Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

Abstract

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.

1. Introduction

There appears to be confusion of various kinds in the modeling of several real world systems, such as trying to characterize a physical system and opinions on its parameters. To deal with this ambiguity, the fuzzy set theory will be used [1]. It is able to handle such linguistic statements mathematically using this theory, such as “large” and “less.” The capacity to investigate fuzzy differential equations (FDEs) in modeling numerous phenomena, including imprecision, is provided by a fuzzy set. In particular, the fuzzy stochastic differential equations (FSDEs), in instance, might be used to investigate a variety of economics and engineering problems that involve two types of uncertainty: randomness and fuzziness.

The fuzzy It stochastic integral was powered in [2, 3]. In [4, 5], the fuzzy stochastic integral is driven by the Wiener process as a fuzzy adapted stochastic process. In [6], Fei et al. studied the existence and uniqueness of solutions to the (FSDEs) under non-Lipschitzian condition. In [7], Jafari et al. study FSDEs driven by fBm. Jialu Zhu et al., in [8], prove existence of solutions to SDEs with fBm. Ding and Nieto [9] investigated analytical solutions of multitime-scale FSDEs driven by fBm. Vas’kovskii et al. [10] prove that the pth moments, , of strong solutions of a mixed-type SDEs are driven by a standard Brownian motion and a fBm. Despite the fact that some research exists on the problem of the uniqueness and existence of solutions to SDEs and FSDEs which are disturbed by Brownian motions or semimartingales [4, 11–15], a kind of the FFSDEs driven by an fBm has not been investigated. Agarwal et al. [16, 17] considered the concept of solution for FDEs with uncertainty and some results on FFDEs and optimal control nonlocal evolution equations. Recently, Zhou et al., in [18–20], gave some important works on the stability analysis of such SFDEs. Our results are inspired by the one in [21] where the existence and uniqueness results for the FSDEs with local martingales under the Lipschitzian conditions are studied. The rest of this paper is given as follows. Section 2 summarizes the fundamental aspects. In Section 3, existence and uniqueness of solutions to the FFSDEs are proved. Moreover, the stability of solutions is studied in Section 4. Finally, in Section 5, a conclusion is given.

2. Preliminaries

This part introduces the notations, definitions, and background information that will be utilized throughout the article.

Let be the family of nonempty convex and compact subsets of . In , the distance is defined by

We denote by the family of -measurable multifunction, taking value in .

Definition 1 (see [21, 22]). A multifunction is called -integrably bounded if such that -a.e, whereWe denote byLet denote the set of the fuzzy such that , for every , where , for , and . Let the metric be , in , ; we have , , and .

Definition 2 (see [23]). Let ; the fuzzy Riemann–Liouville integral of is given by

Definition 3 (see [23]). Let . The fuzzy fractional Caputo differentiability of is given byNow, we define the Henry–Gronwall inequality [24], which can be used in the proof of our result.

Lemma 1. Let , be continuous functions. If is nondecreasing and there exists constants and asthen

If is constant on , the previous inequality is transformed intowhere is given by

Remark 1 (see [24]). For all , does not depend on such that .

Definition 4 (see [21, 22]).  A function is said fuzzy random variable if is an -measurable random variable  A fuzzy random variable is said -integrably bounded, , if , Let denote the set of all fuzzy random variables; they are -integrally bounded.
For the notion of an fBm, we referred to [25].
Let us define a sequence of partitions of by such that as . If, in , converge to the same limit for all this sequences , then this limit is said a Stratonovich-type stochastic integral and noted by . Let , where .

Definition 5 (see [21, 22]).  A function is called fuzzy stochastic process; if , is a fuzzy random variable A fuzzy stochastic process is continuous; if are continuous, and it is -adapted if for every and for all , is -measurable

Definition 6 (see [21, 22]).  The function is called measurable if is a -measurable, for all  The function is said to be nonanticipating if it is -adapted and measurable

Remark 2. The process is nonanticipating if and only if is measurable with respect to , where, for , .

Definition 7 (see [21, 22]). A fuzzy process is said -integrally bounded if .
We denote by the set of all -integrally bounded and nonanticipating fuzzy stochastic processes.

Proposition 1 (see [4]). For and , we have and -continuous.

Proposition 2 (see [4]). For and , we have

Proposition 3 (see [26]). Let ; then, for ,Let us define the embedding of to as :

Proposition 4 (see [4]). Assume that the function satisfies . Then,  The fuzzy stochastic It integral   For , we have, for ,

3. Main Result

Now, we investigate the FFSDEs driven by an fBm given bywhereand is a fBm defined on with Hirst index .

Definition 8. A process is said to be a solution to equation (14) if the following holds: .  is -continuous.  We haveWe will assume that all through this paper, is -measurable. Let the following assumptions be introduced.  If is -measurable, we have  For and , we have for every .  For all ,where is equal to one in .
Let us now introduce the main theorem in this part.

Theorem 1. Under assumptions – and , the equation (14) has a unique solution .

Proof. The method of successive approximations will be used to demonstrate the existence of a solution to (1). Therefore, define a sequence as follows:and for ,It is clear that s are in and -continuous. Indeed, we have and is -continuous.
Let us define for and . Then, from Propositions 2 and 3 and –, we havewhere . Moreover, similarly, we haveThus, we obtainwhere .
Hence, from Chebyshev’s inequality and (24), we obtainSince the series converges, according to Borel–Cantelli lemma, we obtainThus, the sequence is uniformly convergent to for , where and . Then,Let us define as follows:We can observe that, for each and , we haveThen, is -measurable. Hence, is nonanticipating. By (27), we havewhich shows that independent of such thatSince , we have . In addition, we can prove that .
Indeed, for all and , let us denoteThen, we obtainBy the triangle inequality, –, and Propositions 2 and 3, we haveWe obtainwhere and .
According to Lemma 1 and Remark 1, there exist a constant independent of such thatDue to , (31), and (36), we obtainwhich impliesThus, we get .
On the contrary, we haveIndeed, we observewhere and . For , by using Propositions 2 and 3, , and (30), we haveHence, we get (39), which implies (16) holds. Hence, from definition (8), is a solution to equation (14).
For the uniqueness of a solution , suppose that are solutions to equation (14). We denote by . So, for each , we obtainThus, by Lemma 1, we have, for , , which implies

4. Stability Result

In this part, we examine the stability of the solution with respect to initial values by using Henry–Gronwall inequality. Indeed, let and denote the solutions of the following FFSDEs:respectively.

Proposition 5. Suppose that and satisfy –. Then,where and . Especially, if .

Proof. Suppose that are solutions to equations (44) and (45), respectively. So, let . Due to Propositions 2 and 3 and , we obtainThen, according to Lemma 1 and Remark 1, there exist a constant independent of such thatThen, if . Therefore, we know that .
Finally, we examine the exponential stability of solutions to the FFSDEs which disturbed an fBm with respect to and . Thus, let and denote solutions to the following FFSDEs:respectively.

Proposition 6. Suppose that and fulfill –. Furthermore, assume thatThen, we havewhere are solutions of equations (49) and (50), respectively.

Proof. According to Theorem 1, the solutions and are unique and exist. From Propositions 3 and 4, we deduce that, for every ,where. From Lemma 2 and Remark 1, is independent of such thatHence, from (51) and (52), we get .

5. Conclusions

In this study, we have proved the existence and uniqueness of solutions to FFSDEs under the Lipschitzian coefficient. On the contrary, the stability of solutions to the FFSDEs is analyzed.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.