Stability Results for a Class of Fractional Itô–Doob Stochastic Integral Equations

Abstract

In this paper, we study the Hyers–Ulam stability of Hadamard fractional Itô–Doob stochastic integral equations by using the Banach fixed point method and some mathematical inequalities. Finally, we exhibit three theoretical examples to apply our theory.

1. Introduction

The concept of fractional derivative first appeared in a correspondence between L’Hôpital and Leibnitz in 1695. Many scientists have explored this idea. To illustrate this notion, we will give an overview overall history of work in this area. We can cite the study of Euler in 1730. Also, one should not overlook the applications of J. L. Lagrange in 1772, nor forget the proposed notion of fractal derivative by Laplace in 1812. Additionally, the work of Abel in the field of fractional calculus significantly contributes to this area. For more details about the extension of fractional calculus, one can refer to the works of Atangana, Baleanu, and other scientists (see [1–9]).

Over the last decades, fractional calculus had played an interesting role. Its importance appears in various areas such as chemistry, physics, economics, biology, and other fields. Over the past decade, fractional calculus has been applied for describing long-memory processes. Many classical techniques are difficult to apply directly to fractional differential equations. It is, therefore, necessary to develop especially new theories and methods whose analysis becomes more difficult. Compared with the classical properties of differential equations, research on the concept of fractional differential equations is still in its initial stage of development.

In stability concept, the Ulam stability was first introduced by Ulam (see [10]) and then was generalized by Hyers and Rassias (see [11, 12]). Many scientists generalized the Ulam–Hyers–Rassias results in various systems; for Hadamard fractional Itô–Doob stochastic integral equations and Caputo-derivative, we can refer to [13–16], and for fractional stochastic differential equation with fractional Brownian motion and pantograph differential equations, see [17–21].

One of the most important classes of fractional differential equations are the fractional Itô–Doob stochastic differential equations which had many applications in describing many phenomena of real life, and the nonlocal conditions describe numerous problems in physics (see [13, 22, 23]), finance (see [24, 25]), and mechanical problem (see [26, 27]). To the best of our knowledge, there is no existing work on the Hyers–Ulam stability of fractional Itô–Doob stochastic integral equations. Motivated by the previous works, in this paper, we will cover this gap. The main contributions of the paper are as follows:(i)Study the existence and uniqueness of the solution of Hadamard fractional Itô–Doob stochastic integral equations.(ii)Investigate the Hyers–Ulam stability of Hadamard fractional Itô–Doob stochastic integral equations.(iii)Extend the work on [13] to summed Hadamard fractional Itô–Doob stochastic integral equations.

The organization of the paper is as follows. We exhibit some preliminaries and basic notions in Section 2. Section 3 is devoted to the fundamental results. In Section 4, we present three examples to show the effectiveness of our results.

2. Basic Notions

Set and as a complete probability space and as a standard Brownian motion.

For , set space of all -measurable and -th integrable functions with

Definition 1. (see [1]). Set and as a continuous function and thus the fractional Hadamard integral of takes the form

Consider the Hadamard fractional Itô–Doob stochastic integral equationwhere , for , , and , are measurable functions.

As we proceed, we take into account .

Now, we consider the following assumptions which are important criteria to prove the main results of the next sections: : there exist such that for all . : there exist such that , for .

3. Main Results

Let be the family of all processes which are measurable and adapted satisfying . Let be the norm on given by

We have, is a Banach space.

Definition 2. Letwhere , for , , and , are measurable functions.
Equation (3) is Ulam–Hyers stable with respect to if there exists such that for each and for each solution , withthere is a solution of (3), with , and

Theorem 3. Assume that and hold. Let , satisfyingwithwhere . Then, there is a solution of (3), withwhereand is some positive constant such that

Proof. Consider norm on defined byWe have, and are equivalent.
Let given byfor every . We will split our proof into the following three steps:
Step 1: is well defined.
Let , we get for ,By the Hölder inequality, we obtainUsing the Hölder inequality and Theorem 7.1 in [28], we getTherefore, .
Step 2: is contractive.
We have for all ,Using and the Hölder inequality, we obtainUsing , the Hölder inequality and Theorem 7.1 in [28], we getTherefore, we getThen,Thus,Therefore, is contractive.
Step 3: An estimation of . It follows from (9) thatThen,In view of Theorem 2.1 in [13], there is a unique solution such thatConsequently, , and we have

4. Illustrative Examples

Three examples are given to show the effectiveness of our results.

Example 1. Let equation (3) be with , , , and .We have,Thus, assumption holds for . Moreover, for , we haveTherefore, assumption holds for .
Assume that satisfiesfor every , where .
Then, it follows from Theorem 3 that there is ,such thatfor every , where .

Example 2. Let equation (3) be with , , , and .We have,Thus, assumption holds for . Moreover, for , we haveTherefore, assumption holds for .
Assume that satisfiesfor every , where .
Then, it follows from Theorem 3 that there is ,such thatfor every, where .

Example 3. Let equation (3) be with , , and .We have,Thus, assumption holds for . Moreover, for , we haveTherefore, assumption holds for .
Assume that satisfiesfor every , where .
Then, it follows from Theorem 3 that there is ,such thatfor every , where .

5. Conclusion

This paper focuses on the Hyers–Ulam stability of Hadamard fractional Itô–Doob stochastic integral equations by employing the Banach fixed point method, some stochastic analysis, and mathematically useful techniques. The applicability of the obtained results is proved through three illustrative examples. Combining with some related research in the literature about the fractional stochastic pantograph equations, we can explore various extensions and stability problems for pantograph Hadamard fractional Itô–Doob stochastic integral equations.

Data Availability

No underlying data were collected or produced to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research has been funded by Scientific Research Deanship at the University of Ha’il, Saudi Arabia, through project number RG-23 041.