A Class of Fractional Stochastic Differential Equations with a Soft Wall
Abstract
:1. Introduction
2. Main Results
- (A)
- Linear growth of f for any and any
- (B)
- Lipschitz continuity of fin space: for any and
- (C)
- Hölder continuity in time: there exists such that for any and any
- (D)
- Function D is strictly monotonic and surjective.
- (E)
- Constant exists, such that
3. Remarks on the Condition (E)
4. Proofs of Theorems
4.1. Proof of Theorems 1 and 2
4.2. Proof of Theorems 4 and 5
5. Example: Fractional Pearson Diffusion Process with Soft Wall
6. Modelling: Fractional Vasicek Process
6.1. Profile of Soft-Wall Resistant Force
6.2. Process Trajectories Simulation under Soft-Wall conditions
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Estimation of Hurst Parameter
Appendix B. Supplementary Results
Appendix B.1. Limit Results on fBm
Appendix B.2. Hölder Continuous Paths
Appendix B.3. Pathwise Integration
Appendix B.4. Gronwall’s lemma
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Kubilius, K.; Medžiūnas, A. A Class of Fractional Stochastic Differential Equations with a Soft Wall. Fractal Fract. 2023, 7, 110. https://doi.org/10.3390/fractalfract7020110
Kubilius K, Medžiūnas A. A Class of Fractional Stochastic Differential Equations with a Soft Wall. Fractal and Fractional. 2023; 7(2):110. https://doi.org/10.3390/fractalfract7020110
Chicago/Turabian StyleKubilius, Kęstutis, and Aidas Medžiūnas. 2023. "A Class of Fractional Stochastic Differential Equations with a Soft Wall" Fractal and Fractional 7, no. 2: 110. https://doi.org/10.3390/fractalfract7020110