Abstract

In this work, we study a class of Hilfer-type fractional stochastic differential equations (HFSDEs). By Laplace transform and its inverse, we obtain a mild solution to HFSDEs. Furthermore, we research the existence and uniqueness of solutions to the addressed equations. Subsequently, we explore the Ulam–Hyers (U–H) stability of the considered system by the Picard iteration. Finally, we verify the correctness of the theoretical results with an example.

1. Introduction

Fractional calculus was born in 1695, when L’Hopital wrote a letter to Leibniz about the nth derivative of a simple linear function . What happened when ? Leibniz replied that this is a seemingly absurd but meaningful question that will one day lead to useful results. Since then, fractional calculus has become a topic of great interest to some mathematicians. The one of earliest systematic study of fractional calculus was done by Liouville, Riemann, and Holmgren in the early and mid-19th century. By the 20th century, Weyl, Hardy, Littlewood, Caputo, Hilfer, and Miller Ross et al. made important contributions to the development of fractional calculus [1–5]. Among them, Hilfer [5] introduced the definition of generalized fractional derivative called Hilfer fractional derivative, which is the interpolated cases of Riemann–Liouville fractional derivatives and Caputo fractional derivatives. Compared with integer calculus, fractional calculus can reflect the properties and objective facts of objects more specifically, so the application prospect of fractional differential equations (FDEs) is broader. Nowadays, fractional calculus has formed a relatively important branch in the field of mathematics and has achieved rapid development of theory and application, and it has been widely used in the fields of electroanalytical chemistry, viscoelasticity, signal processing, and the unzipping of polymer materials [6–14].

There are various random disturbances and random phenomena in nature, and it is no longer appropriate to use a deterministic model to describe this system affected by random factors. However, it becomes more reasonable to describe the development law of things through stochastic differential equations (SDEs), which also promote the development of SDEs theory. More than half a century ago, the idea of Itô stochastic analysis established by Itô [15] and the theory of SDEs developed based on it enabled researchers to have a deeper understanding of many random phenomena in nature. Scholars are increasingly aware that SDEs are more accurate and objective in describing various random phenomena in real life. Since then, SDEs have been widely used in chemistry, physics, biology, finance, control, and other fields [16–18]. In recent years, many monographs have discussed the main achievements in the theory and application of SDEs [19–22].

In general, it is not easy to solve the explicit solutions of FDEs, so it is necessary to research the existence and uniqueness of FDEs. Since Hilfer introduced the Hilfer fractional derivative, more and more scholars have begun to study various properties of HFSDEs. There are some articles that explore the existence and uniqueness of HFSDEs. Saravanakumar and Balasubramaniam [23] proved the existence of solutions to HFSDEs with noninstantaneous impulsive by utilizing the Mönch fixed-point theorem. In [24], the authors used the same method to study HFSDEs. However, the systems of [23, 24] are different. Lv and Yang [25] investigated the existence of HFSDEs by applying the fixed-point theorem and verified its uniqueness by using the contraction mapping principle. The authors employed Sadovskii fixed-point theorem to explore the existence of HFSDEs [26]. To the best of our knowledge, compared with other types of fractional differential equations, there are few articles about the existence and uniqueness of solutions to HFSDEs. Therefore, there is still a lot of works waiting for researchers to further explore the existence and uniqueness of HFSDEs. For the methods and results of the existence and uniqueness of other types to FDEs, readers can refer to [27–31] and its references. We will also draw on the methods in these articles to study the existence and uniqueness of HFSDEs.

Moreover, for a U–H stable system, when it is difficult to get an exact solution, the system can be approximately solved, and U–H stability can ensure the reliability of approximate solution. The U–H stability originated from a lecture by Ulam [32] at Wisconsin University in 1940, and then, Hyers [33] gave the first affirmative answer to this question in Banach space in the following year. The U–H stability theory has evolved significantly over time. Nowadays, there are a lot of studies using many methods to study the U–H stability of FDEs. The authors obtained the U–H stability of FDEs with a generalized Caputo derivative by the direct method and Grönwall inequality in [34]. With the help of weighted maximum norm and Itô isometry, the researchers explored the U–H stability of Caputo-type fractional stochastic neutral differential equations in [35]. In [36], Vivek et al. investigated the U–H stability and U–H–Rassias stability of Hilfer fractional implicit differential equations with nonlocal conditions. However, there are still very few articles on the U–H stability of HFSDEs, so this field still has great prospects.

In view of the fact that there are few articles about the existence, uniqueness, and U–H stability of the solutions to HFSDEs, most of them prove these theories by the fixed-point theorem. In this study, we intend to use Picard’s type successive approximations to obtain the existence and stability results of HFSDEs, and the system is different from those abovementioned studies. Motivated by the above articles, we aim to study the following system:where is the Hilfer fractional derivative with and , is a matrix of dimensions , the state vector is a stochastic process, and are the continuous functions, and is an -dimensional Brownian motion on a complete probability space and .

The rest of this article is divided into the following sections. We will give some definitions and properties of fractional integrals and derivatives, as well as the lemmas and conditions in Section 2. The purpose of Section 3 is to investigate the existence and uniqueness of HFSDEs by Picard iteration and contradiction, respectively. With the aid of Picard iteration and the properties of convergent series, we derive the U–H stability of system (1) in Section 4. In Section 5, we give an example to test the validity of our theoretical results.

2. Preliminaries

For a vector function and a matrix , we define the standard Euclidean norm as follows:where denote the function square Lebesgue integrable and represents the largest eigenvalue of .

Definition 1. (see [37]). For the function , the Riemann–Liouville fractional integral of order with the lower limit 0 is defined aswhere is the Gamma function.

Definition 2. (see [37]). The Riemann–Liouville fractional derivative of order of the function is defined byIn particular, for ,

Definition 3. (see [37]). For the function , the Hilfer fractional derivative of order and with the lower limit 0 is defined as

Definition 4. (see [38]). The Mittag–Leffler function is defined aswhere stands for the set of complex numbers.
The Laplace integral of is defined aswhere is an identity matrix.

Lemma 1. We consider the system of the following form:Simultaneously integrating both sides of system (9) yieldsBased onwe getBy using the Laplace transform, we havewhere and represent the Laplace transform of and , respectively. Furthermore, we getThrough the inverse Laplace transform, the solution of equation (9) isSimilarly, the mild solution of equation (1) can be obtained as follows:where are the matrix Mittag–Leffler functions. Since and are bounded, for convenience, we assume

Lemma 2. (see [39]). (Grönwall–Bellman inequality) For continuous functions and positive constant , ifholds, then

Lemma 3. (see [40]). For any , is an -value stochastic process and . Then, we can get

Definition 5. (see [41]). For an -value stochastic process , if the following conditions hold, then is a unique solution to equation (1).(1) and (2) is -adapted and continuous(3)For , we have(4)For other solutions , we get

Assumption 1. For and each fixed , the inequalityholds and “” is defined as .

Assumption 2. Let , it follows thatwhere is a constant and represents the set of continuous functions on interval .

3. Existence and Uniqueness

In this section, the existence and uniqueness results of system (1) are established.

Theorem 1. Suppose that Assumptions 1-2 and are satisfied, then system (1) has a unique solution in .

Proof.  

Step 1. We prove the existence of equation (1). To achieve this, we consider the following Picard iteration:For , and , obviously, . By Jensen’s inequality, we havewhich with the aid of Lemma 3, Jensen’s inequality, Cauchy–Schwarz inequality, and Assumptions 1-2 givesFor an arbitrary integer , we can conclude thatWe should notice thatHence,where and .
Based on Grönwall–Bellman inequality, we obtainSince the integer is arbitrary, it is easy to getfor and .
Next, for , it can be seen from equation (24) thatBased on Lemma 3, Jensen’s inequality, Cauchy–Schwarz inequality, and Assumptions 1-2, we infer thatThen,For , by Lemma 3, Jensen’s inequality, Cauchy–Schwarz inequality and Assumptions 1-2, we getSimilarly, for , we can getAssume for some ,Then,Therefore, according to the above induction, for each , we haveFurthermore,Indeed, use Doob’s martingale inequality to yieldNote the seriesThen,From above,converges uniformly on . Denote the limit of by . Note that is -adapted and continuous. From (41), we know that is a Cauchy sequence in , and .
Let on both sides of inequality (31), we can getwhich means thatThis shows that .
Next, we prove that satisfies system (1) on interval . Note thatSimilarly, we haveTherefore, we obtain thatThen, taking the limit on both sides of (24), we can getwhich implies that is a solution of equation (1).

Step 2. We prove the uniqueness of system (1). Suppose and are two different solutions of system (1). For all , by Jensen’s inequality Lemma 3 and Assumption 1, we havewhich, with the aid of Grönwall–Bellman inequality, yields . Therefore, the solution of equation (1) is unique. The proof of this theorem is completed.

4. Ulam–Hyers Stability Analysis of HFSDEs

In this section, we consider the U–H stability of the HFSDEs:

Let , and we consider the following inequality:

Definition 6. (see [34]). Equation (1) is called U–H stable, if there exists a real number , such that , and for each solution of the inequality (53), there exists a solution of equation (1) with

Remark 1. (see [34]). A function is a solution of the inequality (53) if there exists a function , such that(i)(ii)Therefore, the solution to inequality (53) is

Theorem 2. If Assumption 1 and hold, then HFSDEs (1) is U–H stable in .

Proof. We consider in given by andFor and , by Cauchy–Schwarz inequality and Jensen’s inequality, we can conclude thatThen,For , by basing on Assumption 1 and mathematical induction, we derive that