Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients
Abstract
1. Introduction
2. Preliminaries
3. An Equivalent mSVIE
4. Existence and Uniqueness of Solution to mSVIE (7)
4.1. A Modified EM Scheme
4.2. Existence and Uniqueness
5. Strong Convergence Analysis of the Modified EM Approximation
5.1. The Mean—Square Convergence Theorem of the Modified EM Method (14)
5.2. Strong Convergence Order Analysis
6. Numerical Experiments
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Proof of Lemma 1
Appendix B. The Proof of Lemma 2
Appendix C. The Proof of Lemma 3
Appendix D. The Proof of Lemma 4
Appendix E. The Proof of Theorem 1
- Step 1: The Picard sequence . For arbitrary , using inequality (15), let . Then, we have
- Step 2: The Picard sequence is a Cauchy sequence almost surely.We define arbitrary and . We need to argue that, for any , for , or
Appendix F. The Proof of Lemma 5
Appendix G. The Proof of Lemma 7
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Yang, Z.; Xu, C. Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients. Fractal Fract. 2025, 9, 296. https://doi.org/10.3390/fractalfract9050296
Yang Z, Xu C. Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients. Fractal and Fractional. 2025; 9(5):296. https://doi.org/10.3390/fractalfract9050296
Chicago/Turabian StyleYang, Zhaoqiang, and Chenglong Xu. 2025. "Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients" Fractal and Fractional 9, no. 5: 296. https://doi.org/10.3390/fractalfract9050296
APA StyleYang, Z., & Xu, C. (2025). Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients. Fractal and Fractional, 9(5), 296. https://doi.org/10.3390/fractalfract9050296