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

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


Introduction
Te concept of fractional derivative frst 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 feld of fractional calculus signifcantly 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][2][3][4][5][6][7][8][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 felds.Over the past decade, fractional calculus has been applied for describing long-memory processes.Many classical techniques are difcult to apply directly to fractional diferential equations.It is, therefore, necessary to develop especially new theories and methods whose analysis becomes more difcult.Compared with the classical properties of diferential equations, research on the concept of fractional diferential equations is still in its initial stage of development.
One of the most important classes of fractional diferential equations are the fractional Itô-Doob stochastic diferential 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]), fnance (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.Te 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.
Te 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 efectiveness of our results.

Basic Notions
as a complete probability space and W(ω) as a standard Brownian motion.
For q ≥ 2, set Y q ω � L q (Y , M ω ,  P) space of all M ω -measurable and q-th integrable functions ϕ � Defnition 1 (see [1]).Set β ∈ (0, 1) and f(ω) as a continuous function and thus the fractional Hadamard integral of f(ω) takes the form Consider the Hadamard fractional Itô-Doob stochastic integral equation As we proceed, we take into account q > max 1≤i≤n 1/β i  .
Now, we consider the following assumptions which are important criteria to prove the main results of the next sections:

Illustrative Examples
Tree examples are given to show the efectiveness of our results.
We have, Tus, assumption

Conclusion
Tis paper focuses on the Hyers-Ulam stability of Hadamard fractional Itô-Doob stochastic integral equations by employing the Banach fxed point method, some stochastic analysis, and mathematically useful techniques.Te 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.