Strong convergence and Mittag-Leffler stability of stochastic theta method for time-changed stochastic differential equations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908625093525504 |
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| author | Chen, Jingwei Ye, Jun Chen, Jinwen Wang, Zhidong |
| author_facet | Chen, Jingwei Ye, Jun Chen, Jinwen Wang, Zhidong |
| contents | We propose the first $α$-parameterized framework for solving time-changed stochastic differential equations (TCSDEs), explicitly linking convergence rates to the driving parameter of the underlying stochastic processes. Theoretically, we derive exact moment estimates and exponential moment estimates of inverse $α$-stable subordinator $E$ using Mittag-Leffler functions. The stochastic theta (ST) method is investigated for a class of SDEs driven by a time-changed Brownian motion, whose coefficients are time-space-dependent and satisfy the local Lipschitz condition. We prove that the convergence order dynamically responds to the stability index $α$ of stable subordinator $D$, filling a gap in traditional methods that treat these factors independently. We also introduce the notion of Mittag-Leffler stability for TCSDEs, and investigate the criterion of Mittag-Leffler stability for both the exact and numerical solutions. Finally, some numerical simulations are presented to illustrate the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong convergence and Mittag-Leffler stability of stochastic theta method for time-changed stochastic differential equations Chen, Jingwei Ye, Jun Chen, Jinwen Wang, Zhidong Probability Numerical Analysis 65C30, 60H10 We propose the first $α$-parameterized framework for solving time-changed stochastic differential equations (TCSDEs), explicitly linking convergence rates to the driving parameter of the underlying stochastic processes. Theoretically, we derive exact moment estimates and exponential moment estimates of inverse $α$-stable subordinator $E$ using Mittag-Leffler functions. The stochastic theta (ST) method is investigated for a class of SDEs driven by a time-changed Brownian motion, whose coefficients are time-space-dependent and satisfy the local Lipschitz condition. We prove that the convergence order dynamically responds to the stability index $α$ of stable subordinator $D$, filling a gap in traditional methods that treat these factors independently. We also introduce the notion of Mittag-Leffler stability for TCSDEs, and investigate the criterion of Mittag-Leffler stability for both the exact and numerical solutions. Finally, some numerical simulations are presented to illustrate the theoretical results. |
| title | Strong convergence and Mittag-Leffler stability of stochastic theta method for time-changed stochastic differential equations |
| topic | Probability Numerical Analysis 65C30, 60H10 |
| url | https://arxiv.org/abs/2503.21653 |