Fractional Brownian Motion for a System of Fuzzy Fractional Stochastic Differential Equation

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 coe ﬃ cients, we investigate the existence-uniqueness of the solution using an approximation method to the fractional stochastic integral. We can solve an equation with linear coe ﬃ cients, for example, in ﬁ nancial models Application to a model of population dynamics is also illustrated. An example is propounded to show the applicability of our results.


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.
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 socalled fractional Black and Scholes model of geometric Brownian motion. The risky asset's market stock price is given by this model: where B H is an FBM with the Hurst parameter H , μ is the mean rate of return, and σ > 0 is the volatility, and at time ν, the price of non-risky assets is e rν , where r 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 H ∈ ð1, 2Þ is used as the Hurst parameter. This method is well suited to the study of phenomena with long-range and scale-invariant correlations. When H ≠ 3/2, 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 existenceuniqueness of strong solutions, we use Liouville form of FBM with parameter H ∈ ð3/2, 2Þ. Furthermore, we discuss using the equations in financial models: where α ∈ ð1, 2Þ, β ∈ ð0, 1Þ, and B H are FBM and f , g : ½0, I ⟶ R m is the continuous function. There has been a recent interest in input noises lacking independent increments and exhibiting long-range depen-dence 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 selfsimilarity 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.

Preliminaries
A Gaussian process B H = fB H ðνÞ is called FBM of Hurst parameter H ∈ ð0, 1Þ 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 H > 3/2, this process' long-range dependence and self-similarity qualities give suitable driving noise in stochastic models like networks, finance, and physics. BecauseB H is not semimartingale ifH ≠ 3/2. In terms of FBM, classical Itô theory cannot be used to generate stochastic integral. Two approaches were used to define stochastic integrals about FBM. In the situation of H > 3/2, 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 [27][28][29][30]). The following is an illustration of B H ðνÞ given in [25]: where W is a Browian motion, α = H − 3/2 and B H ðνÞ = Ð ν 0 ðν − sÞ α dWðsÞ. A FBM in Liouville form (LFBM) is the process B H ðνÞ with H ∈ ð1, 2Þ, which has many of the same 2 Journal of Mathematics qualities as the FBM except for the non-stationary increments. In [27], a semimartingale process was utilized to approximate B H ðνÞ using the Malliavin calculus technique: Furthermore, where The process B H ,ε ðνÞ converges to B H ðνÞ in L 2 ðΩÞ when ε 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 R m is denoted by GðR m Þ. The Hausdorff metric, abbreviated as d H , is defined as follows: With regard to d H , the space GðR m Þ is a full and separable metric space. If N , B, and Q are equal to GðR m Þ, then A probability space is defined ðΩ, A,℘Þ. If mapping F : Ω ⟶ GðR m Þ satisfies the following conditions, it is called N -measurable.
F ∈ M is known to be L ℘ -integrably bounded if and only if kjFjk ∈ L ℘ ðΩ, N ,℘;R m + Þ (see [31]). Let us put it this way: For fuzzy set u ∈ R m , membership function u : R m ⟶ ½1, 2 is defined, where uðxÞ denotes degree of membership of x in fuzzy set u. Assume fuzzy sets u : Therefore, in FðR m Þ, d ∞ is metric, and ðFðR m Þ, d ∞ Þ is complete metric space. We have below properties for any u , v, w, z ∈ FðR m Þ, λ ∈ R m : Assume metric ρ in set FðR m Þ and η-algebra B ρ , which is created by topology induced by N fuzzy random variable (FRV) can be thought of as measurable mapping between two measurable spaces ðΩ, N Þ and ðFðR m Þ, B ρ Þ, which we refer to as X is N jB ρ -measurable. Take a look at the below metric: where ∧ represents set of strictly increasing continuous functions: λ : ½1, 2 ⟶ ½1, 2, where λð1Þ = 1, λð2Þ = 2, and X u , X v : ½1, 2 ⟶ FðR m Þ are cÃ dlÃ g representations for fuzzy sets u, v ∈ FðR m Þ (see [34]). The space ðFðR m Þ, d s Þ is a Polish metric space, and space ðFðR m Þ, d ∞ Þ is complete and non-separable. We have â€"X is FRV if and only if X is N jB d s -measurable for mapping X : ⟶FðR m Þ on probability space ðΩ, N ,℘Þ.
If X is N jB d s -measurable, it is FRV, but not the other way.
Assume j ≔ ½0, I, and ðΩ, N ,℘Þ be complete probability space with filtration fN ν g ν∈j satisfying hypotheses, an increasing and right continuous family of sub ρ-algebras of N , and containing all ℘-null sets.
A process X is nonanticipating if and only if for every α ∈ ½1, 2, multifunction ½X α is measurable with respect to ρ-algebra A, where it is defined as follows: where N ν = fω : ðν, ωÞ ∈ N g.
We define fuzzy stochastic Itô integral by using FRV as Ð I 0 XðsÞdWðsÞ, where W is Wiener process. [5] will be beneficial for the following properties.

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 Q q ðνÞ and G q ðνÞ are continuous with Cð0Þ = I and Kð0Þ = I, jQ q ðνÞj ≤ c, c > 1 and jG q ðνÞj ≤ c, c > 1, ∀ν ∈ ½0, T. Equation (20) shows that the Liouville form of FBM with H ∈ ð1, 2Þ, Journal of Mathematics ⟶ R m . The approximate corresponding equation (20) is Assumption 10. Consider the below assumptions about coefficients of equation: A2Þ There exists constant L > 0∀u, v ∈ FðR m Þ and every ν ∈ j such that A3Þ There exists constant Q > 0∀u, v ∈ FðR m Þ and every ν ∈ j, Proposition 11. See [5]. Suppose X, Y ∈ L 2 ðj × Ω, A ; R m Þ, then for every ν ∈ j. Proof. Assume SDE (22), By Equation (6), we can write Let us consider the Picard iterations for n =1,2,..., and for every ν ∈ j, and X 0 ðνÞ = X 0 . For ν ∈ j and n ∈ A we denote for α = H − 3/2 > 1. Hence
We suppose that X ε , Y ε : j × Ω ⟶ FðR m Þ are strong solutions. Let Then, using computations similar to those used in existing case, we obtain When the Gronwall inequality is applied, ℓðνÞ = 0 is obtained for ν ∈ j. Therefore, which completes the uniqueness proof.
Proof. Assume the following approximation of equations based on Equation (20):

Journal of Mathematics
We can write ð52Þ To get the solution, consider Equation (6): Then, Apply the Holder inequality, Doob inequality, and Itô isometry property to gain We conclude that using similar arguments to (32), from Hence,