawing the attention of numerous researchers (see [10,11]). In [12], the authors analyzed the existence and uniqueness of solutions using PIT and the averaging principle of FHIDSIE.

Inspired by the analysis above, this paper investigates the averaging principle for FHIDSIE. In the literature, most published articles that study the averaging principle focus on ordinary stochastic differential equations. In this sense, this paper extends the results in [12] to the neutral delay FHIDSIE. Therefore, in this paper, our main aims are:

• Analyze the existence and uniqueness of the solution of FHIDSIE by employing the PIT;
• Investigate the averaging principle of FHIDSIE using the famous stochastic calculus techniques.

The structure of this article is summarized as follows. Section 2 is devoted to several definitions and classical results. In Section 3, we prove the existence and uniqueness result using the PIT. In Section 4, we present two theorems related to demonstrating the averaging principle of FHIDSIE. Section 5 is dedicated to showcasing an application related to the studied problem. Finally, we provide a conclusion in Section 6.

2. Basic Notions

Let $\mathcal{D}=\{\mathcal{N},\mathfrak{Z},{({\mathfrak{Z}}_{\varkappa })}_{\varkappa \ge 1},\mathfrak{P}\}$ be a complete probability space and $\mathcal{W}(\varkappa )$ be a standard Brownian motion with respect the space $\mathcal{D}$. Denote by $C([1-\varphi ,1];{\mathbb{R}}^n)$ the family of functions $\phi$ from $[1-\varphi ,1]$ to ${\mathbb{R}}^n$ that are right-continuous and have limits on the left. $C([1-\varphi ,1];{\mathbb{R}}^n)$ is equipped with the norm $\parallel \phi \parallel ={sup}_{1-\varphi \le r\le 1}\left|\phi (r)\right|$ and $\left|y\right|=\sqrt{y^{T}y}$ for any $y\in {\mathbb{R}}^n$. Denote by $L_{{\mathfrak{F}}_{\varkappa }}^2([1-\varphi ,1];{\mathbb{R}}^n)$, $\varkappa \ge 1$ the set of all ${\mathfrak{F}}_{\varkappa }$-measurable, $C([1-\varphi ,1];{\mathbb{R}}^n)$-valued random variables $\phi =\{\phi (\tau ):1-\varphi \le \tau \le 1\}$ satisfying $\mathbb{E}{sup}_{1-\varphi \le \tau \le 1}{\left|\phi (\tau )\right|}^2<\infty$.

Definition 1. 

The Hadamard fractional integral of order ς for a function h is defined as

$$I^{\varsigma }h(\vartheta )=\frac{1}{\Gamma (\varsigma )}{\int }_1^{\vartheta }{\left(log\frac{\vartheta }{\nu }\right)}^{\varsigma -1}\frac{h(\nu )}{\nu }d\nu ,\varsigma >0,$$

provided the integral exists.

For a more comprehensive and in-depth understanding of fractional calculus, we highly recommend consulting the authoritative work in [1,13,14], which offers a wealth of detailed insights and thorough explanations on this subject matter.

Let

$$H:C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n,$$

$$b:[1,T]\times C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n,$$

$${\sigma }_1:[1,T]\times C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n,$$

$${\sigma }_2:[1,T]\times C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$$

all be Borel measurable. Consider the following FHIDSIE:

$$\zeta (\varkappa )-H({\zeta }_{\varkappa })=\zeta (1)-H({\zeta }_1)+{\int }_1^{\varkappa }b\left(s,{\zeta }_s\right)ds+{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s\right)d\mathcal{W}(s)+\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_s\right)}{s}ds$$

(1)

with initial condition

$${\zeta }_1=\alpha =\{\alpha (\lambda );1-\varphi \le \lambda \le 1\}\in L_{{\mathfrak{Z}}_1}^2\left([1-\varphi ,1],{\mathbb{R}}^n\right),$$

(2)

$\frac{1}{2}<\iota <1$ and $\varphi >0$.

Take the following hypotheses:

${\mathcal{H}}_1$

: There exists $\underline{K}>0$ that satisfies

$$|b(\rho ,\theta )-b(\rho ,\tilde{\theta }){|}^2\vee |{\sigma }_1(\rho ,\theta )-{\sigma }_1(\rho ,\tilde{\theta }){|}^2\vee |{\sigma }_2(\rho ,{\alpha }_1)-{\sigma }_2(\rho ,\tilde{\theta }){|}^2\le \underline{K}\left|\right|\theta -\tilde{\theta }{\left|\right|}^2,$$

(3)

for all $(\rho ,\theta ,\tilde{\theta })\in [1,T]\times C([1-\varphi ,1],{\mathbb{R}}^n)\times C([1-\varphi ,1],{\mathbb{R}}^n)$.

${\mathcal{H}}_2$

: There exists $K>0$ that satisfies

$${\left|b(\rho ,\theta )\right|}^2\vee |{\sigma }_1{(\rho ,\theta )|}^2\vee |{\sigma }_2{(\rho ,\theta )|}^2\le {K(1+|\left|\theta \right||}^2),$$

(4)

for all $(\rho ,\theta )\in [1,T]\times C([1-\varphi ,1],{\mathbb{R}}^n)$.

${\mathcal{H}}_3$

: Assume that $H(0)=0$ and there is $\underline{\varrho }\in (0,1)$ such that

$$|H(\theta )-H(\tilde{\theta })|\le \underline{\varrho }\left|\right|\theta -\tilde{\theta }\left|\right|,$$

(5)

for all $(\theta ,\tilde{\theta })\in C([1-\varphi ,1],{\mathbb{R}}^n)\times C([1-\varphi ,1],{\mathbb{R}}^n)$.

3. Existence and Uniqueness of Solution

Theorem 1. 

Under assumptions ${\mathcal{H}}_1$–${\mathcal{H}}_3$, there exists a unique solution $\zeta (\varkappa )$ of system (1) that satisfies $\zeta (\varkappa )\in {\mathcal{M}}_2\left([1-\varphi ,T];{\mathbb{R}}^n\right)$.

Before demonstrating Theorem 1, we require the following lemma.

Lemma 1. 

Assume that assumptions ${\mathcal{H}}_2$ and ${\mathcal{H}}_3$ hold. If $\zeta (\varkappa )$ is a solution to Equation (1) with initial condition (2), then

$$\mathbb{E}\left(\underset{1-\varphi \le \varkappa \le T}{sup}{\left|\zeta (\varkappa )\right|}^2\right)\le L_1,$$

(6)

where

$$L_1=\left(1+\frac{4+\underline{\varrho }\sqrt{\underline{\varrho }}}{(1-\sqrt{\underline{\varrho }})(1-\underline{\varrho })}{\mathbb{E}\left|\right|\alpha \left|\right|}^2\right)exp\left[\frac{3K(T-1)\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right)}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}\right].$$

Moreover, $\zeta (\varkappa )\in {\mathcal{M}}_2\left([1-\varphi ,T];{\mathbb{R}}^n\right)$.

Proof. 

For any $n\in {\mathbb{N}}^{*}$, denote by ${\tau }_n=T\wedge inf\{\varkappa \in [1,T];||{\zeta }_{\varkappa }\left|\right|\ge n\}$ a stopping time. It is obvious that ${\tau }_n\uparrow T$ a.s. Let ${\zeta }^n(\varkappa )=\zeta (\varkappa \wedge {\tau }_n)$, for $\varkappa \in [1-\varphi ,T]$. Thus, for $1\le \varkappa \le T$,

$${\zeta }^n(\varkappa )=H({\zeta }_{\varkappa }^n)-H(\alpha )+O^n(\varkappa ),$$

(7)

where

$$O^n(\varkappa )=\alpha (1)+{\int }_1^{\varkappa }b\left(s,{\zeta }_s^n\right)1_{[1,{\tau }_n]}(s)ds+{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^n\right)1_{[1,{\tau }_n]}(s)d\mathcal{W}(s)$$

(8)

$$+\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa \wedge {\tau }_n}{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_s^n\right)}{s}{1}_{[1,{\tau }_n]}(s)ds.$$

By Lemma 2.3 in [15] and assumption ${\mathcal{H}}_3$, we can derive that

$$\begin{array}{ccc}\hfill |{\zeta }^n{(\varkappa )|}^2& \le & \frac{1}{\underline{\varrho }}|H({\zeta }_{\varkappa }^n)-H(\alpha ){|}^2+\frac{1}{1-\underline{\varrho }}{\left|O^n(\varkappa )\right|}^2\hfill \\ & \le & \underline{\varrho }\left|\right|{\zeta }_{\varkappa }^n-{\alpha \left|\right|}^2+\frac{1}{1-\underline{\varrho }}{\left|O^n(\varkappa )\right|}^2\hfill \\ & \le & \sqrt{\underline{\varrho }}\left|\right|{\zeta }_{\varkappa }^n{\left|\right|}^2+\frac{\underline{\varrho }}{1-\sqrt{\underline{\varrho }}}{\left|\right|\alpha \left|\right|}^2+\frac{1}{1-\underline{\varrho }}{\left|O^n(\varkappa )\right|}^2.\hfill \end{array}$$

(9)

Consequently,

$$\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \sqrt{\underline{\varrho }}\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)+\frac{\underline{\varrho }}{1-\sqrt{\underline{\varrho }}}{\mathbb{E}\left|\right|\alpha \left|\right|}^2+\frac{1}{1-\underline{\varrho }}\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O^n(s)\right|}^2\right).$$

It is not hard to see that ${\displaystyle \underset{1-\varphi \le s\le \varkappa }{sup}|{\zeta }^n{(s)|}^2\le {\left|\right|\alpha \left|\right|}^2+\underset{1\le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2}$. Then,

$${\displaystyle \mathbb{E}\underset{1-\varphi \le s\le \varkappa }{sup}|{\zeta }^n{(s)|}^2\le {\mathbb{E}\left|\right|\alpha \left|\right|}^2+\mathbb{E}\underset{1\le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2.}$$

Therefore,

$$\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \sqrt{\underline{\varrho }}\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)+\frac{1+\underline{\varrho }-\sqrt{\underline{\varrho }}}{1-\sqrt{\underline{\varrho }}}{\mathbb{E}\left|\right|\alpha \left|\right|}^2+\frac{1}{1-\underline{\varrho }}\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O^n(s)\right|}^2\right).$$

Hence,

$$\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \frac{1+\underline{\varrho }-\sqrt{\underline{\varrho }}}{{(1-\sqrt{\underline{\varrho }})}^2}{\mathbb{E}\left|\right|\alpha \left|\right|}^2+\frac{1}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O^n(s)\right|}^2\right).$$

(10)

Using the Cauchy–Schwartz inequality and assumption ${\mathcal{H}}_2$, we obtain

$$\begin{array}{ccc}\hfill |O^n{(\varkappa )|}^2& \le & {3\left|\alpha (1)\right|}^2+3(\varkappa -1){\int }_1^{\varkappa }{\left|b\left(s,{\zeta }_s^n\right)\right|}^{2}ds\hfill \\ & +& 3|{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^n\right)1_{[1,{\tau }_n]}{(s)d\mathcal{W}(s)|}^2\hfill \\ & +& 3{\iota }^2\left({\int }_1^{\varkappa }\frac{1}{s^2}{\left(log\frac{\varkappa }{s}\right)}^{2\iota -2}ds\right)\left({\int }_1^{\varkappa }{\left|{\sigma }_2\left(s,{\zeta }_s^n\right)\right|}^{2}ds\right)\hfill \\ & \le & {3\left|\alpha (1)\right|}^2+3K(\varkappa -1){\int }_1^{\varkappa }(1+||{\zeta }_s^n{\left|\right|)}^{2}ds\hfill \\ & +& 3|{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^n\right)1_{[1,{\tau }_n]}{(s)d\mathcal{W}(s)|}^2\hfill \\ & +& \frac{3{\iota }^{2}K}{2\iota -1}{(log\varkappa )}^{2\iota -1}{\int }_1^{\varkappa }(1+||{\zeta }_s^n{\left|\right|)}^{2}ds.\hfill \end{array}$$

(11)

Therefore, applying Theorem $7.2$ (p. 40 in [15]) and hypothesis ${\mathcal{H}}_2$, one can obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O^n(s)\right|}^2\right)& \le & {3\mathbb{E}\left|\right|\alpha \left|\right|}^2+3K(T-1){\int }_1^{\varkappa }(1+\mathbb{E}||{\zeta }_s^n{\left|\right|}^2)ds\hfill \\ & +& 12\mathbb{E}{\int }_1^{\varkappa }{\left|{\sigma }_1\left(s,{\zeta }_s^n\right)\right|}^2{1}_{[1,{\tau }_n]}(s)ds\hfill \\ & +& \frac{3{\iota }^{2}K}{2\iota -1}{(logT)}^{2\iota -1}{\int }_1^{\varkappa }(1+\mathbb{E}||{\zeta }_s^n{\left|\right|}^2)ds\hfill \\ & \le & {3\mathbb{E}\left|\right|\alpha \left|\right|}^2+3K\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right){\int }_1^{\varkappa }(1+\mathbb{E}||{\zeta }_s^n{\left|\right|}^2)ds.\hfill \end{array}$$

(12)

Plugging (12) into (10), we can obtain

$$\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \frac{4+\underline{\varrho }\sqrt{\underline{\varrho }}}{(1-\sqrt{\underline{\varrho }})(1-\underline{\varrho })}{\mathbb{E}\left|\right|\alpha \left|\right|}^2+\frac{3K\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right)}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}{\int }_1^{\varkappa }(1+\mathbb{E}||{\zeta }_s^n{\left|\right|}^2)ds.$$

(13)

Hence,

$$\begin{array}{ccc}\hfill 1+\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)& \le & 1+\frac{4+\underline{\varrho }\sqrt{\underline{\varrho }}}{(1-\sqrt{\underline{\varrho }})(1-\underline{\varrho })}{\mathbb{E}\left|\right|\alpha \left|\right|}^2\hfill \\ & +& \frac{3K\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right)}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}{\int }_1^{\varkappa }\left(1+\mathbb{E}\left(\underset{1-\varphi \le l\le s}{sup}{\left|{\zeta }^n(l)\right|}^2\right)\right)ds.\hfill \end{array}$$

By Gronwall’s inequality, one has

$$1+\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \left(1+\frac{4+\underline{\varrho }\sqrt{\underline{\varrho }}}{(1-\sqrt{\underline{\varrho }})(1-\underline{\varrho })}{\mathbb{E}\left|\right|\alpha \left|\right|}^2\right)exp\left[\frac{3K(T-1)\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right)}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}\right].$$

Consequently,

$$\mathbb{E}\left(\underset{1-\varphi \le s\le \varkappa }{sup}{\left|{\zeta }^n(s)\right|}^2\right)\le \left(1+\frac{4+\underline{\varrho }\sqrt{\underline{\varrho }}}{(1-\sqrt{\underline{\varrho }})(1-\underline{\varrho })}{\mathbb{E}\left|\right|\alpha \left|\right|}^2\right)exp\left[\frac{3K(T-1)\left(T+3+\frac{3{\iota }^{2}K{(logT)}^{2\iota -1}}{2\iota -1}\right)}{(1-\underline{\varrho })(1-\sqrt{\underline{\varrho }})}\right].$$

Therefore, we can deduce the desired result when $n⟶\infty$. □

Proof of Theorem 1:

Proof. 

Uniqueness:

Consider $\zeta (\varkappa )$ and $\overline{\zeta }(\varkappa )$ to be two solutions to Equation (1). Using Lemma 1, $\zeta (\varkappa )$ and $\overline{\zeta }(\varkappa )$ belong to ${\mathcal{M}}_2\left([1-\varphi ,T];{\mathbb{R}}^n\right)$. Thus,

$$\zeta (\varkappa )-\overline{\zeta }(\varkappa )=H({\zeta }_{\varkappa })-H({\overline{\zeta }}_{\varkappa })+O(\varkappa ),$$

(14)

where

$$O(\varkappa )={\int }_1^{\varkappa }\left(b\left(s,{\zeta }_s\right)-b\left(s,{\overline{\zeta }}_s)\right)\right)ds+{\int }_1^{\varkappa }\left({\sigma }_1\left(s,{\zeta }_s\right)-{\sigma }_1\left(s,{\overline{\zeta }}_s)\right)\right)d\mathcal{W}(s)$$

$$+\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left({\sigma }_2\left(s,{\zeta }_s\right)-{\sigma }_2\left(s,{\overline{\zeta }}_s)\right)\right)}{s}ds.$$

By Lemma 2.3 in [15] and assumption ${\mathcal{H}}_3$, we can see that

$$|\zeta (\varkappa )-\overline{\zeta }{(\varkappa )|}^2\le \underline{\varrho }\left|\right|{\zeta }_{\varkappa }-{\overline{\zeta }}_{\varkappa }{\left|\right|}^2+\frac{1}{1-\underline{\varrho }}{\left|O(\varkappa )\right|}^2.$$

Hence,

$$\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|\zeta (s)-\overline{\zeta }(s)|}^2\right)\le \underline{\varrho }\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|\zeta (s)-\overline{\zeta }(s)|}^2\right)+\frac{1}{1-\underline{\varrho }}\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O(s)\right|}^2\right).$$

(15)

Using (15), it yields that

$$\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|\zeta (s)-\overline{\zeta }(s)|}^2\right)\le \frac{1}{{(1-\underline{\varrho })}^2}\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O(s)\right|}^2\right).$$

Moreover, by Hölder’s inequality, Theorem $7.2$ (p. 40 in [15]), and ${\mathcal{H}}_1$, using a similar method as in the proof of Lemma 1, we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{\left|O(s)\right|}^2\right)& \le & 3\underline{K}\left(T+4+\frac{{\iota }^2{(logT)}^{2\iota -1}}{2\iota -1}\right){\int }_1^{\varkappa }\mathbb{E}\left|\right|{\zeta }_s-{\overline{\zeta }}_s{\left|\right|}^{2}ds\hfill \\ & \le & 3\underline{K}\left(T+4+\frac{{\iota }^2{(logT)}^{2\iota -1}}{2\iota -1}\right){\int }_1^{\varkappa }\mathbb{E}\left(\underset{1\le w\le s}{sup}{|\zeta (w)-\overline{\zeta }(w)|}^2\right)ds.\hfill \end{array}$$

Therefore,

$$\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|\zeta (s)-\overline{\zeta }(s)|}^2\right)\le \frac{3\underline{K}\left(T+4+\frac{{\iota }^2{(logT)}^{2\iota -1}}{2\iota -1}\right)}{{(1-\underline{\varrho })}^2}{\int }_1^{\varkappa }\mathbb{E}\left(\underset{1\le w\le s}{sup}{|\zeta (w)-\overline{\zeta }(w)|}^2\right)ds.$$

Using the Gronwall inequality, we can derive that: $\mathbb{E}\left({sup}_{1\le \varkappa \le T}{|\zeta (\varkappa )-\overline{\zeta }(\varkappa )|}^2\right)=0$. Consequently, for all $1-\varphi \le \varkappa \le T$, $\zeta (\varkappa )=\overline{\zeta }(\varkappa )$, a.s.

Existence: Let $\lambda$ be sufficiently small such that

$$\vartheta =\underline{\varrho }+\frac{3\underline{K}\left(\lambda +5+\frac{{\iota }^2{(log(\lambda +1))}^{2\iota -1}}{2\iota -1}\right)\lambda }{1-\underline{\varrho }}<1,$$

(16)

and write $[1,T]={\bigcup }_{k=1}^r[1+(k-1)\lambda ,1+k\lambda ]\bigcup [1+r\lambda ,T]$. We split the proof into two steps:

Step 1: Let ${\zeta }_1^0=\alpha$ and ${\zeta }^0(\varkappa )=\alpha (1)$ for $1\le \varkappa \le 1+\lambda$. For each $n=1,2,3,\dots$, let ${\zeta }_1^n=\alpha$, and we define, using the Picard iterations:

$${\zeta }^n(\varkappa )-H({\zeta }_{\varkappa }^{n-1})=\alpha (1)-H(\alpha )+{\int }_1^{\varkappa }b\left(s,{\zeta }_s^{n-1}\right)ds+{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^{n-1}\right)d\mathcal{W}(s)+\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_s^{n-1}\right)}{s}ds.$$

(17)

It is not hard to see that ${\zeta }^n(·)\in {\mathcal{M}}_2\left([1-\varphi ,1+\lambda ];{\mathbb{R}}^n\right)$. Noting that for $1\le \varkappa \le 1+\lambda$,

$$\begin{array}{ccc}\hfill {\zeta }^1(\varkappa )-{\zeta }^0(\varkappa )& =& {\zeta }^1(\varkappa )-\alpha (1)=H({\zeta }_{\varkappa }^0)-H(\alpha )+{\int }_1^{\varkappa }b\left(s,{\zeta }_s^0\right)ds\hfill \\ & +& {\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^0\right)d\mathcal{W}(s)+\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_s^0\right)}{s}ds.\hfill \end{array}$$

Proceeding as in the proof of the uniqueness, we have

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|{\zeta }^1(\varkappa )-{\zeta }^0(\varkappa )|}^2\right)& \le & \underline{\varrho }\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}\left|\right|{\zeta }_{\varkappa }^0-\alpha {\left|\right|}^2\right)+\frac{3K\left(T+4+\frac{{\iota }^2{(logT)}^{2\iota -1}}{2\iota -1}\right)}{1-\underline{\varrho }}\mathbb{E}{\int }_1^{\varkappa }\left(1+\left|\right|{\zeta }_{\varkappa }^0{\left|\right|}^2\right)d\varkappa \hfill \\ & \le & C_1\hfill \end{array}$$

(18)

where

$$C_1=2\underline{\varrho }{\mathbb{E}\left|\right|\alpha \left|\right|}^2+\frac{3K\left(\lambda +5+\frac{{\iota }^2{(log(\lambda +1))}^{2\iota -1}}{2\iota -1}\right)}{1-\underline{\varrho }}\left(1+{\mathbb{E}\left|\right|\alpha \left|\right|}^2\right)\lambda .$$

Noting that for $n\ge 1$ and $1\le \varkappa \le 1+\lambda$, we have

$$\begin{array}{ccc}\hfill {\zeta }^{n+1}(\varkappa )-{\zeta }^n(\varkappa )& =& H({\zeta }_{\varkappa }^n)-H({\zeta }_{\varkappa }^{n-1})+{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_s^n\right)-b\left(s,{\zeta }_s^{n-1}\right)\right]ds\hfill \\ & +& {\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_s^n\right)-{\sigma }_1\left(s,{\zeta }_s^{n-1}\right)\right]d\mathcal{W}(s)\hfill \\ & +& \iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_s^n\right)-{\sigma }_2\left(s,{\zeta }_s^{n-1}\right)\right]}{s}ds.\hfill \end{array}$$

Proceeding as in the proof of the uniqueness and using (18), we can obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\underset{1\le \varkappa \le 1+\lambda }{sup}{|{\zeta }^{n+1}(\varkappa )-{\zeta }^n(\varkappa )|}^2\right)& \le & \underline{\varrho }\mathbb{E}\left(\underset{1\le \varkappa \le 1+\lambda }{sup}{|{\zeta }^n(\varkappa )-{\zeta }^{n-1}(\varkappa )|}^2\right)\hfill \\ & +& \frac{3\underline{K}\left(\lambda +5+\frac{{\iota }^2{(log(\lambda +1))}^{2\iota -1}}{2\iota -1}\right)}{1-\underline{\varrho }}{\int }_1^{\varkappa }\mathbb{E}\left(\underset{1\le s\le \varkappa }{sup}{|{\zeta }^n(s)-{\zeta }^{n-1}(s)|}^2\right)d\varkappa \hfill \\ & \le & \vartheta \mathbb{E}\left(\underset{1\le \varkappa \le 1+\lambda }{sup}{|{\zeta }^n(\varkappa )-{\zeta }^{n-1}(\varkappa )|}^2\right)\hfill \\ & \le & {\vartheta }^n\mathbb{E}\left(\underset{1\le \varkappa \le 1+\lambda }{sup}{|{\zeta }^1(\varkappa )-{\zeta }^0(\varkappa )|}^2\right)\hfill \\ & \le & C_1{\vartheta }^n.\hfill \end{array}$$

(19)

By (16), we can prove from (19) that there exists a solution ${}^1\zeta$ to Equation (1) on $[1-\varphi ,1+\lambda ]$ in the same way as in the proof of Theorem 3.1 in [12].

Step 2: For $2\le k\le r$ and $\varkappa \in [1+(k-1)\lambda ,1+k\lambda ]$: let ${\zeta }^0(\varkappa )={}^{k-1}\zeta (\varkappa )$ for $1-\varphi \le \varkappa \le 1+(k-1)\lambda$ and ${\zeta }^0(\varkappa )={}^{k-1}\zeta (1+(k-1)\lambda )$ for $1+(k-1)\lambda \le \varkappa \le 1+k\lambda$. For each $n=1,2,3,\dots$, let ${\zeta }^n(\varkappa )={}^{k-1}\zeta (\varkappa )$ for $\varkappa \in [1-\varphi ,1+(k-1)\lambda ]$. We define the following Picard iterations:

$$\begin{array}{ccc}\hfill {\zeta }^n(\varkappa )-H({\zeta }_{\varkappa }^{n-1})& =& \alpha (1)-H(\alpha )+{\int }_1^{1+(k-1)\lambda }b\left(s,{}^{k-1}{\zeta }_s\right)ds+{\int }_1^{1+(k-1)\lambda }{\sigma }_1\left(s,{}^{k-1}{\zeta }_s\right)d\mathcal{W}(s)\hfill \\ & +& \iota {\int }_1^{1+(k-1)\lambda }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{}^{k-1}{\zeta }_s\right)}{s}ds\hfill \\ & +& {\int }_{1+(k-1)\lambda }^{\varkappa }b\left(s,{\zeta }_s^{n-1}\right)ds+{\int }_{1+(k-1)\lambda }^{\varkappa }{\sigma }_1\left(s,{\zeta }_s^{n-1}\right)d\mathcal{W}(s)\hfill \\ & +& \iota {\int }_{1+(k-1)\lambda }^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_s^{n-1}\right)}{s}ds.\hfill \end{array}$$

(20)

Proceeding as in step 1, we obtain the existence of the solution ${}^k\zeta$ on $[1-\varphi ,1+k\lambda ]$.

Repeating this on $[1+r\lambda ,T]$, we can observe that there exists a solution to Equation (1) on the entire interval $[1-\varphi ,T]$, which complete the proof. □

4. Averaging Principle

Take the following standard form of Equation (1):

$$\begin{array}{ccc}\hfill {\zeta }_{ϵ}(\varkappa )& =& \zeta (1)-H({\zeta }_1)+H({\zeta }_{ϵ,\varkappa })+ϵ{\int }_1^{\varkappa }b\left(s,{\zeta }_{ϵ,s}\right)ds+\sqrt{ϵ}{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)d\mathcal{W}(s)\hfill \\ & +& {ϵ}^{\iota }\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)}{s}ds,\hfill \end{array}$$

(21)

where $b,$${\sigma }_1$, and ${\sigma }_2$ satisfy ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, and $ϵ\in (0,{ϵ}_0]$, with ${ϵ}_0\in (0,\frac{1}{2})$, is a fixed number. Using Theorem 1, Equation (21) has a unique global solution ${\zeta }_{ϵ}(\varkappa ),$$\varkappa \in [1-\varphi ,T]$, for any $ϵ\in (0,{ϵ}_0]$.

Suppose that the following hypothesis holds true:

${\mathcal{H}}_4$

: Let the measurable functions $\overline{b}(\zeta ):C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$, $\overline{{\sigma }_1}(\zeta ):C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$ and $\overline{{\sigma }_2}(\zeta ):C([1-\varphi ,1],{\mathbb{R}}^n)\to {\mathbb{R}}^n$ exist, satisfying ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$. For any $(\zeta ,T_1)\in C([1-\varphi ,1],{\mathbb{R}}^n)\times [1,T]$, we have

• ${\displaystyle \frac{1}{T_1}{\int }_1^{T_1}|b(s,\zeta )-\overline{b}{(\zeta )|}^{2}ds\le {\psi }_1(T_1){(1+|\left|\zeta \right||}^2);}$
• ${\displaystyle \frac{1}{T_1}{\int }_1^{T_1}|{\sigma }_1(s,\zeta )-\overline{{\sigma }_1}{(\zeta )|}^{2}ds\le {\psi }_2(T_1){(1+|\left|\zeta \right||}^2);}$
• ${\displaystyle \frac{1}{T_1}{\int }_1^{T_1}|{\sigma }_2(s,\zeta )-\overline{{\sigma }_2}{(\zeta )|}^{2}ds\le {\psi }_3(T_1){(1+|\left|\zeta \right||}^2);}$

with ${\psi }_j(T_1)$ as positive bounded functions satisfying ${\displaystyle \underset{x⟶\infty }{lim}{\psi }_j(x)=0}$ for $j=1,2,3$.

Our objective is to establish that the solution ${\zeta }_{ϵ}(\varkappa )$ can be approximated by the solution $y_{ϵ}(\varkappa )$ of the next equation

$$\begin{array}{ccc}\hfill y_{ϵ}(\varkappa )& =& \zeta (1)-H({\zeta }_1)+H(y_{ϵ,\varkappa })+ϵ{\int }_1^{\varkappa }\overline{b}\left(y_{ϵ,s}\right)ds+\sqrt{ϵ}{\int }_1^{\varkappa }{\overline{\sigma }}_1\left(y_{ϵ,s}\right)d\mathcal{W}(s)\hfill \\ & +& {ϵ}^{\iota }\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\overline{\sigma }}_2\left(y_{ϵ,s}\right)}{s}ds,\hfill \end{array}$$

(22)

for $\varkappa \in [1,T]$.

Now, we will show the first main theorem in this section.

Theorem 2. 

Assume that ${\mathcal{H}}_1$–${\mathcal{H}}_4$ hold. For a given arbitrary small constant ${\pi }_1>0$, there are two constants $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ that satisfy $\forall ϵ\in (0,{ϵ}_1]$,

$$\mathbb{E}\left(\underset{\varkappa \in [1,\frac{L}{ϵ}]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2\right)\le {\pi }_1.$$

Proof. 

Using Lemma 2.3 in [15] and assumption ${\mathcal{H}}_3$, one can derive

$$\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2\le \frac{\mathbb{E}{sup}_{\varkappa \in [1,u]}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )-\left(H({\zeta }_{ϵ,\varkappa })-H(y_{ϵ,\varkappa })\right)|}^2}{{(1-\underline{\varrho })}^2}.$$

(23)

Note that

$$\begin{array}{ccc}\hfill {\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )-\left(H({\zeta }_{ϵ,\varkappa })-H(y_{ϵ,\varkappa })\right)& =& ϵ{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)\right]ds\hfill \\ & +& \sqrt{ϵ}{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right]d\mathcal{W}(s)\hfill \\ & +& \iota {ϵ}^{\iota }{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)\right]}{s}ds.\hfill \end{array}$$

(24)

According to the elementary inequality, one has

$$\begin{array}{ccc}\hfill & & \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )-\left(H({\zeta }_{ϵ,\varkappa })-H(y_{ϵ,\varkappa })\right)|}^2\hfill \\ & & \le {ϵ}^2\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & & +3ϵ\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & & +3{\iota }^2{ϵ}^{2\iota }\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & ={\mathcal{J}}_1+{\mathcal{J}}_2+{\mathcal{J}}_3,\hfill \end{array}$$

(26)

where

$$\begin{array}{ccc}\hfill {\mathcal{J}}_1& =& 3{ϵ}^2\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & \le & 6{ϵ}^2\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_{ϵ,s}\right)-b\left(s,y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & +& 6{ϵ}^2\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,y_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & \le & 6{ϵ}^2{\mathcal{J}}_{11}+6{ϵ}^2{\mathcal{J}}_{12}.\hfill \end{array}$$

(27)

By the Hölder inequality and ${\mathcal{H}}_1$, one has

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{11}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,{\zeta }_{ϵ,s}\right)-b\left(s,y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & \le & (u-1)\mathbb{E}{\int }_1^u{|b\left(s,{\zeta }_{ϵ,s}\right)-b\left(s,y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & \underline{K}u\mathbb{E}{\int }_1^u\left|\right|{\zeta }_{ϵ,s}-y_{ϵ,s}{\left|\right|}^{2}ds\hfill \\ & \le & \underline{K}u{\int }_1^u\mathbb{E}\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^{2}ds.\hfill \end{array}$$

(28)

By ${\mathcal{H}}_4$, one can obtain

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{12}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[b\left(s,y_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)\right]ds\right|}^2\hfill \\ & \le & u\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\int }_1^{\varkappa }{|b\left(s,y_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & u^2\mathbb{E}\underset{\varkappa \in [1,u]}{sup}\frac{1}{\varkappa }{\int }_1^{\varkappa }{|b\left(s,y_{ϵ,s}\right)-\overline{b}\left(y_{ϵ,s}\right)|}^{2}ds.\hfill \end{array}$$

(29)

As in the proof of Theorem 1 in [16], one can derive

$${\mathcal{J}}_{12}\le u^2\left(\underset{\varkappa \in [1,u]}{sup}{\psi }_1(\varkappa )\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).$$

(30)

Therefore,

$$\begin{array}{ccc}\hfill {\mathcal{J}}_1& \le & 6{ϵ}^2\underline{K}u{\int }_1^u\mathbb{E}\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^{2}ds\hfill \\ & +& 6{ϵ}^2{u}^2\left(\underset{\varkappa \in [1,u]}{sup}{\psi }_1(\varkappa )\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\mathcal{J}}_2& =& 3ϵ\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & =& 3ϵ\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[\left({\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_1\left(s,y_{ϵ,s}\right)\right)+\left({\sigma }_1\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & \le & 6ϵ\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_1\left(s,y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & +& 6ϵ\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & \le & 6ϵ{\mathcal{J}}_{21}+6ϵ{\mathcal{J}}_{22}.\hfill \end{array}$$

(32)

By employing Theorem $7.2$ (p. 40 in [15]) and ${\mathcal{H}}_1$, one can obtain

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{21}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_1\left(s,y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & \le & 4\mathbb{E}{\int }_1^u{|{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_1\left(s,y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & 4\underline{K}\mathbb{E}{\int }_1^u\left|\right|{\zeta }_{ϵ,s}-y_{ϵ,s}{\left|\right|}^{2}ds\hfill \\ & \le & 4\underline{K}{\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds.\hfill \end{array}$$

(33)

Using ${\mathcal{H}}_4$, one can derive

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{22}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }\left[{\sigma }_1\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)\right]d\mathcal{W}(s)\right|}^2\hfill \\ & \le & 4\mathbb{E}{\int }_1^u{|{\sigma }_1\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & 4u\mathbb{E}\frac{1}{u}{\int }_1^u{|{\sigma }_1\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_1\left(y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & 4u\left(\underset{\varkappa \in [1,u]}{sup}{\psi }_2(\varkappa )\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).\hfill \end{array}$$

(34)

Consequently,

$$\begin{array}{ccc}\hfill {\mathcal{J}}_2& \le & 24ϵ\underline{K}{\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds\hfill \\ & +& 24ϵu\left(\underset{\varkappa \in [1,u]}{sup}{\psi }_2(\varkappa )\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).\hfill \end{array}$$

(35)

Moreover,

$$\begin{array}{ccc}\hfill {\mathcal{J}}_3& =& 3{\iota }^2{ϵ}^{2\iota }\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \\ & \le & 6{\iota }^2{ϵ}^{2\iota }\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_2\left(s,y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \\ & +& 6{\iota }^2{ϵ}^{2\iota }\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \\ & \le & 6{\iota }^2{ϵ}^{2\iota }{\mathcal{J}}_{31}+6{\iota }^2{ϵ}^{2\iota }{\mathcal{J}}_{32}.\hfill \end{array}$$

(36)

By Hölder’s inequality and ${\mathcal{H}}_1$, one can derive

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{31}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_2\left(s,y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \\ & \le & \underset{\varkappa \in [1,u]}{sup}\left({\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{2\iota -2}\frac{1}{s}ds\right)\mathbb{E}{\int }_1^u{|{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)-{\sigma }_2\left(s,y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & \frac{1}{2\iota -1}{(logu)}^{2\iota -1}\underline{K}\mathbb{E}{\int }_1^u\left|\right|{\zeta }_{ϵ,s}-y_{ϵ,s}{\left|\right|}^{2}ds\hfill \\ & \le & \frac{1}{2\iota -1}{(logu)}^{2\iota -1}\underline{K}{\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds.\hfill \end{array}$$

(37)

Using Hölder’s inequality and ${\mathcal{H}}_4$, one has

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{32}& =& \mathbb{E}\underset{\varkappa \in [1,u]}{sup}{\left|{\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{\left[{\sigma }_2\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)\right]}{s}ds\right|}^2\hfill \\ & \le & \frac{u}{2\iota -1}{(logu)}^{2\iota -1}\mathbb{E}\frac{1}{u}{\int }_1^u{|{\sigma }_2\left(s,y_{ϵ,s}\right)-{\overline{\sigma }}_2\left(y_{ϵ,s}\right)|}^{2}ds\hfill \\ & \le & \frac{u}{2\iota -1}{(logu)}^{2\iota -1}\left({\psi }_3(u)\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).\hfill \end{array}$$

(38)

Therefore,

$$\begin{array}{ccc}\hfill {\mathcal{J}}_3& \le & \frac{6{\iota }^2{ϵ}^{2\iota }}{2\iota -1}{(logu)}^{2\iota -1}\underline{K}{\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds\hfill \\ & +& \frac{6{\iota }^2{ϵ}^{2\iota }u}{2\iota -1}{(logu)}^{2\iota -1}\left({\psi }_3(u)\right)\left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right).\hfill \end{array}$$

(39)

Plugging (31)–(39) into (25), using Lemma 1 and (23), one can derive that

$$\mathbb{E}\underset{\varkappa \in [1,u]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2$$

$$\begin{array}{ccc}\hfill & \le & \frac{1}{{(1-\underline{\varrho })}^2}6ϵ\underline{K}\left(ϵu+4+\frac{{\iota }^2{ϵ}^{2\iota -1}u}{2\iota -1}{(logu)}^{2\iota -1}\right){\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds\hfill \\ & +& 6ϵu\left(ϵu\left(\underset{l\in [1,u]}{sup}{\psi }_1(l)\right)+4\left(\underset{l\in [1,u]}{sup}{\psi }_2(l)\right)+\frac{{\iota }^2{ϵ}^{2\iota -1}u}{2\iota -1}{(logu)}^{2\iota -1}{\psi }_3(u)\right)\hfill \\ & \times & \left(1+\mathbb{E}\underset{\varkappa \in [1-\varphi ,u]}{sup}{\left|y_{ϵ}(\varkappa )\right|}^2\right)\times \frac{1}{{(1-\underline{\varrho })}^2}\hfill \\ & \le & ϵL_3+ϵL_2{\int }_1^u\mathbb{E}\left(\underset{v\in [1,s]}{sup}{|{\zeta }_{ϵ}(v)-y_{ϵ}(v)|}^2\right)ds,\hfill \end{array}$$

(40)

with

$$L_2=\frac{6\underline{K}\left(ϵT+4+\frac{{\iota }^2{ϵ}^{2\iota -1}u}{2\iota -1}{(logT)}^{2\iota -1}\right)}{{(1-\underline{\varrho })}^2},$$

and

$$\begin{array}{ccc}\hfill L_3& =& \frac{6T\left(ϵT{sup}_{l\in [1,T]}{\psi }_1(l)+4{sup}_{l\in [1,T]}{\psi }_2(l)+\frac{{\iota }^2{ϵ}^{2\iota -1}}{2\iota -1}{(logT)}^{2\iota -1}{sup}_{l\in [1,T]}{\psi }_3(l)\right)\left(1+L_1\right)}{{(1-\underline{\varrho })}^2}.\hfill \end{array}$$

(41)

By Gronwall’s inequality, one can obtain

$$\mathbb{E}\left(\underset{\varkappa \in [1,u]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2\right)\le ϵL_3{e}^{ϵL_{2}u}.$$

Therefore, given any number ${\pi }_1>0$, there are two constants $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ that satisfy, for every $ϵ\in (0,{ϵ}_1]$,

$$\mathbb{E}\left(\underset{\varkappa \in [1,\frac{L}{ϵ}]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2\right)\le {\pi }_1.$$

□

Next, we show the convergence in probability between ${\zeta }_{ϵ}(\varkappa )$ and $y_{ϵ}(\varkappa )$.

Theorem 3. 

Suppose that the FHIDSIE (21) and (22) both satisfy ${\mathcal{H}}_1$–${\mathcal{H}}_4$. For a given arbitrary small constant ${\pi }_1>0$, there are two constants $L>0$ and ${ϵ}_1\in (0,{ϵ}_0]$ that satisfy $\forall ϵ\in (0,{ϵ}_1]$,

$$\underset{ϵ\to 0}{lim}\mathbb{P}\left(\underset{\varkappa \in [1,\frac{L}{ϵ}]}{sup}|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|>{\pi }_1\right)=0.$$

(42)

Proof. 

Given any constant ${\pi }_1>0$, using Theorem $4.1$ and the inequality of Chebyshev, we can obtain

$$\begin{array}{ccc}\hfill \mathbb{P}\left(\underset{\varkappa \in [1,\frac{L}{ϵ}]}{sup}|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|>{\pi }_1\right)& \le & \frac{1}{{\pi }_1^2}\mathbb{E}\left(\underset{\varkappa \in [1,\frac{L}{ϵ}]}{sup}{|{\zeta }_{ϵ}(\varkappa )-y_{ϵ}(\varkappa )|}^2\right)\hfill \\ & \le & \frac{1}{{\pi }_1^2}ϵL_3{e}^{ϵL_{2}T}.\hfill \end{array}$$

(43)

Letting $ϵ\to 0$, (42) holds. □

5. Application

In this section, we will present an application that illustrates the key findings outlined in the preceding section.

Consider the following perturbed Malthusian FHIDSIE model of population growth (see [15,17]):

$$\begin{array}{ccc}\hfill {\zeta }_{ϵ}(\varkappa )& =& \zeta (1)-H({\zeta }_1)+H({\zeta }_{ϵ,\varkappa })+ϵ{\int }_1^{\varkappa }b\left(s,{\zeta }_{ϵ,s}\right)ds+\sqrt{ϵ}{\int }_1^{\varkappa }{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)d\mathcal{W}(s)\hfill \\ & +& {ϵ}^{\iota }\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)}{s}ds,\hfill \end{array}$$

(44)

with initial condition

$${\zeta }_1=\alpha =\{\alpha (\lambda );1-\varphi \le \lambda \le 1\}\in L_{{\mathfrak{Z}}_1}^2\left([1-\varphi ,1],{\mathbb{R}}^n\right),$$

(45)

where

$$H({\zeta }_{ϵ,\varkappa })=0.1{\zeta }_{ϵ}(-\varphi ),b\left(s,{\zeta }_{ϵ,s}\right)=c_1{\zeta }_{ϵ}(-\varphi ),{\sigma }_1\left(s,{\zeta }_{ϵ,s}\right)=c_2{\zeta }_{ϵ}(-\varphi ),{\sigma }_2\left(s,{\zeta }_{ϵ,s}\right)=c_3{cos}^2\left(s\right){\zeta }_{ϵ}(-\varphi ).$$

The assumptions ${\mathcal{H}}_1$, ${\mathcal{H}}_2$, and ${\mathcal{H}}_3$ are satisfied for $\underline{K}=K=c_1^2\vee c_2^2\vee c_3^2$ and $\underline{\varrho }=0.1$.

We obtain the corresponding averaged equation

$$\begin{array}{ccc}\hfill y_{ϵ}(\varkappa )& =& \zeta (1)-H({\zeta }_1)+H(y_{ϵ,\varkappa })+ϵ{\int }_1^{\varkappa }\overline{b}\left(y_{ϵ,s}\right)ds+\sqrt{ϵ}{\int }_1^{\varkappa }{\overline{\sigma }}_1\left(y_{ϵ,s}\right)d\mathcal{W}(s)\hfill \\ & +& {ϵ}^{\iota }\iota {\int }_1^{\varkappa }{\left(log\frac{\varkappa }{s}\right)}^{\iota -1}\frac{{\overline{\sigma }}_2\left(y_{ϵ,s}\right)}{s}ds,\hfill \end{array}$$

(46)

where

$$\overline{b}\left(y_{ϵ,s}\right)=c_1{y}_{ϵ}(-\varphi ),{\overline{\sigma }}_1\left(y_{ϵ,s}\right)=c_2{y}_{ϵ}(-\varphi ),{\overline{\sigma }}_2\left(y_{ϵ,s}\right)=0.5c_3{y}_{ϵ}(-\varphi ).$$

Now, we present a numerical simulation of Equations (44) and (46), initialized with the condition $\alpha (\varkappa )=exp(-{\varkappa }^2/2)$, while employing the selected parameters $\varphi =0.5$ and $ϵ=0.04$. Subsequently, in Figure 1 and Figure 2, we offer a comparative analysis between the precise solution ${\zeta }_{ϵ}$ for Equation (44) and the averaged solution $y_{ϵ}$ for Equation (46), each evaluated for two distinct values of $\iota$.

6. Conclusions

In this work, we delve into the intricacies surrounding the existence and uniqueness of FHIDSIE with an order denoted by $\iota$, where $\iota$ belongs to the interval $(\frac{1}{2},1)$. Our investigation in this paper harnesses the power of the PIT method to unravel these fundamental aspects of FHIDSIE. Moreover, we establish the profound averaging principle associated with FHIDSIE, leveraging moment inequalities to provide a rigorous and comprehensive proof.