Abstract

We study fractional Brownian motion– (FBM–) driven fuzzy stochastic fractional evolution equations. These equations can be used to model fuzziness, long-range dependence, and unpredictability in hybrid real-world systems. Under various assumptions regarding the coefficients, we investigate the existence-uniqueness of the solution using an approximation method to the fractional stochastic integral. We can solve an equation with linear coefficients, for example, in financial models Application to a model of population dynamics is also illustrated. An example is propounded to show the applicability of our results.

1. Introduction

Fractional stochastic differential equations (FSDEs) play an important role in the modeling of numerous complicated processes in several sectors of science and engineering. FSDE theory and applications were examined. Furthermore, numerous academics have produced interest in systems with memory or aftereffects.

There appears to be some difficulty in modeling a variety of modern-world systems, such as trying to characterize the physical system and differing viewpoints on its properties. The fuzzy set theory will be utilized to resolve this issue [1]. It can handle linguistic claims like “big” and “less” mathematically using this approach. A fuzzy set provides the ability to examine fuzzy differential equations (FDEs) in representing a variety of phenomena, including imprecision. For example, fuzzy stochastic differential equations (FSDEs) could be used to explore a wide range of economic and technical problems involving two types of uncertainty: randomness and fuzziness.

Deterministic fuzzy differential equations were developed as a result of research into dynamic systems with inadequate or ambiguous parameter information. They are increasingly used in system models in biology, engineering, civil engineering, bioinformatics, and computational biology, quantum optics and gravity, hydraulic and mechanical system modeling. Many studies in this field have been conducted utilizing various ways of expressing differential problems in a fuzzy framework. The Hukuhara derivative of a set-valued function was used as the foundation for the first approach to deterministic FDEs.

There are several articles on FSDEs, each of which takes a different approach. The fuzzy stochastic Itô integral was defined by the author in [2]. The fuzzy It stochastic integral was driven by fuzzy non-anticipating stochastic processes and the Wiener process. Malinowski [35] worked on the application, properties, Ito type strong solutions, and with delay to stochastic fuzzy differential equations. To construct a fuzzy random variable, the method involves embedding crisp It stochastic integral into fuzzy space. Guo et al. [6], Li et al. [7], Deng et al. [8], and Ahmad [9] worked on the stability of fractional stochastic differential equation. Hussain et al. [10] studied stochastic modeling of COVID-19. Abbas et al. [11, 12] solve ordinary differential equation. In the study of Niazi et al. [13], Iqbal et al. [14], Shafqat et al. [15], Alnahdi [16], Khan [17], and Abuasbeh et al. [1821], existence and uniqueness of the fuzzy fractional evolution equations were investigated.

FBM has been used to describe the behavior of asset prices and stock market volatility. This process is a good fit for describing these values because of its long-range dependence on self-similarity qualities. For a general discussion of the applications of FBM to model financial quantities, see Shiryaev [22]. Several writers have proposed a fractional Black and Scholes model to replace the traditional Black and Scholes model, which is memoryless and depends on the so-called fractional Black and Scholes model of geometric Brownian motion. The risky asset’s market stock price is given by this model: where is an FBM with the Hurst parameter is the mean rate of return, and is the volatility, and at time , the price of non-risky assets is , where is the interest rate.

In modeling many stochastic systems, on the other hand, the FBM, which exhibits long-range dependency, is proposed to replace Brownian motion as a driving mechanism. The FBM is a Gaussian process with favorable qualities such as long-range dependency, self-similarity, and increment stationarity when is used as the Hurst parameter. This method is well suited to the study of phenomena with long-range and scale-invariant correlations. When , however, FBM is not semimartingale.

Jafari et al. [23] worked on FSDEs driven by FBM. Inspired by the above paper, we introduce fuzzy fractional stochastic differential equation (FFSDEs) in relation to FBM in this study for order (1,2). These equations can be used to simulate unpredictability, fuzziness, and long-range dependence in hybrid dynamic systems. To determine the explicit answers, we use an approximation approach to fractional stochastic integrals. To investigate existence-uniqueness of strong solutions, we use Liouville form of FBM with parameter . Furthermore, we discuss using the equations in financial models: where , , and are FBM and is the continuous function.

There has been a recent interest in input noises lacking independent increments and exhibiting long-range dependence and self-similarity qualities, which has been motivated by some applications in hydrology, telecommunications, queueing theory, and mathematical finance. When the covariances of a stationary time series converge to zero like a power function and diverge so slowly that their sums diverge, this is known as long-range dependence. The self-similarity property denotes distribution invariance when the scale is changed appropriately. FBM, a generalization of classical Brownian motion, is one of the simplest stochastic processes that are Gaussian, self-similar, and exhibit stationary increments. When the Hurst parameter is more than 1/2, the FBM exhibits long-range dependency, as we will see later. In this note, we look at some of the features of FBM and discuss various strategies for constructing a stochastic calculus for this process. We will also go through some turbulence and mathematical finance applications. The remaining of this paper is as follows. Section 2 discusses the definition of FBM and Liouville form of this process. Then, some introductory material on fuzzy stochastic processes and fuzzy stochastic integrals is reviewed. Section 3 introduces a class of FFSDEs driven by FBM. Furthermore, the existence-uniqueness of solutions is proven using an approximation approach. In Section 4, some findings are presented. In section 5, application to a model of population dynamics is also illustrated. Finally, in Section 6, a conclusion is given.

2. Preliminaries

A Gaussian process is called FBM of Hurst parameter if it has mean zero and the covariance function:

This phenomenon was first described in [24] and investigated in [25], where a Brownian motion-based stochastic integral representation was constructed. For , this process’ long-range dependence and self-similarity qualities give suitable driving noise in stochastic models like networks, finance, and physics. Becauseis not semimartingale if. In terms of FBM, classical It theory cannot be used to generate stochastic integral. Two approaches were used to define stochastic integrals about FBM. In the situation of , Young’s integral [26] can be used to define the Riemann-Stieltjes stochastic integral. The Malliavin calculus is used in a second way to define stochastic integral concerning FBM (see [2730]). The following is an illustration of given in [25]: where is a Browian motion, and . A FBM in Liouville form (LFBM) is the process with , which has many of the same qualities as the FBM except for the non-stationary increments. In [27], a semimartingale process was utilized to approximate using the Malliavin calculus technique:

Furthermore, where

The process converges to in when tends to zero [31].

Preliminaries on FRVs, fuzzy stochastic processes (FSP), and fuzzy stochastic integrals are provided in this section (see [4, 29, 32]). The family of all nonempty, compact, and convex subsets of is denoted by . The Hausdorff metric, abbreviated as , is defined as follows:

With regard to , the space is a full and separable metric space. If , and are equal to , then

A probability space is defined . If mapping satisfies the following conditions, it is called -measurable.

for every closed set, let denote a set of -measurable multifunctions with values. For , a multifunction is said to be -integrably bounded if exists that is , and

is known to be -integrably bounded if and only if (see [31]). Let us put it this way:

For fuzzy set , membership function is defined, where denotes degree of membership of in fuzzy set . Assume fuzzy sets denoted by that is for every , where . Define by

Therefore, in , is metric, and is complete metric space. We have below properties for any : (i)(ii)(iii)(iv)

We use as , where for if and if .

Definition 1 See ([33]). A probability space is defined as . If mapping is an -measurable multifunction for all , FRV is function .
Assume metric in set and -algebra , which is created by topology induced by fuzzy random variable (FRV) can be thought of as measurable mapping between two measurable spaces and , which we refer to as is -measurable. Take a look at the below metric: where represents set of strictly increasing continuous functions: , where , and are cà dlà g representations for fuzzy sets (see [34]). The space is a Polish metric space, and space is complete and non-separable.
We have
– is FRV if and only if is -measurable for mapping on probability space .

If is -measurable, it is FRV, but not the other way.

Definition 2. If , for any , FRV is -integrably bounded for .
Let us denote by set of all -integrably bounded FRVs. The random variables are identical if . For FRV , and , the below conditions are equivalent: Assume , and be complete probability space with filtration satisfying hypotheses, an increasing and right continuous family of sub -algebras of , and containing all -null sets.

Definition 3. If mapping , for every , is FRV, then is FSP.

Definition 4. A FSPis-continuous, if almost all its trajectories, that is mappings are -continuous functions.
A FSP is measurable, if is -measurable multifunction for all , where denotes Borel -algebra of subsets of .

A process is nonanticipating if and only if for every , multifunction is measurable with respect to -algebra , where it is defined as follows: where .

Definition 5. A FSP is called -integrably bounded , if there exists real-valued stochastic process , the Fubini theorem, fuzzy integral is defined by for , where and . The fuzzy integral can be defined level-wise. For every , and , Aumann integral belongs to , so FRV belongs to , so FRV belongs to .

Definition 6. The fuzzy stochastic Lebesgue-Aumann integral of is defined as

Proposition 7 See ([5]). The properties of the integral can be demonstrated as follows:
Suppose . If , then . (i)Suppose , then is -continuous(ii)Suppose , for , then(iii)Let us define an embedding of into by , which is for , If is random variable on probability space , now is FRV. The same property exists for stochastic processes.
We define fuzzy stochastic It integral by using FRV as , where is Wiener process. [5] will be beneficial for the following properties.

Proposition 8. Assume , then is FSP, and we have .

Proposition 9. Suppose , then is-continuous.

3. Application to Fuzzy Stochastic Differential Equation

The following is a list of FFSDEs driven by FBM that we will investigate in this section: such that where and are continuous with and , and . Equation (20) shows that the Liouville form of FBM with is an FRV, . The approximate corresponding equation (20) is

Assumption 10. Consider the below assumptions about coefficients of equation:
The mappings and are -measurable and -measurable, respectively
There exists constant and every such that There exists constant and every ,

Proposition 11. See [5]. Suppose , then for every .

Theorem 12. Assume and as mappings satisfy assumptions and . Then, Equation (22) has a strong unique solution.

Proof. Assume SDE (22), By Equation (6), we can write

Let us consider the Picard iterations for n =1,2,..., and for every , and . For and we denote for . Hence

We have

By applying (7)–(31) and Holder inequality, we have

Hence, from (30) and (32), we obtain for every . Then, similarly,

Therefore,

Using Chebyshev inequality, we can determine

The series is convergent. Using Borel-Cantelli lemma, we have

There is a such that for approximately every ,

The sequence is uniformly convergent to -continuous fuzzy process for every , in which and . We can define the mapping , as if and as freely chosen fuzzy function when . For every and every with a.e., we have

Hence, will be continuous FSP. Therefore, by , we have . Consequently, as approaches infinity, we can prove tends to zero. Now,

Therefore,

This demonstrates existence of a strong solution.

We suppose that are strong solutions. Let

Then, using computations similar to those used in existing case, we obtain

When the Gronwall inequality is applied, is obtained for . Therefore, which completes the uniqueness proof.

Lemma 13. For every and we have

Proof. To obtain the function , we use the finite-increments formula: