Proportional Itô–Doob Stochastic Fractional Order Systems
Abstract
:1. Introduction
- ⋄
- Unlike other works in the literature, this paper presents a new theory by the proportional fractional integral.
- ⋄
- We study the existence and uniqueness of the solutions of PIDSFOS using the Picard iteration technique.
- ⋄
- We discuss the convergence of the solution of the averaged PIDSFOS towards that of the standard PIDSFOS in the sense of the mean square.
2. Preliminaries and Definitions
- : There is such that
- : There is such that
3. Existence and Uniqueness Results
- According to Lemma 1, and belong to . Thus,
- By the Gronwall inequality, we can obtain . Hence, , a.s.
- Existence:– Set . The iterative Picard method is defined by the sequence as follows:
- By Equation (10), it yields that for any :
- Using the Gronwall inequality, we have
- Note that
- We can see that
- Therefore, .
- We can see that
4. Averaging Principle
- : Suppose that the measurable functions
- (1)
- (2)
- (3)
- where are bounded positive functions such that for .
- We will show that the solution can be approximated by the solution of the standard equation
5. Illustrative Example
- Consequently, - hold with . Finally, Theorem 2 is satisfied.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ben Makhlouf, A.; Mchiri, L.; Othman, H.A.; Rguigui, H.M.S.; Boulaaras, S. Proportional Itô–Doob Stochastic Fractional Order Systems. Mathematics 2023, 11, 2049. https://doi.org/10.3390/math11092049
Ben Makhlouf A, Mchiri L, Othman HA, Rguigui HMS, Boulaaras S. Proportional Itô–Doob Stochastic Fractional Order Systems. Mathematics. 2023; 11(9):2049. https://doi.org/10.3390/math11092049
Chicago/Turabian StyleBen Makhlouf, Abdellatif, Lassaad Mchiri, Hakeem A. Othman, Hafedh M. S. Rguigui, and Salah Boulaaras. 2023. "Proportional Itô–Doob Stochastic Fractional Order Systems" Mathematics 11, no. 9: 2049. https://doi.org/10.3390/math11092049
APA StyleBen Makhlouf, A., Mchiri, L., Othman, H. A., Rguigui, H. M. S., & Boulaaras, S. (2023). Proportional Itô–Doob Stochastic Fractional Order Systems. Mathematics, 11(9), 2049. https://doi.org/10.3390/math11092049