Analytical and Computational Investigations of Stochastic Functional Integral Equations: Solution Existence and Euler–Karhunen–Loève Simulation

Abstract

This paper presents a comprehensive investigation into the solution existence of stochastic functional integral equations within real separable Banach spaces, emphasizing the establishment of sufficient conditions. Leveraging advanced mathematical tools including probability measures of noncompactness and Petryshyn’s fixed-point theorem adapted for stochastic processes, a robust analytical framework is developed. Additionally, this paper introduces the Euler–Karhunen–Loève method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly suited for handling continuous-time processes with an infinite number of random variables. By conducting thorough analysis and computational simulations, which also involve implementing the Euler–Karhunen–Loève method, this paper effectively highlights the practical relevance of the proposed methodology. Two specific instances, namely, the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay model, are utilized as illustrative examples to underscore the effectiveness of this approach in tackling real-world challenges encountered in the realms of finance and stochastic dynamics.

1. Introduction

Recent advancements in stochastic differential equations (SDEs) have underscored their versatility and efficacy in modeling complex systems across diverse domains. In population dynamics modeling, Thieu et al. introduced a coupled system of reflecting Skorokhod-type SDEs, offering insights into the dynamics of interacting populations with applications in neuroscience [1]. Tang et al. proposed a Bayesian model combining phylodynamic inference and stochastic epidemic models, facilitating the estimation of epidemiological parameters and understanding of disease transmission dynamics [2]. Almalki et al. developed numerical techniques for approximating the probability distribution of random variable transformations, extending the applicability of SDEs in modeling complex systems such as the heat equation with stochastic inputs [3]. Xu et al. investigated stochastic resonance phenomena in neural systems using the Hindmarsh–Rose neural model, providing insights into brain dynamics and neurological disorders [4]. Shamsipour et al. discussed modern approaches to stochastic modeling, including non-stationary simulation methods and conditional generative adversarial networks (GANs), highlighting their applications in the earth science and data science domains [5]. Recent studies by Chalishajar et al. [6] and Kasinathan et al. [7] have further advanced the field by exploring exponential stability and controllability of fractional neutral stochastic differential equations, providing new insights into the behavior of complex stochastic systems with delays and impulses. These investigations emphasize the flexibility and efficiency of SDEs in capturing random fluctuations and variability across a wide range of disciplines, thus facilitating the progression of stochastic modeling and simulation techniques.

Furthermore, stochastic integral equations (SIEs) have emerged as indispensable tools with wide-ranging implications across various disciplines. In electrical engineering, Kolářová [8] demonstrated their utility in modeling RLC electrical circuits with noisy parameters, enhancing our understanding of complex network behaviors. Similarly, Hromadka [9] utilized stochastic integral equations to analyze modeling errors in rainfall–runoff hydrograph models, improving the accuracy of hydrological predictions. Kolářová and Brančík[10] extended this application to electrical networks, providing analytic solutions to stochastic integral equations and enhancing predictive capabilities. Postan [11] explored their use in modeling computer networks, shedding light on information distribution within network nodes.

Moreover, recent research has focused on establishing the existence and uniqueness of solutions for these equations. Balachandran and Kim [12] generalized results by utilizing admissibility theory and fixed point theorems. Similarly, Balachandran et al. [13] extended their findings using the measure of noncompactness and the Darbo fixed point theorem. Gacki et al. [14] explored the stability of solutions, while Cahlon [15] investigated general stochastic functional integral equations. Subramaniam et al. [16] broadened the scope of their study, employing similar techniques. Additionally, works by Hu and Lerner [17], Turo [18], and Mao [19] have contributed insights into specific classes of stochastic equations. These collective efforts have significantly advanced our understanding of stochastic integral equations and their applications. These studies have expanded the usefulness of established methods and offered valuable perspectives on the characteristics of stochastic equations across different contexts.

In this manuscript, we establish and confirm sufficient conditions for the existence of a solution to the following stochastic functional integral equation within a real separable Banach space $\mathcal{B}$, over the probability space $(\Lambda ,\mathcal{F},\mathbb{P})$, which we denote as $(\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$. The operator $\mathcal{L}$ is defined as:

$$\begin{array}{cc}\hfill X_t& =\mathcal{L}(t,X\left(\alpha \left(t\right)\right)),{\int }_0^{T_f}{h}_1(\xi ,X_{\xi },X\left({\alpha }_1\left(\xi \right)\right))d\xi ,\hfill \\ \hfill & {\int }_0^{\phi \left(t\right)}{h}_2(\xi ,X_{\xi },X\left({\alpha }_2\left(\xi \right)\right))d\xi ,{\int }_0^t{h}_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right))dW\left(\xi \right)),\hfill \end{array}$$

(1)

where $X_t$ is a stochastic process and $t\in \Omega :=[0,T_f]$. The operator $\mathcal{L}$ is in $C(\Omega \times {\mathbb{R}}^4,\mathbb{R})$, and the functions $h_1$, $h_2$, and $h_3$ belong to $C({\Omega }^2\times \mathbb{R},\mathbb{R})$, $C(\Omega \times [0,L]\times \mathbb{R},\mathbb{R})$, and $C({\Omega }^2\times \mathbb{R},\mathbb{R})$, respectively. Moreover, the functions $\alpha$, ${\alpha }_1$, ${\alpha }_3:\Omega \to \Omega$, ${\alpha }_2:[0,L]\to \Omega$, and $\phi :\Omega \to {\mathbb{R}}_{+}$ satisfy $\phi \left(t\right)\le L$.

In this context, the principal techniques employed in these analyses involve the measure of noncompactness methodology and the utilization of Petryshyn’s fixed point theorem. Petryshyn’s fixed point theorem, renowned as a powerful tool in nonlinear analysis, has found widespread application in various fields, including mathematical biology, nonlinear dynamics, and integral equations. Originating from the pioneering works of Petryshyn [20] and further refined by subsequent researchers [21,22,23,24,25], this theorem has proven instrumental in establishing the existence of solutions for a broad range of functional and nonlinear Volterra integral equations. Notably, Deep et al. [26] demonstrated the theorem’s efficacy in solving functional integral equations in Banach algebras, underscoring its applicability across diverse contexts. The utility of the theorem extends beyond theoretical realms, with Deep et al. proposing a numerical method based on the modified homotopy perturbation approach for resolving nonlinear Volterra integral equations [27]. Furthermore, Alsaadi et al. [25] generalized the theorem to accommodate the product of multiple nonlinear integral equations, showcasing its versatility and robustness. Such advancements highlight the theorem’s significance in addressing complex mathematical problems and its potential for further exploration in nonlinear analysis and its applications.

This paper is structured as follows: Section 2 provides essential groundwork by introducing key concepts in stochastic processes, including the Wiener process, Itô integral, and measures of noncompactness. Additionally, it discusses stochastic contraction mappings and Petryshyn’s fixed-point theorem, adapted for stochastic processes. In Section 3, the focus is on establishing the existence of solutions for stochastic functional integral equations under specific conditions, using a modified Petryshyn’s fixed-point theorem. Section 4 introduces the Euler–Karhunen–Loève Method, which utilizes the Karhunen–Loève expansion to represent stochastic processes, particularly useful for continuous-time processes with an infinite number of random variables. Section 5 demonstrates the practical implications of the proposed methods through numerical simulations, evaluating accuracy and performance, and exploring the influence of mean-reversion speed on model dynamics. Furthermore, sufficient conditions for solution existence for certain models are examined. Finally, Section 6 concludes the paper.

2. Preliminaries

In this section, we introduce the foundational concepts and mathematical structures that are essential for the subsequent analysis. We begin by discussing the properties of real separable Banach spaces, followed by an exploration of Wiener processes within these spaces. These preliminary concepts lay the groundwork for understanding the advanced stochastic processes and integral equations addressed in later sections.

A real separable Banach space is a complete normed vector space $\mathcal{B}$ over the real numbers that contains a countable dense subset. Specifically, this means that there exists a countable set $D\subset \mathcal{B}$ such that for every element $x\in \mathcal{B}$ and every $ϵ>0$, there is an element $d\in D$ satisfying $\parallel x-d\parallel <ϵ$. The completeness property ensures that every Cauchy sequence in $\mathcal{B}$ converges to a limit within $\mathcal{B}$, and the separability condition implies that the space can be densely approximated by a countable set, which is advantageous for various analytical and computational techniques. For instance, ${\mathcal{L}}^2\left([0,T_f]\right)$, the space of square-integrable functions over the interval $[0,T_f]$, is a real separable Banach space. It is also a Hilbert space due to its inner product structure, which is defined by ${\langle f,g\rangle }_{{\mathcal{L}}^2}={\int }_0^{T_f}f\left(t\right)g\left(t\right)dt$. The norm in ${\mathcal{L}}^2\left([0,T_f]\right)$ is given by ${\parallel f\parallel }_{{\mathcal{L}}^2}={\left({\int }_0^{T_f}{\left|f\left(t\right)\right|}^{2}dt\right)}^{1/2}$, and the space has a countable dense subset, making it a practical example of a real separable Banach space used in various applications. In the following sections, we will consider stochastic processes and their properties within ${\mathcal{L}}^2\left([0,T_f]\right)$, leveraging its structure as a real separable Banach space to facilitate our analysis.

2.1. Wiener Process in a Real Separable Banach Space

A fundamental element in a model depicting stochastic evolution is the foundational concept of Brownian motion or the Wiener process. Various approaches exist to define and characterize the Wiener process $W=\left\{W\right(t),t\ge 0\}$, and one such method involves the following properties:

Definition 1

([28]). A covariance operator $R:{\mathcal{B}}^{*}\to \mathcal{B}$ is a bounded linear operator that characterizes the second-order properties of the Gaussian distribution of the Wiener process. Here, ${\mathcal{B}}^{*}$ denotes the dual space of $\mathcal{B}$. For a Wiener process ${\left\{W\left(t\right)\right\}}_{t\in [0,1]}$, the covariance operator R satisfies:

$$\mathbb{E}[\langle W(t),f\rangle \langle W(t),g\rangle ]=t\langle Rf,g\rangle$$

for all $f,g\in {\mathcal{B}}^{*}$, where $\langle ·,·\rangle$ denotes the duality pairing between $\mathcal{B}$ and ${\mathcal{B}}^{*}$. This means that for any $t\in [0,1]$, the random variable $W\left(t\right)$ is Gaussian with mean 0 and its covariance structure is given by $tR$.

Definition 2

([28]). A random process ${\left\{W\left(t\right)\right\}}_{t\in [0,1]}$ in a separable Banach space $\mathcal{B}$ is termed a Wiener process if it satisfies the following conditions:

• $W\left(0\right)=0$ almost surely.
• The increments $W\left(t_2\right)-W\left(t_1\right),W\left(t_3\right)-W\left(t_2\right),\dots ,W\left(t_n\right)-W\left(t_{n-1}\right)$ are independent for any $0\le t_1<t_2<\dots <t_n\le 1$.
• For any $t\in [0,1]$, $W\left(t\right)$ is Gaussian with mean 0 and covariance operator $tR$, where $R:{\mathcal{B}}^{*}\to \mathcal{B}$ is a bounded linear operator satisfying $\mathbb{E}[\langle W(t),f\rangle \langle W(t),g\rangle ]=t\langle Rf,g\rangle$ for all $f,g\in {\mathcal{B}}^{*}$.
• The paths of the Wiener process ${\left\{W\left(t\right)\right\}}_{t\in [0,1]}$ are continuous.

Theorem 1

([29]). If $X_t$ is a continuous adapted process, then the Itô integral ${\int }_0^{t}X\left(\xi \right)dW\left(\xi \right)$ exists. In particular, the integral ${\int }_0^{t}f\left(W\left(\xi \right)\right)dW\left(\xi \right)$, where f is a continuous function on $\mathbb{R}$, is well-defined.

2.2. Stochastic Petryshyn’s Fixed-Point Theorem

In this subsection, we define the “Probability Measure of Noncompactness” and the “Hausdorff Probability Measure of Noncompactness,” adapting concepts initially introduced in works such as [30,31]. Additionally, we modify the properties of the “Hausdorff Measure of Noncompactness,” as outlined in [32], to suit the context of probability. It is evident that the foundational principles and fundamental findings discussed below are universally applicable across all measures of noncompactness. These results offer invaluable insights into the nature of noncompactness in various mathematical contexts and serve as the cornerstone for more intricate investigations and developments within the field. Their significance and ubiquity in mathematical discourse underscore their importance as essential components of mathematical analysis.

Definition 3

(Probability Measure of Noncompactness). Let $\mu \left(A\right)$ denote the probability measure of noncompactness for a bounded subset A within a real separable Banach space $\mathcal{B}$, where $(\Lambda ,\mathcal{F},\mathbb{P})$ denotes the underlying probability space. This measure quantifies the probability that a stochastic process $X_t$ associated with the probability space lies outside a compact subset of $\mathcal{B}$. Formally, it is expressed as:

$$\mu \left(A\right)=inf\left\{ϵ>0:\mathbb{P}(X_t\notin K_{ϵ})>0\right\},$$

(2)

where $K_{ϵ}$ is a compact subset of $\mathcal{B}$ with diameter less than ϵ, and $\mathbb{P}$ denotes the probability measure associated with the stochastic process $X_t$.

It is evident that the fundamental findings outlined below hold universally across all measures of noncompactness. These results serve as foundational principles within the domain of mathematical analysis, offering invaluable insights into the nature of noncompactness in various contexts. Furthermore, they form the cornerstone for more intricate investigations and developments within the field, exemplifying their significance and ubiquity in mathematical discourse.

Definition 4

(Hausdorff Probability Measure of Noncompactness). Let $\mu \left(A\right)$ denote the Hausdorff probability measure of noncompactness for a bounded subset A within the real separable Banach space $\mathcal{B}$, defined over the probability space $(\Lambda ,\mathcal{F},\mathbb{P})$. This measure quantifies the probability that a random variable chosen uniformly from A does not belong to a finite ϵ-net in $\mathcal{B}$. Formally, it is defined as:

$$\mu \left(A\right)=inf\left\{ϵ>0:thereexists\mathrm{a}finiteϵ-netforthesetAin\mathcal{B}\right\},$$

(3)

where a finite ϵ-net for A in $\mathcal{B}$ consists of a finite set $\{z_1,z_2,\dots ,z_m\}\subset \mathcal{B}$ such that the balls $B_{ϵ}\left(z_1\right),B_{ϵ}\left(z_2\right),\dots ,B_{ϵ}\left(z_m\right)$ cover A.

This definition provides a probabilistic framework for analyzing the noncompactness of sets within the real separable Banach space $\mathcal{B}$, defined over the probability space $(\Lambda ,\mathcal{F},\mathbb{P})$, allowing for a deeper understanding of their geometric properties in the context of stochastic processes.

Proposition 1 

(Properties of the Hausdorff Probability Measure of Noncompactness). Assume that $(\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$ represents a real separable Banach space for the stochastic process $X_t$, where Λ is the sample space, $\mathcal{F}$ is the Borel σ-algebra of $\mathcal{B}$, and $\mathbb{P}$ is the probability measure. Considering $v,\alpha \in \mathbb{R}$, and $A,B$ being bounded subsets of this space, we can state the following assertions:

• Monotonicity: For any two bounded sets $A,B\subset \mathcal{B}$ such that $A\subset B$, we have $\mu \left(A\right)\le \mu \left(B\right)$.
• Translation Invariance: The measure $\mu \left(A\right)$ is invariant under translations. That is, for any bounded set A and any fixed vector $v\in \mathcal{B}$, $\mu (A+v)=\mu \left(A\right)$.
• Scaling Property: If A is a bounded set and $\alpha >0$, then $\mu \left(\alpha A\right)=\alpha \mu \left(A\right)$.
• Continuity: The measure $\mu \left(A\right)$ is continuous with respect to the Hausdorff metric. That is, if $A_n$ is a sequence of bounded sets converging to A in the Hausdorff metric, then $\mu \left(A_n\right)\to \mu \left(A\right)$ as $n\to \infty$.
• Equivalence with Other Measures: The Hausdorff probability measure of noncompactness $\mu \left(A\right)$ is equivalent to other measures of noncompactness, such as the Kuratowski measure $\beta \left(A\right)$. Specifically, for any bounded set $A\subset \mathcal{B}$, we have $\mu \left(A\right)\le \beta \left(A\right)\le 2\mu \left(A\right)$.

In the forthcoming discussion, we delve into the function space $(\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$, encompassing stochastic processes characterized by real-valued functions that are continuous over the interval $\Omega$. This space is endowed with the standard norm defined as $\parallel X_t{\parallel }_{\mathcal{B}}=sup\left\{\right|X_t|:t\in \Omega \}$, reflecting the supremum of the process over $\Omega$. It is crucial to define the modulus of continuity for a stochastic process $X_t\in (\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$, given by

$$\omega (X,ϵ)=sup\left\{\right|X_t-X_s|:|t-s|\le ϵ\}.$$

This modulus measures the rate of change of the process over small intervals. Notably, as $ϵ\to 0$, $\omega (X_t,ϵ)$ tends to zero due to the uniform continuity of $X_t$ over $\Omega$. Furthermore, if this convergence holds uniformly for all processes $X_t$ within a bounded set $A\subset \mathcal{B}$, then A is said to be equicontinuous. Conversely, equicontinuity implies the uniform convergence of $\omega (X,ϵ)$ as $ϵ\to 0$ for all $X_t\in A$.

The measures of noncompactness defined within the space $(\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$ can be expressed as:

$$\mu \left(A\right)=\underset{ϵ\to 0}{lim}sup\left\{\omega (X_t,ϵ)|X_t\in A,ϵ>0\right\},$$

(4)

for all bounded sets $A\subset \mathcal{B}$.

In the forthcoming sections, we will employ Equation (4) extensively. Moreover, closely associated with the measures of noncompactness is the concept of k-set contraction. This concept will hold considerable importance in our subsequent discussions and analyses.

Definition 5

(Contraction Mapping). Consider a mapping $\mathcal{F}:\mathcal{B}\to \mathcal{B}$ defined on the real separable Banach space $\mathcal{B}$. $\mathcal{F}$ is a stochastic contraction mapping if there exists $0<k<1$ such that

$$\parallel \mathcal{F}\left(X_t\right)-\mathcal{F}\left(Y_t\right){\parallel }_{\mathcal{B}}\le k·{\parallel X_t-Y_t\parallel }_{\mathcal{B}},$$

(5)

holds for all stochastic processes $X_t,Y_t\in \mathcal{B}$.

Definition 6

(Densifying Property). A stochastic mapping $\mathcal{F}$ on a real separable Banach space $\mathcal{B}$ has a densifying property if, for any bounded subset $B\subset \mathcal{B}$, the image $\mathcal{F}\left(B\right)$ remains bounded and the probability measure of $\mathcal{F}\left(B\right)$ satisfies $\mathbb{P}\left(\mathcal{F}\right(B\left)\right)<\mathbb{P}\left(B\right)$ for all $\mathbb{P}\left(B\right)>0$.

Theorem 2

(Petryshyn’s Fixed Point Theorem for Stochastic Processes). Let $\mathcal{F}:{\overline{B}}_r\to \mathcal{B}$ be a mapping that densifies the space and satisfies the following boundary condition:

$$If\mathcal{F}\left(X_t\right)=k{X}_{t}forsome{X}_{t}ontheboundary\partial B_r,thrnk\le 1.$$

(6)

Under these conditions, Petryshyn’s fixed point theorem guarantees the existence of fixed points for $\mathcal{F}$ within the closed ball ${\overline{B}}_r$ when considering the stochastic process $X_t$.

Proof. 

The proof of this theorem follows from Petryshyn’s original work [32] on Banach spaces. It is important to note that the theorem can be adapted to apply to the setting of $(\mathcal{B},\Lambda ,\mathcal{F},\mathbb{P})$ and mappings involving stochastic processes without any significant difficulties.    □

3. Exploring Solution Existence: Analytical Framework

The stochastic process $X_t$ belongs to the real separable Banach space $\mathcal{B}$ equipped with a norm ${\parallel ·\parallel }_{\mathcal{B}}$. This process is defined over the probability space $(\Lambda ,\mathcal{F},\mathbb{P})$.

We proceed to explore the existence of solutions to the stochastic functional integral equation represented by (1) based on the following conditions:

(K1)

Local Lipschitz Condition: There exist nonnegative constants $L_c$, $c_1$, $c_2$, $c_3$, with $L_c<1$, ensuring the Lipschitz continuity of $\mathcal{L}$:

$$\left|\mathcal{L}(t,V_t,U_1,U_2,U_3)-{\mathcal{L}}_{\mathcal{c}}(t,{\overline{V}}_t,{\overline{U}}_1,{\overline{U}}_2,{\overline{U}}_3)\right|\le L_c\parallel V_t-{\overline{V}}_t{\parallel }_{\mathcal{B}}+\sum _{i=1}^3{c}_i|U_i-{\overline{U}}_i|.$$

Herein, ${\parallel ·\parallel }_{\mathcal{B}}$ denotes the norm in the real separable Banach space $\mathcal{B}$, ensuring the Lipschitz continuity property of $\mathcal{L}$ with respect to the stochastic process $X_t$.

(K2)

Bounded Condition: There exists $r_0\ge 0$ such that $\mathcal{L}$ satisfies the bounded condition:

$$\begin{array}{cc}& \left|\mathcal{L}(t,V_t,U_1,U_2,U_3)\right|\hfill \\ & =sup\left\{\right|\mathcal{L}(t,V_t,U_1,U_2,U_3)|:t\in \Omega ,\parallel X_t{\parallel }_{\mathcal{B}}\le r_0,|U_1|\le T_f{M}_1,\left|U_2\right|\le L{M}_2,\hfill \\ & |U_3|\le \lambda M_3\}\le r_0,\hfill \end{array}$$

where $M_1$, $M_2$, $M_3$, and $\lambda$ are defined as follows:

$$\begin{array}{cc}\hfill & M_1=sup\left\{|h_1(t,X_t,X_{{\alpha }_1})|:\mathrm{for}\mathrm{all}t\in \Omega ,\mathrm{and}\parallel X_t{\parallel }_{\mathcal{B}}\le r_0\right\},\hfill \\ \hfill & M_2=sup\left\{|h_2(t,X_t,X_{{\alpha }_2})|:\mathrm{for}\mathrm{all}t\in [0,L]\subseteq \Omega ,\mathrm{and}{\parallel X_t\parallel }_{\mathcal{B}}\le r_0\right\},\hfill \\ \hfill & M_3=sup\left\{|h_3(t,X_t,X_{{\alpha }_3})|:\mathrm{for}\mathrm{all}t\in \Omega ,\mathrm{and}\parallel X_t{\parallel }_{\mathcal{B}}\le r_0\right\},\hfill \\ \hfill & \lambda =sup\left\{\left|W\right(t\left)\right|:t\in \Omega \right\}.\hfill \end{array}$$

These conditions lead to the subsequent theorem.

Theorem 3.

Under the assumptions (K1) and (K2), there exists at least one solution to Equation (1) within the real separable Banach space $\mathcal{B}$ defined over the probability space $(\Lambda ,\mathcal{F},\mathbb{P})$.

Proof. 

We will establish this existence result through a systematic application of fixed point theory. Specifically, we will use Theorem 2, which provides conditions for the existence of fixed points for operators in Banach spaces. The proof will proceed in three main steps: (1) defining and establishing the continuity of an appropriate operator, (2) verifying the densifying condition, and (3) checking boundary conditions.    □

To establish this result employing Theorem 2 as our principal tool, we define the operator $\mathcal{F}:{\overline{B}}_r\to \mathcal{B}$ as follows:

$$\begin{array}{cc}\hfill \mathcal{F}{X}_t& :=\mathcal{L}(t,X\left(\alpha \left(t\right)\right),{\int }_0^{T_f}{h}_1(\xi ,X_{\xi },X\left({\alpha }_1\left(\xi \right)\right))d\xi ,\hfill \\ \hfill & {\int }_0^{\phi \left(t\right)}{h}_2(\xi ,X_{\xi },X\left({\alpha }_2\left(\xi \right)\right))d\xi ,{\int }_0^t{h}_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right))dW\left(\xi \right)).\hfill \end{array}$$

(7)

Here, the operator $\mathcal{F}$ combines several key components: a local operator $\mathcal{L}$, multiple delay terms through functions ${\alpha }_i$, and various integral terms including both deterministic and stochastic integrals. The presence of the stochastic integral term necessitates careful treatment under the probability measure.

Presently, we establish the continuity of the operator $\mathcal{F}$ within the ball $B_{r_0}$. This is accomplished by considering $ϵ>0$ and selecting arbitrary $X_t$ and $Y_t$ from $B_{r_0}$ with $\parallel X_t-Y_t{\parallel }_{\mathcal{B}}\le ϵ$. The ball $B_{r_0}$ is chosen specifically to ensure the boundedness of our operator, where $r_0$ is determined by the assumptions (K1) and (K2). Subsequently, for $t\in \Omega$, we obtain the following:

$$\begin{array}{cc}\hfill |\mathcal{F}{X}_t-\mathcal{F}{Y}_t|\le & L_c{\parallel X_t-Y_t\parallel }_{\mathcal{B}}\hfill \\ \hfill & +c_1{\int }_0^{T_f}|h_1(\xi ,X_{\xi },X\left({\alpha }_1\left(\xi \right)\right))-h_1(\xi ,Y_{\xi },Y\left({\alpha }_1\left(\xi \right)\right))|d\xi \hfill \\ \hfill & +c_2{\int }_0^{\phi \left(t\right)}|h_2(\xi ,X_{\xi },X\left({\alpha }_2\left(\xi \right)\right))-h_2(\xi ,Y_{\xi },Y\left({\alpha }_2\left(\xi \right)\right))|d\xi \hfill \\ \hfill & +c_3{\int }_0^t|h_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right))-h_3(\xi ,Y_{\xi },Y\left({\alpha }_3\left(\xi \right)\right))|dW\left(\xi \right).\hfill \end{array}$$

The above inequality is derived by applying the triangle inequality and using the Lipschitz continuity of $\mathcal{L}$ with constant $L_c$. The constants $c_1$, $c_2$, and $c_3$ arise from the boundedness conditions in (K1).

Now, utilizing Theorem 1 concerning stochastic integrals, we arrive at the following inequality:

$$|\mathcal{F}{X}_t-\mathcal{F}{Y}_t|\le L_c{\parallel X_t-Y_t\parallel }_{\mathcal{B}}+c_1{T}_f\omega (h_1,ϵ)+c_{2}L\omega (h_2,ϵ)+c_3\lambda T_f\omega (h_3,ϵ),$$

(8)

herein, $\parallel X_t{\parallel }_{\mathcal{B}}\le ϵ,{\parallel Y_t\parallel }_{\mathcal{B}}\le ϵ$, and $\parallel X_t-Y_t{\parallel }_{\mathcal{B}}\le ϵ$ for $ϵ>0$, we define

$$\begin{array}{cc}\hfill \omega (h_1,ϵ)& =sup\left\{|h_1(t,X_t,X_{{\alpha }_1}))-h_1(t,Y_t,Y_{{\alpha }_1})\left)\right|:t\in \Omega \right\},\hfill \\ \hfill \omega (h_2,ϵ)& =sup\left\{|h_2(t,X_t,X_{{\alpha }_2}))-h_2(t,Y_t,Y_{{\alpha }_2})\left)\right|:t\in [0,L]\subseteq \Omega \right\},\hfill \\ \hfill \omega (h_3,ϵ)& =sup\left\{|h_3(t,X_t,X_{{\alpha }_3}))-h_3(t,Y_t,Y_{{\alpha }_3})\left)\right|:t\in \Omega \right\}.\hfill \end{array}$$

These moduli of continuity $\omega (h_i,ϵ)$ are crucial in establishing the uniform continuity of our operator. The supremum is taken over all possible values within our domain, ensuring that our estimates hold uniformly.

These definitions apply to diverse parameters $ϵ$, measuring the maximum differences among corresponding functions $h_i=h_i(t,X_t,X_{{\alpha }_i}),i=1,2,3,$ under specific conditions. Given that the functions $h_i,i=1,2,3,$ are uniformly continuous, it follows that $\omega (h_i,ϵ)\to 0$ as $ϵ\to 0$. This convergence property is essential for establishing the continuity of our operator $\mathcal{F}$. Consequently, the provided estimates demonstrate the continuity of the operator $\mathcal{F}$ on the ball $B_{r_0}$.

Next, we aim to establish that the operator $\mathcal{F}[·]$ adheres to the densifying condition concerning the measure $\mu (·)$ as outlined in (4). For this purpose, we select a predetermined $ϵ>0$. Consider $X_t\in A$, where A denotes a bounded subset of the real separable Banach space $\mathcal{B}$, and let $s,t\in \Omega$ be such that, without loss of generality, we can assume $s\le t$ with $t-s\le ϵ$. This leads us to:

$$\begin{array}{cc}\hfill |\mathcal{F}{X}_t-\mathcal{F}{X}_s|& \le sup\left\{\right|\mathcal{L}(t,V,U_1,U_2,U_3)-\mathcal{L}(s,V,U_1,U_2,U_3)|:|t-s|\le ϵ,\hfill \\ & t,s\in \Omega ,\parallel X_t{\parallel }_{\mathcal{B}}\le r_0,|U_1|\le T_f{M}_1,|U_2|\le L{M}_2,\left|U_3\right|\le \lambda M_3\}\hfill \\ & +L_c{\parallel X_t-X_s\parallel }_{\mathcal{B}}\hfill \\ & +c_2\left|{\int }_{\phi \left(s\right)}^{\phi \left(t\right)}{h}_2(\xi ,X_{\xi },X\left({\alpha }_2\left(\xi \right)\right))d\xi \right|\hfill \\ & +c_3\left|{\int }_s^t{h}_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right))dW\left(\xi \right)\right|.\hfill \end{array}$$

To simplify, we introduce the following notations:

$$\begin{array}{c}\hfill {\omega }_{{\mathcal{L}}_1}(\Omega ,ϵ)=sup\left\{\right|\mathcal{L}(t,V,U_1,U_2,U_3)-\mathcal{L}(s,V,U_1,U_2,U_3)|:|t-s|,\le ϵ\\ \hfill t,s\in \Omega ,\parallel X_t{\parallel }_{\mathcal{B}}\le r_0,|U_1|\le T_f{M}_1,|U_2|\le L{M}_2,|U_3|\le \lambda M_3\},\end{array}$$

$${\omega }_{\phi }(\Omega ,ϵ)=sup\left\{\right|\phi \left(t\right)-\phi \left(s\right)|:|t-s|\le ϵ,t,s\in \Omega \},$$

$${\omega }_W(\Omega ,ϵ)=sup\left\{\right|W\left(t\right)-W\left(s\right)|:|t-s|\le ϵ,t,s\in \Omega \}.$$

Utilizing the aforementioned relationship, we derive the approximation:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |\mathcal{F}{X}_t-\mathcal{F}{X}_s|& \le {\omega }_{{\mathcal{L}}_1}(\Omega ,ϵ)+L_c{\parallel X_t-X_s\parallel }_{\mathcal{B}}+c_2{M}_2{\omega }_{\phi }(\Omega ,ϵ)+c_3{M}_3\lambda {\omega }_W(\Omega ,ϵ).\hfill \end{array}\end{array}$$

(9)

As the limit $ϵ\to 0$ is taken, the expression $|\mathcal{F}{X}_t-\mathcal{F}{X}_s|\to 0$ is obtained, indicating that the difference between $\mathcal{F}{X}_t$ and $\mathcal{F}{X}_s$ tends to zero as $ϵ$ approaches zero. Furthermore, in the limit as $ϵ\to 0$ and referring to Equation (8), it follows that $\omega (\mathcal{F}{X}_t,ϵ)\le L_c\omega (X_t,ϵ)$ holds true for $X_t$ belonging to A. This deduction plays a crucial role in deriving the inequality $\mu \left(\mathcal{F}A\right)\le L_c\mu \left(A\right)$.

According to Definition 6, when we adopt the probability measure $\mu (·)=\mathbb{P}(·)$ associated with the Hausdorff probability measure of noncompactness, the inequality $\mathbb{P}\left(\mathcal{F}A\right)\le L_c\mathbb{P}\left(A\right)$ emerges. This inequality suggests that the probability content of $\mathcal{F}\left(A\right)$ is constrained by a fixed multiple of the probability content of A. Consequently, it implies that $\mathcal{F}[·]$ operates as a densifying map.

Finally, addressing condition (6), we consider $X_t\in \partial {\overline{B}}_{r_0}$ and $k=L_c$. Since $\mathcal{F}{X}_t=k{X}_t$, then

$$\parallel \mathcal{F}{X}_t{\parallel }_{\mathcal{B}}=k{\parallel X_t\parallel }_{\mathcal{B}}$$

According to condition (K2), we can deduce that:

$$|\mathcal{F}{X}_t|\le r_0,$$

for all $t\in \Omega$, hence $\parallel \mathcal{F}{X}_t{\parallel }_{\mathcal{B}}\le r_0$, indicating that $k\le 1$. The final step of verifying $k\le 1$ is crucial as it ensures the contraction property of our operator $\mathcal{F}$. This, combined with the previously established continuity and densifying conditions, allows us to apply Theorem 2, thereby proving the existence of at least one fixed point, which corresponds to a solution of our original Equation (1). Thus, the proof is concluded.

4. Euler–Karhunen–Loève Method

The Karhunen–Loève expansion is a mathematical method employed to depict a stochastic process through a series that incorporates independent random variables and associated coefficients. When applied to the Wiener process (Brownian motion), the Karhunen–Loève expansion expresses the process using a sequence of independent random variables and their corresponding coefficients. This approach is especially advantageous for addressing continuous-time processes with an infinite number of random variables, as seen in the Wiener process, which comprises an uncountable set of Gaussian variables. The frequency-domain Karhunen-0Loeve method, introduced by Kim [33], extends this framework to linear dynamic systems, providing a powerful tool for analyzing stochastic processes in the frequency domain. Additionally, the convergence properties and rates of generic Karhunen–Loève expansions have been rigorously studied by Steinwart [34], offering insights into sample path properties and further enhancing the applicability of this method.

The representation of the Wiener process through the Karhunen–Loève expansion involves independent and identically distributed (i.i.d.) Gaussian random variables along with orthogonal basis functions. Furthermore, it is emphasized that the trajectories of the Wiener process reside in the ${\mathcal{L}}^2\left([0,T_f]\right)$ space for almost all sample paths.

Let $X\left(t\right)$ denote the trajectory of the stochastic process $X(t,\omega )$ for a given $\omega$. Additionally, it is noted that $W\left(t\right)$ possesses trajectories within the ${\mathcal{L}}^2\left([0,T_f]\right)$ space for almost all $\omega$.

The Karhunen–Loève expansion for the Wiener process is expressed as follows:

$$W(t,\omega )=\sum _{i=0}^{\infty }{Z}_i\left(\omega \right){\Psi }_i\left(t\right),0\le t\le T_f,$$

(10)

where the functions ${\Psi }_i\left(t\right)$ are defined as orthogonal basis functions:

$${\Psi }_i\left(t\right)={\displaystyle \frac{2\sqrt{2T_f}}{(2i+1)\pi }}sin\left({\displaystyle \frac{(2i+1)\pi t}{2T_f}}\right),$$

(11)

and the coefficients $Z_i\left(\omega \right)$ constitute a sequence of independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance, i.e., $Z_i\left(\omega \right)\sim \mathcal{N}(0,1)$ [35]. The basis functions ${\Psi }_i\left(t\right)$ satisfy the orthogonality condition:

$${\int }_0^{T_f}{\Psi }_i\left(t\right){\Psi }_j\left(t\right)dt={\delta }_{ij},$$

where ${\delta }_{ij}$ is the Kronecker delta.

To incorporate the Karhunen–Loève expansion into the given nonlinear stochastic integral Equation (1), we can utilize a representation of the Wiener process through this expansion. Consequently, the stochastic term in Equation (1) can be articulated using the Karhunen–Loève expansion as follows:

$${\int }_0^t{h}_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right))dW\left(\xi \right)=\sum _{i=0}^{\infty }{Z}_i{\int }_0^t{h}_3(\xi ,X_{\xi },X\left({\alpha }_3\left(\xi \right)\right)){\Psi }_i\left(\xi \right)d\xi .$$

Afterward, the solution to Equation (1) can be acquired using standard numerical methods such as Euler, Runge-Kutta, Adams-Bashforth, etc. Without loss of generality, we restrict our consideration to the class of processes discussed in this section, namely diffusion processes, which are solutions to SDEs of the form:

$$dX\left(t\right)=b(t,X\left(t\right))dt+\sigma (t,X\left(t\right))dW\left(t\right),X\left(0\right)=X_0.$$

(12)

Here, $X\left(t\right)$ represents the diffusion process, $b(t,X(t\left)\right)$ is the drift term, $\sigma (t,X(t\left)\right)$ is the diffusion term, $dW\left(t\right)$ is the increment of a Wiener process, and $X\left(0\right)$ is the initial condition. As is customary, (12) is interpreted in the Itô sense:

$$X\left(t\right)=X\left(0\right)+{\int }_0^{t}b(\xi ,X_{\xi })d\xi +{\int }_0^t\sigma (\xi ,X_{\xi })dW\left(\xi \right).$$

(13)

The iteration formula, derived using the Euler method with the Karhunen–Loève expansion, can be expressed as follows:

$$X_{i+1}=X_i+b(t_i,X_i)\Delta t+\sum _{i=1}^M{Z}_i{\int }_0^{t_i}\sigma (\xi ,X_{\xi }){\Psi }_i\left(\xi \right)d\xi ,$$

(14)

where $\Delta t=t_{i+1}-t_i,i=0,...,N$ represents the step size in the numerical discretization, M denotes the number of terms, and $Z_i$ signifies the coefficients associated with the Karhunen–Loève expansion.

5. Numerical Simulations and Results

Within this section, we present a range of stochastic functional integral equations to elucidate the practical significance of our results. These examples not only serve to highlight the applicability of our findings but also provide a concrete demonstration of the broader implications and utility of the results in various contexts. The computational experiments were carried out utilizing MATLAB R2019a on a machine equipped with an Intel (R) Core (TM) i3-8145U CPU operating at 2.30 GHz. Moreover, we define the discrete mean square norm as follows:

Definition 7.

For a sequence ${\left\{X_n\right\}}_{n=1}^N$ of random variables, where N is finite, the discrete mean square norm is defined as:

$${\parallel X\parallel }_{\mathrm{MS}}={\displaystyle \frac{1}{N}}\sum _{n=1}^N{\left|X_n\right|}^2$$

The discrete mean square norm is defined with the purpose of assessing the performance and accuracy of the proposed method. By quantifying the “average size” of the sequence, it allows for a systematic evaluation of the method’s effectiveness in capturing the variability and magnitude of the random variables.

Example 1

(Benchmark Problem). To evaluate the accuracy of our proposed method, we introduce a test problem solved using the Euler–Karhunen–Loève technique. The problem involves a linear SDE:

$$dX\left(t\right)=\lambda X\left(t\right)dt+\mu X\left(t\right)dW\left(t\right),X\left(0\right)=X_0,$$

(15)

where λ and μ are constants. This equation is commonly used in financial mathematics to model asset prices. The exact solution to (15) is expressed as:

$$X\left(t\right)=X\left(0\right)exp\left((\lambda -{\displaystyle \frac{1}{2}}{\mu }^2)t+\mu W\left(t\right)\right).$$

(16)

Employing the Karhunen–Loève (KL) expansion, we generate paths of the Wiener process $W\left(t\right)$ over the interval $[0,1]$ using varying numbers of terms $M=[50,100,200,300,500]$ in the expansion. Additionally, we compute the mean square error (MSE) of $X_M\left(t\right)$, which represents the approximate values of $X\left(t\right)$ obtained using different values of M, with a step size of $\Delta t=0.001$.

In our test scenario, we model the stock price of a technology company. Based on historical data analysis, we select $\lambda =0.08$ to reflect an average annual return of 8%, and $\mu =0.2$ to represent the stock’s volatility, a value commonly observed for technology stocks. The initial stock price is set to $X_0=100$, representing an initial value of $100. For reproducibility, the random number generator is fixed using rng(123, ‘twister’).

The exact solution $X\left(t\right)$ is computed using the reference Wiener process, while the Euler–Maruyama method is applied with the truncated Wiener process to approximate $X\left(t\right)$. The mean square error between the exact solution and the approximate solution is computed at 11 equidistant time points to evaluate the accuracy of the method.

The results are visualized in two subplots. The first subplot compares the trajectories of the Euler–Karhunen–Loève approximation with the exact solution for different values of M. The close alignment between the trajectories demonstrates the accuracy of the method. The second subplot shows the evolution of the mean square error over time for different values of M, plotted on a logarithmic scale to highlight the convergence behavior.

Figure 1 illustrates the effectiveness of the Euler–Karhunen–Loève method in estimating the stock price dynamics of a technology company. In the first subplot, a comparison is made between the stock price paths generated by the Euler–Karhunen–Loève method and the exact solution. The close alignment between these trajectories underscores the precision of the Euler–Karhunen–Loève method in capturing the underlying dynamics. In the second subplot, the mean square error at evenly spaced intervals highlights the discrepancies between the exact and simulated paths. Collectively, these visualizations provide a comprehensive understanding of the efficacy and reliability of the Euler–Karhunen–Loève method in modeling stock price dynamics.

Example 2.

Delay Cox–Ingersoll–Ross process

• The Cox–Ingersoll–Ross (CIR) process is a stochastic process that is commonly used to model interest rates. It was introduced as a modification of the Ornstein–Uhlenbeck process to capture the mean-reverting behavior of interest rates. The Delay Cox–Ingersoll–Ross process extends the standard CIR process to incorporate a time delay:

  $$dX\left(t\right)=\kappa (\theta -X\left(t\right))dt+\sigma \sqrt{X(t-\tau )}dW\left(t\right),$$

  (17)

  with the constraint $X\left(t\right)=X_0$ for $-\tau \le t\le 0$. This stochastic differential equation can be derived from its integral representation, providing a clear example of the relationship between differential and integral equations that is central to our work. In the context of the Delay CIR process, $X\left(t\right)$ denotes the stochastic variable. The parameter κ represents the rate at which the process reverts to its mean, θ signifies the long-term average, and σ characterizes the volatility of the process. The delay parameter τ introduces a time lag in the influence of past interest rates on the current state of the process. From a physical perspective, this process shares similarities with systems exhibiting mean-reverting behavior under random forces, such as particle motion in a potential well with noise, though our focus here is on its mathematical properties and financial applications. The Delay CIR process ensures that the process remains non-negative, which is important for modeling interest rates. The mean-reverting nature of the process is controlled by the term $\kappa (\theta -X(t\left)\right)$, which tends to pull $X\left(t\right)$ back towards the long-term mean θ. The volatility term $\sigma \sqrt{X(t-\tau )}$ introduces randomness and ensures that the process exhibits variability over time. The stochastic nature of the Delay CIR process arises from the Wiener process $W\left(t\right)$, which introduces random fluctuations in the dynamics of $X\left(t\right)$. Each simulated trajectory represents a possible realization of the stochastic process, highlighting the inherent variability due to the random term $\sigma \sqrt{X(t-\tau )}dW\left(t\right)$. The Delay CIR process is often used in the valuation of interest rate derivatives and for risk management purposes. The parameters κ, θ, and σ play crucial roles in determining the characteristics of the Delay CIR process. Adjusting these parameters allows the model to capture different aspects of interest rate behavior, such as the speed of mean reversion and the level of volatility.

Equation (17) is interpreted in the Itô sense:

$$X\left(t\right)=X\left(0\right)+{\int }_0^t\kappa (\theta -X\left(\xi \right))d\xi +{\int }_0^t\sigma \sqrt{X(\xi -\tau )}dW\left(\xi \right).$$

(18)

This integral equation demonstrates how our theoretical framework naturally encompasses both differential and integral representations. The objective is to examine the existence of a solution to this equation within the Banach space. It is evident that condition (K1) is satisfied. We only need to demonstrate the validity of condition (K2). The following inequality further supports condition (K2). It is evident that:

$$X\left(t\right)=\mathcal{L}\left({\int }_0^t\kappa (\theta -X\left(\xi \right))d\xi ,{\int }_0^t\sigma \sqrt{X(\xi -\tau )}dW\left(\xi \right)\right),$$

so, we have:

$$\begin{array}{cc}\hfill {\parallel X\left(t\right)\parallel }_{\mathcal{B}}& =sup\left\{\right|\mathcal{L}(U_2,U_3)|:t\in [0,T_f],|X_t|\le r_0,|U_2|\le L{M}_2,|U_3|\le \lambda T_f{M}_3\}\hfill \\ \hfill & =sup\left\{\left|X_0+\kappa (\theta -r_0)t+\sigma \lambda \sqrt{r_0}\right|;t\in [0,T_f],\mathrm{where}\phi \left(t\right)=t\le L\le T_f\right\}\hfill \\ \hfill & =\left|X_0+{\kappa }_{\max }({\theta }_{\max }-r_0)T_f+{\sigma }_{\max }\lambda \sqrt{r_0}\right|,\hfill \end{array}$$

for every t in the interval $\Omega$, condition (K2) holds if:

$$\left|X_0+{\kappa }_{\max }({\theta }_{\max }-r_0)T_f+{\sigma }_{\max }\lambda \sqrt{r_0}\right|\le r_0.$$

(19)

Note: When utilizing the Delay CIR interest rate model, practitioners often consider common parameter values derived from market conventions and empirical studies. In this framework, the mean-reversion speed ($\kappa$) is typically chosen within the range of 0.1 to 1.0, with higher values indicating a faster mean reversion. The long-term mean ($\theta$) is commonly set between 0% and 5%, aligning with historical averages or expected long-term interest rates. For the volatility parameter ($\sigma$), practitioners often opt for values in the range of 0.1 to 0.5, reflecting the degree of variability in interest rates. These values serve as initial benchmarks, and practitioners often refine them through calibration to historical data or specific market observations tailored to their modeling objectives.

The contour plot in Figure 2 reveals specific regions within the $(r_0,\lambda )$ plane where particular inequalities determine the existence and uniqueness of solutions for the Delay CIR model within the specified parameter space. The plot is generated by systematically varying the mean-reversion speed $\kappa$ in reverse order. Each region on the plot corresponds to a different $\kappa$ value, distinguished by a specific color. The scatter points and contour lines mark areas in the $(r_0,\lambda )$ plane where the inequality condition (19) holds. The color of each region visually signifies how the mean-reversion speed influences the existence and uniqueness of solutions in the Delay CIR model.

Figure 3 presents a dynamic visualization of simulated stock price trajectories under varying mean-reversion speeds ($\kappa$) within the Delay CIR model. The simulations employ the Euler numerical method, incorporating the Karhunen–Loève expansion for accurate representation. Each trajectory signifies the stochastic evolution of stock prices over time, encapsulating the intricate interplay between interest rates and asset values. The color-coded representation assigns distinct hues to trajectories corresponding to different $\kappa$ values, offering an immediate visual grasp of how changes in mean-reversion speed influence stock price behavior. The trajectories not only serve as visualizations of stock price dynamics but also convey the model’s capacity to capture the nuanced relationship between interest rates and asset prices. The simulation results provide a valuable lens for interpreting the influence of $\kappa$ on the tendency of interest rates to revert to their long-term mean, ultimately shaping the stochastic behavior of stock prices within the Delay CIR model.

Example 3.

Stochastic Dynamics of Stock Prices: A Modified Black–Scholes with proportional delay model

• In examining the stochastic behavior of financial assets, particularly stock and electricity prices, we delve into a sophisticated modification of the Black–Scholes model, a renowned tool in the field [36,37,38]. This modified model, incorporating a time delay, is described by the following nonlinear SDE with proportional delay:

  $$d{\mathcal{S}}_p\left(t\right)=\kappa {\mathcal{S}}_p\left(t\right)(\overline{\mu }-ln\left({\mathcal{S}}_p\left(\tau t\right)\right))dt+\overline{\sigma }{\mathcal{S}}_p\left(t\right)dW\left(t\right),$$

  (20)

  subject to the condition ${\mathcal{S}}_p\left(0\right)={\mathcal{S}}_0$ where $0\le \tau \le 1$ represents a proportional delay in the dynamics of the process. The term ${\mathcal{S}}_p\left(\tau t\right)$ represents a proportional delay in the dynamics of the process, where the value of ${\mathcal{S}}_p\left(t\right)$ at time t influences the process at a time $\tau t$. Here, the constants $\kappa >0$, $\overline{\sigma }$, and $\overline{\mu }$ play pivotal roles, influencing the dynamics of the stochastic process, and τ represents the time delay. This model captures the intricate interplay between deterministic and stochastic components, offering a nuanced perspective on the evolution of stock prices over time. The stochastic nature of the model arises from the Wiener process $W\left(t\right)$, which introduces random fluctuations in the stock price dynamics. Each simulated trajectory represents a different realization of the stochastic process, emphasizing the impact of the random term $\overline{\sigma }{\mathcal{S}}_p\left(t\right)dW\left(t\right)$. The chosen parameter values, grounded in financial theory and empirical analysis, contribute to the robustness of this enhanced Black–Scholes framework, making it a valuable instrument for comprehending and predicting real-world financial phenomena. The analytical solution for this equation remains elusive.

Let ${\mathcal{S}}_p\left(t\right)=X\left(t\right)+1$, transforming the stochastic model (20) into the following nonlinear stochastic It$\widehat{o}$–Volterra integral equation:

$$X\left(t\right)=X_0+\kappa {\int }_0^t\left(X\left(\xi \right)+1\left)\right(\overline{\mu }-ln(1+X\left(\tau \xi \right))\right)d\xi +\overline{\sigma }{\int }_0^t(X\left(\xi \right)+1)dW\left(\xi \right),t\in [0,T_f],$$

(21)

where $X_0={\mathcal{S}}_0-1$. Utilizing Theorem 3 for the consideration of the problem (21), we explore the solution within the space $C[0,T_f]$. It can be demonstrated that the functions adhere to the stipulations of assumption (K1). To substantiate the validity of assumption (K2), as illustrated in a similar preceding example, we examine the expression:

$$X\left(t\right)=\mathcal{L}\left({\int }_0^t\kappa (X\left(\xi \right)+1)\left(\overline{\mu }-ln(1+X\left(\tau \xi \right))\right)d\xi ,\overline{\sigma }{\int }_0^t(X\left(\xi \right)+1)dW\left(\xi \right)\right),$$

which yields:

$$\begin{array}{cc}\hfill {\parallel X\left(t\right)\parallel }_{\mathcal{B}}& =sup\left\{\right|\mathcal{L}(U_2,U_3)|:t\in [0,T_f],|X_t|\le r_0,|U_2|\le L{M}_2,|U_3|\le \lambda T_f{M}_3\}\hfill \\ \hfill & =\left|X_0+{\kappa }_{\max }(r_0+1)\left({\overline{\mu }}_{\max }-r_0\right)T_f+{\overline{\sigma }}_{\max }(r_0+1)\lambda \right|,\hfill \end{array}$$

for all $t\in \Omega$, condition (K2) is fulfilled if

$$\left|X_0+{\kappa }_{\max }(r_0+1)\left({\overline{\mu }}_{\max }-r_0\right)T_f+{\overline{\sigma }}_{\max }(r_0+1)\lambda \right|\le r_0.$$

(22)

Consequently, a distinct stochastic solution exists for Equation (21).

• Note: In the context of modeling stock price dynamics using the Geometric Brownian Motion (GBM) model with a drift term, parameter choices are often guided by established empirical studies and market insights. The drift parameter ($\kappa$), typically within the range of 0.1 to 0.3, signifies the influence of mean-reverting forces. The mean return ($\overline{\mu }$), commonly set around 0.05 for annual returns, encapsulates the anticipated average return of the stock. Volatility ($\overline{\sigma }$), chosen between 0.1 and 0.3, dictates the degree of price fluctuations. Additionally, the initial stock price (${\mathcal{S}}_0$) serves as a critical consideration, influencing the starting point of the modeled price trajectory. These prescribed values act as preliminary benchmarks, with practitioners exercising discretion to refine them based on specific financial instruments, prevailing market conditions, and meticulous calibration to historical data, ensuring an accurate representation of stock dynamics.

The contour plot in Figure 4 visually represents how changing the mean-reversion speed ($\kappa$) influences the regions in the $(r_0,\lambda )$ plane according to the condition (K2). Each contour line corresponds to a specific $\kappa$ value, ranging from $0.1$ to $0.3$. The areas enclosed by these contour lines indicate where the condition is satisfied. Notably, as $\kappa$ decreases, the enclosed region expands, and the contour lines become larger, implying a broader range of parameter values for $(r_0,\lambda )$ that fulfill the condition. Specifically, the region associated with $\kappa =0.1$ is larger than the region corresponding to $\kappa =0.2$, and similarly, the region for $\kappa =0.2$ is larger than that for $\kappa =0.3$. This interpolation is indicative of how a lower mean-reversion speed allows for a wider set of parameter combinations that satisfy the condition (K2). As $\kappa$ increases, the condition becomes more restrictive, resulting in smaller enclosed regions on the contour plot.

Figure 5 provides a comprehensive visualization of the simulated trajectories of stock prices within the framework of the Modified Black–Scholes with proportional delay model, offering insights into the impact of varying mean-reversion speeds ($\kappa$). Each trajectory on the plot represents the stochastic evolution of stock prices over time, with a distinct color assigned to different values of $\kappa$ ranging from $0.1$ to $0.3$. The color interpolation serves as a dynamic representation of how the mean-reversion speed influences the behavior of stock prices. As $\kappa$ decreases, the trajectories exhibit increased variability and volatility, resulting in a broader spectrum of potential stock price paths. Conversely, higher $\kappa$ values lead to more controlled trajectories, converging towards a smoother and more predictable behavior. To aid in the interpretation of the plot, a colorbar on the bottom provides a reference for the corresponding $\kappa$ values associated with each trajectory’s color. This color-coded scale offers a visual guide to understanding the relationship between mean-reversion speed and the simulated dynamics of stock prices. The trajectories are generated using the Euler numerical method in tandem with the Karhunen–Loève expansion. This computational approach efficiently approximates the solution to the SDE governing the Modified Black–Scholes with proportional delay model, allowing for the simulation of stock price dynamics over discrete time points within the specified time horizon.

6. Conclusions

In conclusion, our paper has achieved significant advancements in two key areas of research. Firstly, we have made substantial progress in the analysis of stochastic functional integral equations by establishing the existence of solutions under stringent conditions. Through rigorous investigations into Lipschitz continuity and boundedness within a specific mathematical space, our research has provided a robust framework for determining solution existence. Utilizing a Petryshyn’s fixed-point theorem, we have demonstrated the continuity and densifying property of the operator involved, thereby confirming the existence of at least one solution within the defined framework. These findings not only enhance our theoretical understanding of stochastic functional integral equations but also offer practical insights into their applicability across various mathematical and scientific domains. However, challenges remain in extending these results to more general cases and exploring further implications of the established conditions, suggesting avenues for future research.

Secondly, our paper has significantly contributed to the analysis of stochastic processes by introducing innovative approaches and achieving insightful results. A notable accomplishment is the development and application of the Euler–Karhunen–Loève Method, which provides a powerful framework for representing and simulating complex stochastic systems numerically. Through rigorous computational experiments, including simulations of the Delay Cox–Ingersoll–Ross process and modified Black–Scholes with proportional delay models, our methodology has demonstrated remarkable effectiveness in capturing and understanding real-world phenomena, particularly in finance and stochastic dynamics. These achievements highlight the practical relevance and versatility of our approach in addressing complex problems in various fields. Nevertheless, ongoing challenges exist in refining and expanding these methodologies to handle more diverse and intricate stochastic processes, indicating avenues for future investigation and development.