Delayed Interval-Valued Symmetric Stochastic Integral Equations

Abstract

In this paper, delayed stochastic integral equations with an initial condition and a drift coefficient given as interval-valued mappings are considered. These equations have a certain symmetric form that distinguishes them from classical single-valued stochastic integral equations and has implications for the properties of the diameter of the values of the solutions of the equations. The main result of the paper is the theorem that there is a unique solution to the equation considered. It was obtained under the assumptions of continuity of the kernels and Lipschitz continuity of the drift and diffusion coefficients. The proof of the existence of the solution is carried out by the method of iterating successive approximations. The paper ends with theorems about the continuous dependence of the solution on the initial function, kernels and nonlinearities.

1. Introduction

Stochastic differential and integral equations (e.g., [1,2,3]) are now a significant field with enormous potential applications for the mathematical modeling of the dynamics of phenomena evolving in an environment requiring a framework of randomness. However, the values of the processes and mappings considered are in the form of a single number. In this paper, stochastic integral equations are considered, but in such a way that it is possible for certain mappings and processes to have values in the form of intervals rather than single numbers. In this way, another source of uncertainty apart from randomness, namely value imprecision, can be included in a single equation. An example of such uncertainty is a situation in which certain physical quantities are measured using a technical device. Then, the measurement result is always subject to a tolerance error and, in fact, the measured quantity can be treated as a value in the form of an interval, not a single number. Given an interval as an initial value, the dynamics of the phenomenon should be described in such a way as to take into account the fact that at each moment in time the state in which the phenomenon is located is an interval. This approach motivates the study of stochastic differential/integral equations in a multi-valued setting. In this context, the author began research on stochastic differential equations in which the processes under consideration have values that are sets or fuzzy sets. The paper [4] starts this study of Itô-type fuzzy stochastic differential equations with integrals on the left side of the equation and shows the existence of a unique solution (i.e., fuzzy stochastic process) under the condition of Lipschitz continuity of nonlinearity coefficients. The author also considered multivalued stochastic differential equations with solutions valued in the hyperspace of subsets of square-integrable random variables. Moreover, the equations can have the integrals written on the left side of the equation, which results in different geometric properties of the solutions, i.e., the diameters of the values may now decrease with time, while previously they increased. Finally, the author proposed to combine both situations and consider equations of symmetric type (with integrals on both sides of the equation), where the diameters of the values of solutions could decrease as well as increase. Following the methods obtained in [4], papers [5,6,7] have been written, which deal with a fractional approach, i.e., on fractional stochastic differential equations, where the main problem is to establish the existence of a solution. In papers [8,9,10], preceding the author’s research, multi-valued stochastic differential equations were studied in which a certain construction of a multi-valued Itô-type integral was used. However, as shown in [11], such an integral is an unbounded set and is not applicable to the construction of multi-valued stochastic differential equations. Taking this into account, the author proposed that in the multi-valued Itô stochastic equations, the drift part should be multi-valued, but the diffusion part in the form of the Itô integral should be single-valued. The current research will also be carried out in this spirit.

Although studies of Itô-type equations have been performed in this fairly new branch of multi-valued equations, there have been no studies on multi-valued stochastic Volterra-type integral equations. This article will fill this gap and present the theoretical foundations for further research in this direction. Therefore, interval-valued symmetric Volterra-type stochastic integral equations with delay will be investigated in the context of theoretical considerations. Delay equations can be useful for modeling phenomena where the current state depends on what happened in the past. However, this article does not provide examples or case studies. It is left as a starting point for one possible further development of this type of equation.

To be a bit more precise in this introduction, it will now be specified what equation will be considered. This will be an equation of the following symmetric form:

$\begin{array}{ccc}& & X\left(t\right)\oplus {\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\}\hfill \\ & & =\xi \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)\right\}\hfill \\ & & \mathrm{for}t\in [0,T],\mathrm{and}X\left(t\right)=\xi \left(t\right)\mathrm{for}t\in [-\tau ,0],\hfill \end{array}$

where $\tau >0$ is a constant delay, the mapping $\xi$ is interval-valued, drift coefficients $F_1,F_2$ are interval-valued, diffusion coefficients $G_1,G_2$ are single-valued, Brownian motions $B_1$ and $B_2$ are real-valued and not necessarily independent, and also all the kernels, $k_1,l_1,k_2$ and $l_2$, are single-valued. These settings make it so that the solution X to such an equation is an interval-valued stochastic process.

Finally, an explanation will be provided for why it is necessary to consider symmetric equations; this is a certain innovation introduced by the author. First of all, they cannot be easily converted into an equation with only two integrals on the right side, as is the case with classic single-valued equations. This is because subtracting sets is problematic. Secondly, symmetric equations and their solutions exhibit certain desirable geometric properties. Namely, the diameter of the solution value is a function that can change the nature of monotonicity. For comparison, it is pointed out here that an equation with only the right side, i.e., equation

$\begin{array}{ccc}& & X\left(t\right)=\xi \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)\right\}\hfill \\ & & \mathrm{for}t\in [0,T],\mathrm{and}X\left(t\right)=\xi \left(t\right)\mathrm{for}t\in [-\tau ,0],\hfill \end{array}$

would have a solution with values whose diameter does not decrease as the variable t increases. However, solutions to equations with only the left side, i.e.,

$\begin{array}{ccc}& & X\left(t\right)\oplus {\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\}=\xi \left(t\right)\hfill \\ & & \mathrm{for}t\in [0,T],\mathrm{and}X\left(t\right)=\xi \left(t\right)\mathrm{for}t\in [-\tau ,0],\hfill \end{array}$

would have a diameter that does not increase with the increase in t. When considering symmetric equations, both types of behavior of the geometric properties of solutions are taken into account, which is an advantage.

The paper is organized as follows. In Section 2, all the necessary information and facts regarding multi-valued analysis, multi-valued random variables, multi-valued stochastic processes and multi-valued stochastic integrals of the Lebesgue type are presented in a concise way. This is for the reader’s convenience. Section 3 is dedicated to the study of the existence of a unique solution to a delayed interval-valued symmetric stochastic integral equation. The method of successive approximations is used to achieve the result of the existence of a solution. Lipschitz’s continuity of nonlinearities guarantees that there is only one solution. The full justification that the considered problems described by the studied equations are well-posed problems is provided in Section 4, where we show that the solution does not change much when the initial value, nonlinearities or kernels in the integrals change only slightly. Finally, Section 5 presents concluding remarks and potential directions for future research.

2. Preliminaries

Let $\mathbb{I}$ denote a family of all nonempty, compact and convex subsets of $\mathbb{R}$ (intervals). It is possible (cf. [12]) to define the addition and scalar multiplication in $\mathbb{I}$ in the following way: for $X,Y\in \mathbb{I}$, $X=[x^{-},x^{+}],Y=[y^{-},y^{+}]$, and $\alpha ⩾0$,

$X\oplus Y=[u^{-}+v^{-},u^{+}+v^{+}],\alpha X=[\alpha x^{-},\alpha x^{+}],(-\alpha )X=[-\alpha x^{+},-\alpha x^{-}].$

In the set $\mathbb{I}$, the Hausdorff metric H will be considered:

$H(X,Y)=max\left\{|x^{-}-y^{-}|,|x^{+}-y^{+}|\right\}\mathrm{for}X=[x^{-},x^{+}],Y=[y^{-},y^{+}].$

It is known (cf. [12,13]) that $(\mathbb{I},H)$ is a complete, separable and locally compact metric space, and also it becomes a semilinear metric space with algebraic operations of addition and non-negative scalar multiplication.

For the metric H, the following properties hold: (cf. [12])

• $H(X\oplus Z,Y\oplus Z)=H(X,Y)$,

• $H(X\oplus Y,Z\oplus W)\le H(X,Z)+H(Y,W)$,

• $H(\alpha X,\alpha Y)=\left|\alpha \right|H(X,Y)$,

for every $X,Y,Z,W\in \mathbb{I}$, and every $\alpha \in \mathbb{R}$.

Let $X,Y\in \mathbb{I}$. If there exists an interval $Z\in \mathbb{I}$ such that $X=Y\oplus Z$, then it is called the Hukuhara difference of X and Y. The interval Z will be denoted by $X\ominus Y$. Notice that $X\ominus Y\ne X\oplus (-1)Y$ and the Hukuhara difference may not exist. For example, $[2,3]\ominus [1,3]$ does not exist.

For $X=[x^{-},x^{+}]\in \mathbb{I}$, denote the diameter and the magnitude of X by

$\mathrm{diam}\left(X\right):=x^{+}-x^{-}{\mathrm{and}\parallel X\parallel }_{\mathbb{I}}:=H(X,\left\{0\right\})=max\left\{\right|x^{+}|,|x^{-}\left|\right\},$

respectively. It is known that $X\ominus Y$ exists in the case $\mathrm{diam}\left(X\right)\ge \mathrm{diam}\left(Y\right)$ (cf. [12]). Also, one can verify the following properties for $X,Y,Z,W\in \mathbb{I}$:

• if $X\ominus Y$ exists, then ${\parallel X\ominus Y\parallel }_{\mathbb{I}}=H(X,Y)$;

• if $X\ominus Y$, $X\ominus Z$ exist, then $H(X\ominus Y,X\ominus Z)=H(Y,Z)$;

• if $X\ominus Y$, $Z\ominus W$ exist, then $H(X\ominus Y,Z\ominus W)⩽H(X,Z)+H(Y,W)$.

Let $\left(\Omega ,\mathcal{A},P\right)$ be a complete probability space and $\mathcal{M}(\Omega ,\mathcal{A};\mathbb{I})$ denote the family of $\mathcal{A}$-measurable interval-valued mappings $F:\Omega \to \mathbb{I}$ (interval-valued random variable) such that

$\left\{\omega \in \Omega :F\left(\omega \right)\cap O\ne \varnothing \right\}\in \mathcal{A}\mathrm{for} \mathrm{every} \mathrm{open} \mathrm{set}O\subset \mathbb{R}.$

An interval-valued random variable $F\in \mathcal{M}(\Omega ,\mathcal{A};\mathbb{I})$ is called $L^p$-integrally bounded, $p⩾1$, if there exists $h\in L^p\left(\Omega ,\mathcal{A},P;\mathbb{R}\right)$ such that $\left|a\right|⩽h\left(\omega \right)$ for any a and $\omega$ with $a\in F\left(\omega \right)$. It is known (see [14]) that F is $L^p$-integrally bounded if $\omega \mapsto {\parallel F\left(\omega \right)\parallel }_{\mathbb{I}}$ is in $L^p(\Omega ,\mathcal{A},P;\mathbb{R})$, where $L^p(\Omega ,\mathcal{A},P;\mathbb{R})$ is a space of equivalence classes (with respect to the equality P-a.e.) of $\mathcal{A}$-measurable random variables $h:\Omega \to \mathbb{R}$ such that ${\mathbb{E}\left|h\right|}^p={\int }_{\Omega }{\left|h\right|}^{p}dP<\infty$. Let us denote

${\mathcal{L}}^p(\Omega ,\mathcal{A},P;\mathbb{I}):=\left\{F\in \mathcal{M}(\Omega ,\mathcal{A};\mathbb{I}):Fis{L}^p-\mathrm{integrally} \mathrm{bounded}\right\},p⩾1.$

The interval-valued random variables $F,G\in {\mathcal{L}}^p(\Omega ,\mathcal{A},P;\mathbb{I})$ are considered to be identical, if $F=G$ holds at P-a.e.

Let $T>0$ and let the system $(\Omega ,\mathcal{A},{\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]},P)$ be a complete, filtered probability space with a filtration ${\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]}$ satisfying the usual hypotheses, i.e., ${\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]}$ is an increasing and right continuous family of sub-$\sigma$-algebras of $\mathcal{A}$, and ${\mathcal{A}}_0$ contains all P-null sets. The mapping $X:[0,T]\times \Omega \to \mathbb{I}$ is called an interval-valued stochastic process, if for every $t\in [0,T]$ a mapping $X\left(t\right):\Omega \to \mathbb{I}$ is an interval-valued random variable. It is said that the interval-valued stochastic process X is H-continuous, if almost all its paths (with respect to the probability measure P), i.e., the mappings $X(·,\omega ):[0,T]\to \mathbb{I}$, are the H-continuous functions. An interval-valued stochastic process X is said to be ${\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]}$-adapted, if for every $t\in [0,T]$ the interval-valued random variable $X\left(t\right):\Omega \to \mathbb{I}$ is ${\mathcal{A}}_t$-measurable. It is called measurable if $X:[0,T]\times \Omega \to \mathbb{I}$ is a $\mathcal{B}\left(\right[0,T\left]\right)\otimes \mathcal{A}$-measurable interval-valued random variable, where $\mathcal{B}\left(\right[0,T\left]\right)$ denotes the Borel $\sigma$-algebra of subsets of $[0,T]$. If $X:[0,T]\times \Omega \to \mathbb{I}$ is ${\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]}$-adapted and measurable, then it will be called nonanticipating. Equivalently, X is nonanticipating if X is measurable with respect to the $\sigma$-algebra $\mathcal{N}$, which is defined as follows

$\mathcal{N}:=\{A\in \mathcal{B}\left([0,T]\right)\otimes \mathcal{A}:A^t\in {\mathcal{A}}_t\mathrm{for} \mathrm{every}t\in [0,T]\},$

where $A^t=\{\omega :(t,\omega )\in A\}$. An interval-valued nonanticipating stochastic process $X:[0,T]\times \Omega \to \mathbb{I}$ is called ${\mathcal{L}}^p$-integrally bounded if there exists a measurable stochastic process $h:[0,T]\times \Omega \to \mathbb{R}$ such that $\mathbb{E}\left({\int }_0^T{\left|h\left(s\right)\right|}^{p}ds\right)<\infty$ and ${\parallel X(t,\omega )\parallel }_{\mathbb{I}}⩽h(t,\omega )$ for a.a. $(t,\omega )\in [0,T]\times \Omega$. According to ${\mathcal{L}}^p([0,T]\times \Omega ,\mathcal{N};\mathbb{I})$, we can denote the set of all equivalence classes (with respect to the equality $\lambda \times P$-a.e., where $\lambda$ denotes the Lebesgue measure) of nonanticipating and ${\mathcal{L}}^p$-integrally bounded interval-valued stochastic processes.

Let $F\in {\mathcal{L}}^p([0,T]\times \Omega ,\mathcal{N};\mathbb{I})$, $p⩾1$. For such a process as F, one can define (see, e.g., [4]) the interval-valued stochastic Lebesgue–Aumann integral, which is an interval-valued random variable

$\Omega \ni \omega \mapsto {\int }_0^{T}F(s,\omega )ds\in \mathbb{I}.$

Then, ${\int }_0^{t}F\left(s\right)ds$ (from now on, argument $\omega$ will not be written) is understood as the following integral: ${\int }_0^T{\mathbf{1}}_{[0,t]}\left(s\right)F\left(s\right)ds$. For the interval-valued stochastic Lebesgue–Aumann integral, the following properties (see [4]) hold.

Proposition 1.

Let $p⩾1$. If $F,G\in {\mathcal{L}}^p([0,T]\times \Omega ,\mathcal{N};\mathbb{I})$, then

• (i) 

  $[0,T]\times \Omega \ni (t,\omega )\mapsto {\int }_0^{t}F(s,\omega )ds\in \mathbb{I}$ belongs to ${\mathcal{L}}^p([0,T]\times \Omega ,\mathcal{N};\mathbb{I})$;

• (ii) 

  the interval-valued stochastic process $(t,\omega )\mapsto {\int }_0^{t}F(s,\omega )ds$ is H-continuous;

• (iii) 

  with probability one, for every $t\in [0,T]$

  $H^p\left({\int }_0^{t}F\left(s\right)ds,{\int }_0^{t}G\left(s\right)ds\right)⩽t^{p-1}{\int }_0^t{H}^p(F\left(s\right),G\left(s\right))ds,$

  ;

• (iv) 

  for every $t\in [0,T]$ it holds that

  $\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^p\left({\int }_0^{z}F\left(s\right)ds,{\int }_0^{z}G\left(s\right)ds\right)⩽t^{p-1}\mathbb{E}{\int }_0^t{H}^p\left(F\left(s\right),G\left(s\right)\right)ds.$

3. Existence of a Unique Solution

This paper is dedicated to examining delayed interval-valued symmetric stochastic integral equations of the Volterra type, which have the following form:

(1)$\begin{array}{ccc}& & X\left(t\right)\oplus {\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\}\hfill \\ & & =\xi \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)\right\}\hfill \\ & & \mathrm{for}t\in [0,T],\mathrm{and}X\left(t\right)=\xi \left(t\right)\mathrm{for}t\in [-\tau ,0],\hfill \end{array}$

where $\tau >0$ is a constant delay,

• $\xi :[-\tau ,T]\to \mathbb{I}$ is a given interval-valued function;

• $F_1,F_2:\mathbb{I}\times \mathbb{I}\to \mathbb{I}$ are the drift coefficients;

• $G_1,G_2:\mathbb{I}\times \mathbb{I}\to \mathbb{R}$ are the diffusion coefficients;

• $B_1$, $B_2$ are the real-valued ${\left\{{\mathcal{A}}_t\right\}}_{t\in [0,T]}$-Brownian motions (not necessarily independent),

  $k_1,l_1,k_2,l_2:[0,T]\times [0,T]\to \mathbb{R}$ are the continuous kernels.

Note that by transferring the integrals to one right-hand side of (1), one can only write

(2)$\begin{array}{ccc}\hfill X\left(t\right)& =& \left[\xi \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\ominus {\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds\hfill \\ & & \oplus \left\{{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\}.\hfill \end{array}$

This form of the equation will require making assumptions about the existence of some Hukuhara differences ⊖, which will be undertaken later in this paper. Now, a description of what is meant by a solution to (1) is provided. Let $\tilde{T}\in (0,T]$.

Definition 1.

An interval-valued stochastic process $X:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ is called the solution to (1) on the interval $[-\tau ,\tilde{T}]$ if:

• (i) 

  $X\left(t\right)=\xi \left(t\right)$ for every $t\in [-\tau ,0]$ with probability one;

• (ii) 

  $X{|}_{[0,\tilde{T}]\times \Omega }\in {\mathcal{L}}^2([0,\tilde{T}]\times \Omega ,\mathcal{N};\mathbb{I})$;

• (iii) 

  X is H-continuous;

• (iv) 

  X verifies (1).

Since one wants to obtain the result where there is a unique solution to (1), a description of what is meant by the fact that the solution is unique is also given.

Definition 2.

A solution $X:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ to (1) is said to be unique if

$P\left(X\left(t\right)=Y\left(t\right)\mathit{for}\mathit{every}t\in [0,\tilde{T}]\right)=1,$

where $Y:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ is any other solution to (1).

The main aim of the paper is to obtain a theorem about the existence of a unique solution. Below, assumptions regarding the initial value $\xi$ and the nonlinearities $F_1,G_1,F_2,G_2$ in Equation (1) will be formulated, which will allow such a result to be obtained.

In addition to the continuity of the kernels $k_1,l_1,k_2,l_2$ in the integrals (throughout this paper, it will be assumed that the kernels in the integrals are continuous), it will be required that

• (A0). ${\displaystyle \underset{t\in [-\tau ,T]}{sup}}{\parallel \xi \left(t\right)\parallel }_{\mathbb{I}}<\infty$;

• (A1). there exists a positive constant C such that for every $X_1,Y_1,X_2,Y_2\in \mathbb{I}$

  $\begin{array}{ccc}& & max\{H^2(F_1(X_1,Y_1),F_1(X_2,Y_2)),{|G_1(X_1,Y_1)-G_1(X_2,Y_2)|}^2,H^2(F_2(X_1,Y_1),F_2(X_2,Y_2)),\hfill \\ & & |G_2(X_1,Y_1)-G_2(X_2,Y_2){|}^2\}⩽C(H^2(X_1,X_2)+H^2(Y_1,Y_2));\hfill \end{array}$

• (A2). there exists a positive constant C such that for every $X,Y\in \mathbb{I}$

  $\begin{array}{c}\hfill max\left\{\parallel F_1{(X,Y)\parallel }_{\mathbb{I}}^2,|G_1{(X,Y)|}^2,\parallel F_2{(X,Y)\parallel }_{\mathbb{I}}^2,{\left|G_2(X,Y)\right|}^2\right\}⩽{C(1+\parallel X\parallel }_{\mathbb{I}}^2+{\parallel Y\parallel }_{\mathbb{I}}^2);\end{array}$

• (A3). there exists $\tilde{T}\in (0,T]$ such that for every $n=1,2,\dots$ the mappings $X^n:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ described as

  $X^n\left(t\right)=\xi \left(t\right)\mathrm{for}t\in [-\tau ,0]\mathrm{with}\mathrm{probability}\mathrm{one}$

  and

  (3)$\begin{array}{cc}\hfill X^n\left(t\right)& =\left[\xi \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2\left(X^{n-1}\left(s\right),X^{n-1}(s-\tau )\right)ds\right]\hfill \\ \hfill & \ominus {\int }_0^t{k}_1(t,s)F_1\left(X^{n-1}\left(s\right),X^{n-1}(s-\tau )\right)ds\hfill \\ \hfill & \oplus \{{\int }_0^t{l}_2(t,s)G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_2\left(s\right)\hfill \\ \hfill & -{\int }_0^t{l}_1(t,s)G_1\left(X^{n-1}\left(s\right),X^{n-1}(s-\tau )\right)d{B}_1\left(s\right)\}\hfill \\ \hfill & \mathrm{for}t\in [0,\tilde{T}]\mathrm{with}\mathrm{probability}\mathrm{one}\hfill \end{array}$

  are well defined (in particular, the Hukuhara differences do exist). Here, $X^0:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ is defined as $X^0\left(t\right)=\xi \left(t\right)$ for $t\in [-\tau ,\tilde{T}]$, with probability one.

The requirement for the existence of Hukuhara differences in (A3) is necessary and ineradicable, and this is due to the representation (2). The interval-valued stochastic processes $X^n$ with $n=0,1,2,\dots$ are intended to be successive approximations to the solution of Equation (1). Before the main theorem of this part of the paper is formulated, a very useful result about the uniform boundedness of the sequence $\left\{X^n\right\}$ is given.

Lemma 1.

Let the assumptions (A0), (A2) and (A3) be satisfied. Then, the sequence ${\left\{X^n\right\}}_{n=0}^{\infty }$ has the property

${\displaystyle \underset{n}{max}}{\displaystyle \underset{t\in [-\tau ,\tilde{T}]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2<\infty .$

Proof. 

For every $n=0,1,2,\dots$ and every $t\in [-\tau ,0]$ we have $X^n\left(t\right)=\xi \left(t\right)$. Thus,

${\displaystyle \underset{t\in [-\tau ,0]}{sup}}\mathbb{E}\parallel X^n{\left(t\right)\parallel }_{\mathbb{I}}^2={\displaystyle \underset{t\in [-\tau ,0]}{sup}}{\parallel \xi \left(t\right)\parallel }_{\mathbb{I}}^2$

which is bounded by assumption (A0). Now, let us fix n and $t\in [0,\tilde{T}]$. In this case, one has

$\begin{array}{ccc}\hfill {\displaystyle {\displaystyle \underset{z\in [0,t]}{sup}}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2& =& {\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}\parallel \left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(X^{n-1}\left(s\right),X^{n-1}(s-\tau )\right)ds\hfill \\ & & \oplus \{{\int }_0^z{l}_2(z,s)G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_2\left(s\right)\hfill \\ & & -{\int }_0^z{l}_1(z,s)G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_1\left(s\right)\}{\parallel }_{\mathbb{I}}^2\hfill \end{array}$

and, according to the Cauchy–Schwarz inequality,

$\begin{array}{ccc}\hfill {\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2& ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}\parallel {\int }_0^z{k}_2(z,s)F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))ds{\parallel }_{\mathbb{I}}^2\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}\parallel {\int }_0^z{k}_1(z,s)F_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))ds{\parallel }_{\mathbb{I}}^2\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}|{\int }_0^z{l}_2(z,s)G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_2\left(s\right){|}^2\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}|{\int }_0^z{l}_1(z,s)G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_1\left(s\right){|}^2.\hfill \end{array}$

According to Proposition 1 (iii), the Itô isometry and the Fubini Theorem,

$\begin{array}{ccc}\hfill {\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2& ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}z\mathbb{E}{\int }_0^z\parallel k_2(z,s)F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}z\mathbb{E}{\int }_0^z\parallel k_1(z,s)F_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\int }_0^z|l_2(z,s)G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds\hfill \\ & & +5{\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\int }_0^z|l_1(z,s)G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds\hfill \\ & ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2+5{\parallel k_2\parallel }_{\infty }^{2}t{\int }_0^t\mathbb{E}\parallel F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +5\parallel k_1{{\parallel }_{\infty }^{2}t{\int }_0^t\mathbb{E}\parallel F_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +5\parallel l_2{\parallel }_{\infty }^2{\int }_0^t\mathbb{E}|G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds\hfill \\ & & +5\parallel l_1{\parallel }_{\infty }^2{\int }_0^t\mathbb{E}|G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds.\hfill \end{array}$

According to assumption (A2),

$\begin{array}{ccc}\hfill {\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2& ⩽& 5{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{\parallel \xi \left(t\right)\parallel }_{\mathbb{I}}^2+\left[5C\parallel k_2{\parallel }_{\infty }^{2}t+5C\parallel k_1{\parallel }_{\infty }^{2}t+5C\parallel l_2{\parallel }_{\infty }^2+5C{\parallel l_1\parallel }_{\infty }^2\right]\hfill \\ & & \times {\int }_0^t\mathbb{E}\left[1+\parallel X^{n-1}{\left(s\right)\parallel }_{\mathbb{I}}^2+{\parallel X^{n-1}(s-\tau )\parallel }_{\mathbb{I}}^2\right]ds\hfill \\ & ⩽& A+B{\int }_0^t\mathbb{E}\left[\parallel X^{n-1}{\left(s\right)\parallel }_{\mathbb{I}}^2+{\parallel X^{n-1}(s-\tau )\parallel }_{\mathbb{I}}^2\right]ds\hfill \\ & ⩽& A+B{\int }_0^t\left[{\displaystyle \underset{z\in [0,s]}{sup}}\mathbb{E}\parallel X^{n-1}{\left(z\right)\parallel }_{\mathbb{I}}^2+{\displaystyle \underset{z\in [0,s]}{sup}}\mathbb{E}{\parallel X^{n-1}(z-\tau )\parallel }_{\mathbb{I}}^2\right]ds,\hfill \end{array}$

where $A=5{sup}_{z\in [0,\tilde{T}]}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2+5C\parallel k_2{\parallel }_{\infty }^2{\tilde{T}}^2+5C\parallel k_1{\parallel }_{\infty }^2{\tilde{T}}^2+5C\parallel l_2{\parallel }_{\infty }^2\tilde{T}+5C{\parallel l_1\parallel }_{\infty }^2\tilde{T}$ and $B=A/\tilde{T}$. Since

${\int }_0^t{\displaystyle \underset{z\in [0,s]}{sup}}\mathbb{E}\parallel X^{n-1}{(z-\tau )\parallel }_{\mathbb{I}}^{2}ds⩽\tilde{T}{\displaystyle \underset{z\in [-\tau ,0]}{sup}}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2+{\int }_0^t{\displaystyle \underset{z\in [0,s]}{sup}}\mathbb{E}\parallel X^{n-1}\left(z\right){)\parallel }_{\mathbb{I}}^{2}ds,$

one can obtain

${\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}\parallel X^n{\left(t\right)\parallel }_{\mathbb{I}}^2⩽A_1+B_1{\int }_0^t{\displaystyle \underset{z\in [0,s]}{sup}}\mathbb{E}{\parallel X^{n-1}\left(z\right)\parallel }_{\mathbb{I}}^{2}ds$

with some new positive constants $A_1,B_1$.

Let us denote $a_n\left(t\right)={\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2$ for $t\in [0,\tilde{T}]$. Notice that for $k⩾1$, one has

${\displaystyle \underset{1\le n\le k}{max}}{a}_n\left(t\right)⩽A_1+B_1{\int }_0^t{\displaystyle \underset{1\le n\le k}{max}}{a}_{n-1}\left(s\right)ds.$

Since ${\displaystyle \underset{1\le n\le k}{max}}{a}_{n-1}\left(s\right)⩽{sup}_{z\in [-\tau ,\tilde{T}]}{\parallel \xi \left(z\right)\parallel }_{\mathbb{I}}^2+a_n\left(s\right)$, one can obtain

${\displaystyle \underset{1\le n\le k}{max}}{a}_n\left(t\right)⩽A_2+B_1{\int }_0^t{\displaystyle \underset{1\le n\le k}{max}}{a}_n\left(s\right)ds$

with a new positive constant $A_2$. Invoking the Gronwall inequality, one can find

${\displaystyle \underset{1\le n\le k}{max}}{a}_n\left(t\right)⩽A_2{e}^{B_{1}t}\mathrm{for}\mathrm{every}t\in [0,\tilde{T}]\mathrm{and}\mathrm{every}k⩾1.$

Hence,

${\displaystyle \underset{1\le n\le k}{max}}{a}_n\left(\tilde{T}\right)⩽A_2{e}^{B_1\tilde{T}}\mathrm{for}\mathrm{every}k⩾1,$

and the proof is completed. □

Using the above calculations, one can find

$\mathbb{E}{\int }_0^{\tilde{T}}\parallel X^n{\left(t\right)\parallel }_{\mathbb{I}}^{2}dt⩽{\int }_0^{\tilde{T}}{\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^{2}dt⩽{\int }_0^{\tilde{T}}{A}_2{e}^{B_1\tilde{T}}dt=\tilde{T}{A}_2{e}^{B_1\tilde{T}}<\infty .$

Hence, one can infer the following fact.

Corollary 1.

Under assumptions of Lemma 1, for every $n=0,1,2,\dots$ the interval-valued stochastic process $X^n{|}_{[0,\tilde{T}]\times \Omega }$ belongs to ${\mathcal{L}}^2([0,\tilde{T}]\times \Omega ,\mathcal{N},\mathbb{I})$.

Theorem 1.

Assume that (A0)–(A3) are satisfied. Then, Equation (1) possesses a unique solution.

Proof. 

Let us fix $t\in [0,\tilde{T}]$. Notice that

$\begin{array}{ccc}\hfill \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X^1\left(z\right),X^0\left(z\right))& =& \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(\xi \left(s\right),\xi (s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(\xi \left(s\right),\xi (s-\tau )\right)ds\hfill \\ & & \oplus \{{\int }_0^z{l}_2(z,s)G_2(\xi \left(s\right),\xi (s-\tau ))d{B}_2\left(s\right)\hfill \\ & & -{\int }_0^z{l}_1(z,s)G_1(\xi \left(s\right),\xi (s-\tau ))d{B}_1\left(s\right)\},\xi \left(z\right))\hfill \end{array}$

and further

$\begin{array}{ccc}\hfill \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X^1\left(z\right),X^0\left(z\right))& ⩽& \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}[4\parallel {\int }_0^z{k}_2(z,s)F_2(\xi \left(s\right),\xi (s-\tau ))ds{\parallel }_{\mathbb{I}}^2\hfill \\ & & +4\parallel {\int }_0^z{k}_1(z,s)F_1(\xi \left(s\right),\xi (s-\tau ))ds{\parallel }_{\mathbb{I}}^2\hfill \end{array}$

$\begin{array}{ccc}& & +4|{\int }_0^z{l}_2(z,s)G_2(\xi \left(s\right),\xi (s-\tau ))d{B}_2\left(s\right){|}^2\hfill \\ & & +4\left|{\int }_0^z{l}_1(z,s)G_1(\xi \left(s\right),\xi (s-\tau ))d{B}_1\left(s\right){|}^2\right].\hfill \end{array}$

According to Proposition 1 (iv), the Doob inequality and assumption (A2),

$\begin{array}{ccc}\hfill {\displaystyle \underset{z\in [0,t]}{sup}}\mathbb{E}{H}^2(X^1\left(z\right),X^0\left(z\right))& ⩽& 4\parallel k_2{\parallel }_{\infty }^{2}t\mathbb{E}{\int }_0^t{\parallel F_2(\xi \left(s\right),\xi (s-\tau ))\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +4\parallel k_1{\parallel }_{\infty }^{2}t\mathbb{E}{\int }_0^t{\parallel F_1(\xi \left(s\right),\xi (s-\tau ))\parallel }_{\mathbb{I}}^{2}ds\hfill \\ & & +16\parallel l_2{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t|G_2(\xi \left(s\right),\xi (s-\tau )){|}^{2}ds\hfill \\ & & +16\parallel l_1{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t|G_1(\xi \left(s\right),\xi (s-\tau )){|}^{2}ds\hfill \\ & ⩽& 4C\left[\parallel k_2{\parallel }_{\infty }^2\tilde{T}+\parallel k_1{\parallel }_{\infty }^2\tilde{T}+4\parallel l_2{\parallel }_{\infty }^2+4{\parallel l_1\parallel }_{\infty }^2\right]\hfill \\ & & \times {\int }_0^t\left[1+{\parallel \xi \left(s\right)\parallel }_{\mathbb{I}}^2+{\parallel \xi (s-\tau ))\parallel }_{\mathbb{I}}^2\right]ds\hfill \\ & ⩽& At,\hfill \end{array}$

where $A=4C\left[\parallel k_2{\parallel }_{\infty }^2\tilde{T}+\parallel k_1{\parallel }_{\infty }^2\tilde{T}+4\parallel l_2{\parallel }_{\infty }^2+4{\parallel l_1\parallel }_{\infty }^2\right](1+2{\displaystyle \underset{t\in [-\tau ,\tilde{T}]}{sup}}{\parallel \xi \left(t\right)\parallel }_{\mathbb{I}}^2)$.

Now, for $n⩾1$ and $t\in [0,\tilde{T}]$,

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X^{n+1}\left(z\right),X^n\left(z\right))\hfill \\ & ⩽& 4\parallel k_2{\parallel }_{\infty }^{2}t\mathbb{E}{\int }_0^t{H}^2(F_2(X^n\left(s\right),X^n(s-\tau )),F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )))ds\hfill \\ & & +4\parallel k_1{\parallel }_{\infty }^{2}t\mathbb{E}{\int }_0^t{H}^2(F_1(X^n\left(s\right),X^n(s-\tau )),F_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau )))ds\hfill \\ & & +16\parallel l_2{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t|G_2(X^n\left(s\right),X^n(s-\tau ))-G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds\hfill \\ & & +16\parallel l_1{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t|G_1(X^n\left(s\right),X^n(s-\tau ))-G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau )){|}^{2}ds.\hfill \end{array}$

Due to assumption (A1),

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X^{n+1}\left(z\right),X^n\left(z\right))\hfill \\ & ⩽& {\displaystyle \frac{B}{2}}\mathbb{E}{\int }_0^t\left[H^2(X^n\left(s\right),X^{n-1}\left(s\right))+H^2(X^n(s-\tau ),X^{n-1}(s-\tau ))\right]ds\hfill \\ & ⩽& B\mathbb{E}{\int }_0^t{H}^2(X^n\left(s\right),X^{n-1}\left(s\right))ds\hfill \\ & ⩽& B{\int }_0^t\mathbb{E}{\displaystyle \underset{z\in [0,s]}{sup}}{H}^2(X^n\left(z\right),X^{n-1}\left(z\right))ds,\hfill \end{array}$

where $B=8C\left[\parallel k_2{\parallel }_{\infty }^2\tilde{T}+\parallel k_1{\parallel }_{\infty }^2\tilde{T}+4\parallel l_2{\parallel }_{\infty }^2+4{\parallel l_2\parallel }_{\infty }^2\right]$. Therefore,

$\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X^n\left(z\right),X^{n-1}\left(z\right))⩽{\displaystyle \frac{A}{B}}·{\displaystyle \frac{{\left(Bt\right)}^{n-1}}{n!}}\mathrm{for}\mathrm{every}t\in [0,\tilde{T}]\mathrm{and}\mathrm{every}n=1,2,\dots$

Using the Chebyshev inequality, one can find

$P\left({\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}{H}^2(X^n\left(t\right),X^{n-1}\left(t\right))>{\displaystyle \frac{1}{4^n}}\right)⩽{\displaystyle \frac{A}{B}}·{\displaystyle \frac{{\left(4B\tilde{T}\right)}^n}{n!}}.$

Since the series ${\sum }_{n=1}^{\infty }{\displaystyle \frac{{\left(4B\tilde{T}\right)}^n}{n!}}$ is convergent, according to the Borel–Cantelli lemma, one can obtain

$P\left({\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}{H}^2(X^n\left(t\right),X^{n-1}\left(t\right))>{\displaystyle \frac{1}{4^n}}\mathrm{infinitely}\mathrm{often}\right)=0$

and, as a consequence,

$P\left({\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}H(X^n\left(t\right),X^{n-1}\left(t\right))>{\displaystyle \frac{1}{2^n}}\mathrm{infinitely}\mathrm{often}\right)=0.$

Thus, the sequence $\left\{X^n(·,\omega )\right\}$ converges uniformly to an H-continuous multivalued function $\tilde{X}(·,\omega ):[0,\tilde{T}]\to \mathbb{I}$ for every $\omega \in {\Omega }_c$, such that $P\left({\Omega }_c\right)=1$. Let us define $X(t,\omega ):=\xi \left(t\right)$ for $(t,\omega )\in [-\tau ,0]\times \Omega$, $X(t,\omega ):=\tilde{X}(t,\omega )$ for $(t,\omega )\in [0,\tilde{T}]\times {\Omega }_c$ and $X(t,\omega )$ as a freely chosen interval in $\mathbb{I}$ for $(t,\omega )\in [0,\tilde{T}]\times (\Omega \setminus {\Omega }_c)$. Due to convergence,

${\displaystyle \underset{n\to \infty }{lim}}H(X^n(t,\omega ),X(t,\omega ))=0\mathrm{for}(t,\omega )\in [0,\tilde{T}]\times {\Omega }_c,$

the mapping $X(t,·):\Omega \to \mathbb{I}$ is an ${\mathcal{A}}_t$-measurable multivalued random variable. Therefore, $X{|}_{[0,\tilde{T}]\times \Omega }$ is an H-continuous $\left\{{\mathcal{A}}_t\right\}$-adapted multivalued stochastic process, and according to this a nonanticipating multivalued stochastic process. Moreover, notice that

$\begin{array}{ccc}\hfill \mathbb{E}{\int }_0^{\tilde{T}}{\parallel X\left(t\right)\parallel }_{\mathbb{I}}^{2}dt& ⩽& 2\mathbb{E}{\int }_0^{\tilde{T}}{H}^2(X\left(t\right),X^n\left(t\right))dt+2\mathbb{E}{\int }_0^{\tilde{T}}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^{2}dt\hfill \\ & ⩽& 2\tilde{T}\mathbb{E}{\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}{H}^2(X\left(t\right),X^n\left(t\right))+2\tilde{T}{\displaystyle \underset{t\in [0,\tilde{T}]}{sup}}\mathbb{E}{\parallel X^n\left(t\right)\parallel }_{\mathbb{I}}^2\hfill \end{array}$

and the first term converges to zero and the second term is uniformly bounded in n by Lemma 1. Hence, $\mathbb{E}{\int }_0^{\tilde{T}}{\parallel X\left(t\right)\parallel }_{\mathbb{I}}^{2}dt<\infty$ and $X\in {\mathcal{L}}^2([0,\tilde{T}]\times \Omega ,\mathcal{N};\mathbb{I})$. It will now be shown that X is a solution to (1). For this purpose, it will be shown that $D=0$, where

$\begin{array}{ccc}\hfill D& =& \mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X\left(z\right),\left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(X\left(s\right),X(s-\tau )\right)ds\hfill \\ & & \oplus \left\{{\int }_0^z{l}_2(z,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^z{l}_1(z,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\}).\hfill \end{array}$

Notice that

$\begin{array}{ccc}\hfill D& ⩽& 2\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X\left(z\right),X^n\left(z\right))\hfill \\ & & +2\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X^n\left(z\right),\left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(X\left(s\right),X(s-\tau )\right)ds\hfill \\ & & \oplus \left\{{\int }_0^z{l}_2(z,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^z{l}_1(z,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\})\hfill \end{array}$

and the first term on the right-hand side of the above inequality converges to zero as n goes to infinity. It will be shown that the same happens for the second term. Notice that

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X^n\left(z\right),\left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(X\left(s\right),X(s-\tau )\right)ds\hfill \\ & & \oplus \left\{{\int }_0^z{l}_2(z,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^z{l}_1(z,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\})\hfill \\ & ⩽& 4\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2({\int }_0^z{k}_2(z,s)F_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))ds,\hfill \\ & & {\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds)\hfill \\ & & +4\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2({\int }_0^z{k}_1(z,s)F_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))ds,\hfill \\ & & {\int }_0^z{k}_1(z,s)F_1(X\left(s\right),X(s-\tau ))ds)\hfill \\ & & +4\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}|{\int }_0^z{l}_2(z,s)G_2(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_2\left(s\right)\hfill \\ & & -{\int }_0^z{l}_2(z,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right){|}^2\hfill \\ & & +4\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}|{\int }_0^z{l}_1(z,s)G_1(X^{n-1}\left(s\right),X^{n-1}(s-\tau ))d{B}_1\left(s\right)\hfill \\ & & -{\int }_0^z{l}_1(z,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right){|}^2\hfill \\ & ⩽& A_1\mathbb{E}{\int }_0^{\tilde{T}}\left[H^2(X^{n-1}\left(s\right),X\left(s\right))+H^2(X^{n-1}(s-\tau ),X(s-\tau ))\right]ds\hfill \\ & ⩽& A_1\mathbb{E}{\int }_0^{\tilde{T}}\left[{\displaystyle \underset{s\in [0,\tilde{T}]}{sup}}{H}^2(X^{n-1}\left(s\right),X\left(s\right))+{\displaystyle \underset{s\in [0,\tilde{T}]}{sup}}{H}^2(X^{n-1}(s-\tau ),X(s-\tau ))\right]ds\hfill \\ & ⩽& B_1\mathbb{E}{\displaystyle \underset{s\in [0,\tilde{T}]}{sup}}{H}^2(X^{n-1}\left(s\right),X\left(s\right))\stackrel{n\to \infty }{⟶}0,\hfill \end{array}$

where $A_1=4C\left[\parallel k_2{\parallel }_{\infty }^2\tilde{T}+\parallel k_1{\parallel }^2\tilde{T}+4\parallel l_2{\parallel }_{\infty }^2+4{\parallel l_1\parallel }_{\infty }^2\right]$ and $B_1=2A_1\tilde{T}$. In this way, one has $D=0$, and this implies

$\begin{array}{ccc}& & P\left({\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}H\right(X\left(z\right),\left[\xi \left(z\right)\oplus {\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\hfill \\ & & \ominus {\int }_0^z{k}_1(z,s)F_1\left(X\left(s\right),X(s-\tau )\right)ds\hfill \\ & & \oplus \{{\int }_0^z{l}_2(z,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)\hfill \\ & & -{\int }_0^z{l}_1(z,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\}=0)=1.\hfill \end{array}$

Therefore, the existence of a solution to (1) has been proved. It remains to show that solution X is unique. If Y were another solution, for every $t\in [0,\tilde{T}]$ one would have

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))\hfill \\ & ⩽& A_1\mathbb{E}{\int }_0^t\left[{\displaystyle \underset{z\in [0,s]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))+{\displaystyle \underset{z\in [0,s]}{sup}}{H}^2(X(z-\tau ),Y(z-\tau ))\right]ds\hfill \\ & ⩽& 2A_1{\int }_0^t\mathbb{E}{\displaystyle \underset{z\in [0,s]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))ds.\hfill \end{array}$

According to the Gronwall inequality,

$\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))⩽0\mathrm{for}t\in [0,\tilde{T}].$

Hence, $\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))=0\mathrm{for}t\in [0,\tilde{T}]$, and this implies that

$P\left({\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))=0\right)=1.$

Thus, X and Y are indistinguishable, and in view of Definition 2 the interval-valued stochastic process X is the only solution to (1). □

The above theorem about the existence of a unique solution is extremely important when we recall that, in general, it is difficult to find explicit forms of solutions to stochastic differential and integral equations. Once one is sure that the solution exists, one can think about future research on finding approximate solutions using, for example, numerical methods.

4. Well-Posedness of Delayed Interval-Valued Symmetric Stochastic Integral Equations

For the problems described by the delayed interval-valued symmetric stochastic integral equations to be well posed, it is necessary that the solutions of these equations are not very sensitive to small changes in the parameters of these equations. Therefore, in this part of the paper it will be shown that a small change in the initial condition $\xi$ does not significantly affect the solution. For this purpose, a study of Equation (1) is conducted with an initial value $\xi$ and the equation

(4)$\begin{array}{ccc}& & Y\left(t\right)\oplus {\int }_0^t{k}_1(t,s)F_1(Y\left(s\right),Y(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_1(t,s)G_1(Y\left(s\right),Y(s-\tau ))d{B}_1\left(s\right)\right\}\hfill \\ & & =\zeta \left(t\right)\oplus {\int }_0^t{k}_2(t,s)F_2(Y\left(s\right),Y(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_2(t,s)G_2(Y\left(s\right),Y(s-\tau ))d{B}_2\left(s\right)\right\},\hfill \\ & & \mathrm{for}t\in [0,T],\mathrm{and}Y\left(t\right)=\zeta \left(t\right),\mathrm{for}t\in [-\tau ,0]\hfill \end{array}$

with initial value $\zeta$.

Theorem 2.

Let Equations (1) and (4) possess solutions $X:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ and $Y:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$, respectively. Suppose condition (A1) is satisfied. Then, there is a positive constant A such that

$\mathbb{E}{\displaystyle \underset{t\in [-\tau ,\tilde{T}]}{sup}}{H}^2(X\left(t\right),Y\left(t\right))⩽A{\displaystyle \underset{[-\tau ,\tilde{T}]}{sup}}{H}^2(\xi \left(t\right),\zeta \left(t\right)).$

Proof. 

For $t\in [-\tau ,0]$, one has $X\left(t\right)=\xi \left(t\right)$ and $Y\left(t\right)=\zeta \left(t\right)$. Hence,

$\mathbb{E}{\displaystyle \underset{t\in [-\tau ,0]}{sup}}{H}^2(X\left(t\right),Y\left(t\right))={\displaystyle \underset{t\in [-\tau ,0]}{sup}}{H}^2(\xi \left(t\right),\zeta \left(t\right)).$

For $t\in [0,\tilde{T}]$, it is noticed that

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))\hfill \\ & =& \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\left[\xi \left(t\right)+{\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds\right]\ominus {\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds\hfill \\ & & +\left\{{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right\},\hfill \\ & & \left[\zeta \left(t\right)+{\int }_0^t{k}_2(t,s)F_2(Y\left(s\right),Y(s-\tau ))ds\right]\ominus {\int }_0^t{k}_1(t,s)F_1(Y\left(s\right),Y(s-\tau ))ds\hfill \\ & & +\left\{{\int }_0^t{l}_2(t,s)G_2(Y\left(s\right),Y(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_1(t,s)G_1(Y\left(s\right),Y(s-\tau ))d{B}_1\left(s\right)\right\})\hfill \\ & ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))\hfill \\ & & +5\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2({\int }_0^z{k}_2(z,s)F_2(X\left(s\right),X(s-\tau ))ds,{\int }_0^z{k}_2(z,s)F_2(Y\left(s\right),Y(s-\tau ))ds)\hfill \\ & & +5\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2({\int }_0^z{k}_1(z,s)F_1(X\left(s\right),X(s-\tau ))ds,{\int }_0^z{k}_1(z,s)F_1(Y\left(s\right),Y(s-\tau ))ds)\hfill \\ & & +5\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{\left|{\int }_0^z[l_2(z,s)G_2(X\left(s\right),X(s-\tau ))-l_2(z,s)G_2(Y\left(s\right),Y(s-\tau ))]d{B}_2\left(s\right)\right|}^2\hfill \\ & & +5\mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{\left|{\int }_0^z[l_1(z,s)G_1(X\left(s\right),X(s-\tau ))-l_1(z,s)G_1(Y\left(s\right),Y(s-\tau ))]d{B}_1\left(s\right)\right|}^2.\hfill \end{array}$

Now, applying Proposition 1 (iv) and the Doob maximal inequality, one can obtain

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))\hfill \\ & ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))\hfill \\ & & +5\parallel k_2{\parallel }^{2}t\mathbb{E}{\int }_0^t{H}^2(F_2(X\left(s\right),X(s-\tau )),F_2(Y\left(s\right),Y(s-\tau )))ds\hfill \\ & & +5\parallel k_1{\parallel }^{2}t\mathbb{E}{\int }_0^t{H}^2(F_1(X\left(s\right),X(s-\tau )),F_1(Y\left(s\right),Y(s-\tau )))ds\hfill \\ & & +20\parallel l_2{\parallel }^2\mathbb{E}{\int }_0^t{\left|G_2(X\left(s\right),X(s-\tau ))-G_2(Y\left(s\right),Y(s-\tau ))\right|}^{2}ds\hfill \\ & & +20\parallel l_1{\parallel }^2\mathbb{E}{\int }_0^t{\left|G_1(X\left(s\right),X(s-\tau ))-G_1(Y\left(s\right),Y(s-\tau ))\right|}^{2}ds.\hfill \end{array}$

According to assumption (A1), one can obtain

$\begin{array}{ccc}& & \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))\hfill \\ & ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))+\left(5\parallel k_2{\parallel }_{\infty }^2\tilde{T}C+5\parallel k_1{\parallel }_{\infty }^2\tilde{T}C+20\parallel l_2{\parallel }_{\infty }^{2}C+20{\parallel l_1\parallel }_{\infty }^{2}C\right)\hfill \\ & & \times \mathbb{E}{\int }_0^t\left(H^2(X\left(s\right),Y\left(s\right))+H^2(X(s-\tau ),Y(s-\tau ))\right)ds\hfill \end{array}$

and further denoting $A_1=2\left(5\parallel k_2{\parallel }_{\infty }^2\tilde{T}C+5\parallel k_1{\parallel }_{\infty }^2\tilde{T}C+20\parallel l_2{\parallel }_{\infty }^{2}C+20{\parallel l_1\parallel }_{\infty }^{2}C\right)$,

$\begin{array}{ccc}\hfill \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))& ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))+A_1\mathbb{E}{\int }_0^t{H}^2(X\left(s\right),Y\left(s\right))ds\hfill \\ & ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))+A_1{\int }_0^t\mathbb{E}{\displaystyle \underset{z\in [0,s]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))ds.\hfill \end{array}$

Using Gronwall’s inequality, one arrives at

$\begin{array}{ccc}\hfill \mathbb{E}{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))& ⩽& 5{\displaystyle \underset{z\in [0,t]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))exp\left\{A_{1}t\right\}\mathrm{for}\mathrm{every}t\in [0,\tilde{T}].\hfill \end{array}$

Hence,

$\mathbb{E}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(X\left(z\right),Y\left(z\right))⩽5exp\left\{A_1\tilde{T}\right\}{\displaystyle \underset{z\in [0,\tilde{T}]}{sup}}{H}^2(\xi \left(z\right),\zeta \left(z\right))$

and the thesis follows easily. □

From the above theorem, it is easy to see the continuous dependence of the solution on the initial condition.

Now, it will be shown that there is also a continuous dependence of the solution on the nonlinearities $F_1,F_2,G_1,G_2$ and the kernels $k_1,k_2,l_1,l_2$ in the integrals. Therefore, one can investigate Equation (1) with kernels $k_1,k_2,l_1,l_2$ and nonlinearities $F_1,F_2,G_1,G_2$, and the sequence of equations of the form

(5)$\begin{array}{ccc}& & Y^n\left(t\right)\oplus {\int }_0^t{k}_1^n(t,s)F_1^n(Y^n\left(s\right),Y^n(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_1^n(t,s)G_1^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_1\left(s\right)\right\}\hfill \\ & & =\xi \left(t\right)\oplus {\int }_0^t{k}_2^n(t,s)F_2^n(Y^n\left(s\right),Y^n(s-\tau ))ds\oplus \left\{{\int }_0^t{l}_2^n(t,s)G_2^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_2\left(s\right)\right\},\hfill \\ & & \mathrm{for}t\in [0,T],Y^n\left(t\right)=\xi \left(t\right),t\in [-\tau ,0]\hfill \end{array}$

for $n=1,2,\dots$ with kernels $k_1^n,k_2^n,l_1^n,l_2^n$ and nonlinearities $F_1^n,F_2^n,G_1^n,G_2^n$

Theorem 3.

Let $X:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ and $Y^n:[-\tau ,\tilde{T}]\times \Omega \to \mathbb{I}$ be the solutions to Equations (1) and (5), respectively. Suppose that condition (A1) is satisfied and ${\displaystyle \underset{n}{max}}\parallel k_1^n{\parallel }_{\infty }+{\displaystyle \underset{n}{max}}\parallel l_1^n{\parallel }_{\infty }+{\displaystyle \underset{n}{max}}\parallel k_2^n{\parallel }_{\infty }+{\displaystyle \underset{n}{max}}{\parallel l_2^n\parallel }_{\infty }<\infty$. If for every $t\in [0,T]$ and for every $X,Y\in \mathbb{I}$, the sequences

(6)$\left\{{\int }_0^T{H}^2(k_1^n(t,s)F_1^n(X,Y),k_1(t,s)F_1(X,Y))ds\right\},$

$\left\{{\int }_0^T{H}^2(k_2^n(t,s)F_2^n(X,Y),k_2(t,s)F_2(X,Y))ds\right\},$

$\left\{{\int }_0^T{\left|l_1^n(t,s)G_1^n(X,Y)-l_1(t,s)G_1(X,Y)\right|}^{2}ds\right\}$

$\left\{{\int }_0^T{\left|l_2^n(t,s)G_2^n(X,Y)-l_2(t,s)G_2(X,Y)\right|}^{2}ds\right\}$

converge to zero as $n\to \infty$, and then

$\mathbb{E}{H}^2(Y^n\left(t\right),X\left(t\right))⟶0\mathit{as}n\to \infty \mathit{for}\mathit{every}t\in [-\tau ,T].$

Proof. 

Obviously, for $t\in [-\tau ,0]$ one has $Y^n\left(t\right)=\xi \left(t\right)$ and $X\left(t\right)=\xi \left(t\right)$, which gives

(7)$\mathbb{E}{H}^2(Y^n\left(t\right),X\left(t\right))=0\mathrm{for}t\in [-\tau ,0].$

Notice that for $t\in [0,T]$, one has

$\begin{array}{ccc}& & \mathbb{E}{H}^2(X\left(t\right),Y^n\left(t\right))\hfill \\ & ⩽& 4\mathbb{E}{H}^2({\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_2^n(t,s)F_2^n(Y^n\left(s\right),Y^n(s-\tau ))ds)\hfill \\ & & +4\mathbb{E}{H}^2({\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_1^n(t,s)F_1^n(Y^n\left(s\right),Y^n(s-\tau ))ds)\hfill \\ & & +4\mathbb{E}{\left|{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_2^n(t,s)G_2^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_2\left(s\right)\right|}^2\hfill \\ & & +4\mathbb{E}{\left|{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)-{\int }_0^t{l}_1^n(t,s)G_1^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_1\left(s\right)\right|}^2.\hfill \end{array}$

Using the triangle inequality, one can obtain

$\begin{array}{ccc}& & \mathbb{E}{H}^2(X\left(t\right),Y^n\left(t\right))\hfill \\ & ⩽& 8\mathbb{E}{H}^2({\int }_0^t{k}_2(t,s)F_2(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_2^n(t,s)F_2^n(X\left(s\right),X(s-\tau ))ds)\hfill \\ & & +8\mathbb{E}{H}^2({\int }_0^t{k}_2^n(t,s)F_2^n(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_2^n(t,s)F_2^n(Y^n\left(s\right),Y^n(s-\tau ))ds)\hfill \\ & & +8\mathbb{E}{H}^2({\int }_0^t{k}_1(t,s)F_1(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_1^n(t,s)F_1^n(X\left(s\right),X(s-\tau ))ds)\hfill \\ & & +8\mathbb{E}{H}^2({\int }_0^t{k}_1^n(t,s)F_1^n(X\left(s\right),X(s-\tau ))ds,{\int }_0^t{k}_1^n(t,s)F_1^n(Y^n\left(s\right),Y^n(s-\tau ))ds)\hfill \\ & & +8\mathbb{E}{\left|{\int }_0^t{l}_2(t,s)G_2(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_2^n(t,s)G_2^n(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)\right|}^2\hfill \\ & & +8\mathbb{E}{\left|{\int }_0^t{l}_2^n(t,s)G_2^n(X\left(s\right),X(s-\tau ))d{B}_2\left(s\right)-{\int }_0^t{l}_2^n(t,s)G_2^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_2\left(s\right)\right|}^2\hfill \\ & & +8\mathbb{E}{\left|{\int }_0^t{l}_1(t,s)G_1(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)-{\int }_0^t{l}_1^n(t,s)G_1^n(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)\right|}^2\hfill \\ & & +8\mathbb{E}{\left|{\int }_0^t{l}_1^n(t,s)G_1^n(X\left(s\right),X(s-\tau ))d{B}_1\left(s\right)-{\int }_0^t{l}_1^n(t,s)G_1^n(Y^n\left(s\right),Y^n(s-\tau ))d{B}_1\left(s\right)\right|}^2.\hfill \end{array}$

Now, according to Proposition 1 (iii) and the Itô isometry,

$\begin{array}{ccc}& & \mathbb{E}{H}^2(X\left(t\right),Y^n\left(t\right))\hfill \\ & ⩽& 8\tilde{T}\mathbb{E}{\int }_0^t{H}^2(k_2(t,s)F_2(X\left(s\right),X(s-\tau )),k_2^n(t,s)F_2^n(X\left(s\right),X(s-\tau )))ds\hfill \\ & & +8\parallel k_2^n{\parallel }_{\infty }^2\tilde{T}\mathbb{E}{\int }_0^t{H}^2(F_2^n(X\left(s\right),X(s-\tau )),F_2^n(Y^n\left(s\right),Y^n(s-\tau )))ds\hfill \\ & & +8\tilde{T}\mathbb{E}{\int }_0^t{H}^2(k_1(t,s)F_1(X\left(s\right),X(s-\tau )),k_1^n(t,s)F_1^n(X\left(s\right),X(s-\tau )))ds\hfill \\ & & +8\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}\mathbb{E}{\int }_0^t{H}^2(F_1^n(X\left(s\right),X(s-\tau )),F_1^n(Y^n\left(s\right),Y^n(s-\tau )))ds\hfill \\ & & +8\mathbb{E}{\int }_0^t{\left|l_2(t,s)G_2(X\left(s\right),X(s-\tau ))-l_2^n(t,s)G_2^n(X\left(s\right),X(s-\tau ))\right|}^{2}ds\hfill \\ & & +8\parallel l_2^n{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t{\left|G_2^n(X\left(s\right),X(s-\tau ))-G_2^n(Y^n\left(s\right),Y^n(s-\tau ))\right|}^{2}ds\hfill \\ & & +8\mathbb{E}{\int }_0^t{\left|l_1(t,s)G_1(X\left(s\right),X(s-\tau ))-l_1^n(t,s)G_1^n(X\left(s\right),X(s-\tau ))\right|}^{2}ds\hfill \\ & & +8\parallel l_1^n{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t{\left|G_1^n(X\left(s\right),X(s-\tau ))-G_1^n(Y^n\left(s\right),Y^n(s-\tau ))\right|}^{2}ds.\hfill \end{array}$

Let us denote

$\begin{array}{ccc}\hfill D_n& :=& 8\tilde{T}\mathbb{E}{\int }_0^{\tilde{T}}{H}^2(k_2(t,s)F_2(X\left(s\right),X(s-\tau )),k_2^n(t,s)F_2^n(X\left(s\right),X(s-\tau )))ds\hfill \\ & & +8\tilde{T}\mathbb{E}{\int }_0^{\tilde{T}}{H}^2(k_1(t,s)F_1(X\left(s\right),X(s-\tau )),k_1^n(t,s)F_1^n(X\left(s\right),X(s-\tau )))ds\hfill \\ & & +8\mathbb{E}{\int }_0^{\tilde{T}}{\left|l_2(t,s)G_2(X\left(s\right),X(s-\tau ))-l_2^n(t,s)G_2^n(X\left(s\right),X(s-\tau ))\right|}^{2}ds\hfill \\ & & +8\mathbb{E}{\int }_0^{\tilde{T}}{\left|l_1(t,s)G_1(X\left(s\right),X(s-\tau ))-l_1^n(t,s)G_1^n(X\left(s\right),X(s-\tau ))\right|}^{2}ds\hfill \end{array}$

and

$\begin{array}{ccc}\hfill C_n\left(t\right)& :=& 8\parallel k_2^n{\parallel }_{\infty }^2\tilde{T}\mathbb{E}{\int }_0^t{H}^2(F_2^n(X\left(s\right),X(s-\tau )),F_2^n(Y^n\left(s\right),Y^n(s-\tau )))ds\hfill \\ & & +8\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}\mathbb{E}{\int }_0^t{H}^2(F_1^n(X\left(s\right),X(s-\tau )),F_1^n(Y^n\left(s\right),Y^n(s-\tau )))ds\hfill \\ & & +8\parallel l_2^n{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t{\left|G_2^n(X\left(s\right),X(s-\tau ))-G_2^n(Y^n\left(s\right),Y^n(s-\tau ))\right|}^{2}ds\hfill \\ & & +8\parallel l_1^n{\parallel }_{\infty }^2\mathbb{E}{\int }_0^t{\left|G_1^n(X\left(s\right),X(s-\tau ))-G_1^n(Y^n\left(s\right),Y^n(s-\tau ))\right|}^{2}ds.\hfill \end{array}$

According to the Lebesgue dominated convergence theorem and assumption (A1), one can find that the sequence $\left\{D_n\right\}$ converges to zero as $n\to \infty$.

Applying assumption (A1) and the Fubini theorem,

$\begin{array}{ccc}\hfill C_n\left(t\right)& ⩽& 8C\left({\displaystyle \underset{n}{max}}\parallel k_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}+{\displaystyle \underset{n}{max}}\parallel l_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}{\parallel l_1^n\parallel }_{\infty }^2\right)\hfill \\ & & \times {\int }_0^t\mathbb{E}\left(H^2(X\left(s\right),Y^n\left(s\right))+H^2(X(s-\tau ),Y^n(s-\tau ))\right)ds\hfill \\ & ⩽& 16C\left({\displaystyle \underset{n}{max}}\parallel k_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}+{\displaystyle \underset{n}{max}}\parallel l_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}{\parallel l_1^n\parallel }_{\infty }^2\right)\hfill \\ & & \times {\int }_0^t\mathbb{E}{H}^2(X\left(s\right),Y^n\left(s\right))ds.\hfill \end{array}$

Therefore,

$\begin{array}{ccc}\hfill \mathbb{E}{H}^2(X\left(t\right),Y^n\left(t\right))& ⩽& D_n+16C\left({\displaystyle \underset{n}{max}}\parallel k_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}+{\displaystyle \underset{n}{max}}\parallel l_2^n{\parallel }_{\infty }^2+{\displaystyle \underset{n}{max}}{\parallel l_1^n\parallel }_{\infty }^2\right)\hfill \\ & & \times {\int }_0^t\mathbb{E}{H}^2(X\left(s\right),Y^n\left(s\right))ds\mathrm{for}\mathrm{every}t\in [0,\tilde{T}].\hfill \end{array}$

Now, according to the Gronwall inequality, one has

$\begin{array}{ccc}\hfill \mathbb{E}{H}^2(Y_n\left(t\right),X\left(t\right))& ⩽& D_{n}exp\left\{At\right\}\mathrm{for}\mathrm{every}t\in [0,\tilde{T}],\hfill \end{array}$

where $A=16C\left({max}_n\parallel k_2^n{\parallel }_{\infty }^2+{max}_n\parallel k_1^n{\parallel }_{\infty }^2\tilde{T}+{max}_n\parallel l_2^n{\parallel }_{\infty }^2+{max}_n{\parallel l_1^n\parallel }_{\infty }^2\right)$. Since $D_n\stackrel{n\to \infty }{⟶}0$,

$\mathbb{E}{H}^2(X\left(t\right),Y_n\left(t\right))\stackrel{n\to \infty }{⟶}0\mathrm{for}\mathrm{every}t\in [0,\tilde{T}],$

This, together with (7), completes the proof. □

Due to the above two theorems, one gains confidence that slightly perturbed data in Equation (1) do not lead to solutions that are far from the solution of Equation (1) with unperturbed data.

5. Concluding Remarks

In this paper, delayed stochastic integral equations are considered in the context of interval-valued mappings. Such a framework allows for the study of symmetric equations, and clearly distinguishes them from classical stochastic integral equations with single-valued mappings. The presented research is of a fundamental nature and constitutes the basis for further theoretical and practical research. In the foundations of research included in this paper, the author presents a theorem about the existence of a unique solution to the equation he is considering. This result is obtained with the assumption of Lipschitz continuity of nonlinearities appearing on both sides of the equation. It is also justified that if the given equation changes only slightly, i.e., the initial value, kernel or nonlinearity changes slightly, then the solution also changes only slightly. Thus, it opens the way for future research on determining approximate solutions and on finding approximate solutions using numerical methods by practitioners. Further theoretical research can also be conducted in the future. One can think about examining the existence of a solution under conditions, e.g., of kernel discontinuity or a condition other than the Lipschitz continuity of the nonlinear coefficients of drift and diffusion in the equation.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The author declares no conflicts of interest.

Footnotes

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References

1. Arnold, L. Stochastic Differential Equations: Theory and Applications; John Wiley & Sons: New York, NY, USA, 1974.

2. Gihman, I.I.; Skorohod, A.V. Stochastic Differential Equations; Springer: Berlin/Heidelberg, Germany, 1972.

3. Øksendal, B. Stochastic Differential Equations: An Introduction with Applications; Springer: Berlin/Heidelberg, Germany, 2003.

4. Malinowski, M.T. Strong solutions to stochastic fuzzy differential equations of Itô type. Math. Comput. Model.; 2012; 55, pp. 918-928. [DOI: https://dx.doi.org/10.1016/j.mcm.2011.09.018]

5. Priyadharsini, J.; Balasubramaniam, P. Existence of fuzzy fractional stochastic differential system with impulses. Comput. Appl. Math.; 2020; 39, 195. [DOI: https://dx.doi.org/10.1007/s40314-020-01229-0]

6. Jafari, H.; Farahani, H. An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion. Stochastics Dyn.; 2023; 23, 2350017. [DOI: https://dx.doi.org/10.1142/S021949372350017X]

7. Luo, D.; Wang, X.; Caraballo, T.; Zhu, Q. Ulam–Hyers stability of Caputo-type fractional fuzzy stochastic differential equations with delay. Commun. Nonlinear Sci. Numer. Simul.; 2023; 121, 107229. [DOI: https://dx.doi.org/10.1016/j.cnsns.2023.107229]

8. Li, J.; Li, S.; Ogura, Y. Strong solution of Itô type set-valued stochastic differential equations. Acta Math. Sin.; 2010; 26, pp. 1739-1748. [DOI: https://dx.doi.org/10.1007/s10114-010-8298-x]

9. Mitoma, I.; Okazaki, Y.; Zhang, J. Set-valued stochastic differential equation in M-type 2 Banach space. Commun. Stoch. Anal.; 2010; 4, pp. 215-237. [DOI: https://dx.doi.org/10.31390/cosa.4.2.06]

10. Zhang, J.; Li, S.; Mitoma, I.; Okazaki, Y. On the solutions of set-valued stochastic differential equations in M-type 2 Banach spaces. Tohoku Math. J.; 2009; 61, pp. 417-440. [DOI: https://dx.doi.org/10.2748/tmj/1255700202]

11. Ogura, Y. On stochastic differential equations with fuzzy set coefficients. Soft Methods for Handling Variability and Imprecision; Dubois, D.; Lubiano, M.A.; Prade, H.; Gil, M.Á.; Grzegorzewski, P.; Hryniewicz, O. Springer: Berlin/Heidelberg, Germany, 2008; Volume 48, pp. 263-270.

12. Lakshmikantham, V.; Bhaskar, T.G.; Devi, J.V. Theory of Set Differential Equations in Metric Spaces; Cambridge Scientific Publishers: Cambrigde, UK, 2006.

13. Hu, S.; Papageorgiou, N.S. Handbook of Multivalued Analysis, Vol. I: Theory; Kluwer Academic Publishers: Boston, MA, USA, 1997.

14. Hiai, F.; Umegaki, H. Integrals, conditional expectation, and martingales of multivalued functions. J. Multivar. Anal.; 1977; 7, pp. 149-182. [DOI: https://dx.doi.org/10.1016/0047-259X(77)90037-9]