Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.18626 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911816709308416 |
|---|---|
| author | Wang, Yu Xiao, Yimin Xu, Lihu |
| author_facet | Wang, Yu Xiao, Yimin Xu, Lihu |
| contents | We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ and $α$-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $α$-stable processes with $α=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $α$-stable process with $α\in (0,2)$, all the $β$-th moments, with $β\in (0,α)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $α\uparrow 2$. We verify our results by simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18626 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Phase transition in the EM scheme of an SDE driven by $α$-stable noises with $α\in (0,2]$ Wang, Yu Xiao, Yimin Xu, Lihu Probability We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ and $α$-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $α$-stable processes with $α=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $α$-stable process with $α\in (0,2)$, all the $β$-th moments, with $β\in (0,α)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $α\uparrow 2$. We verify our results by simulations. |
| title | Phase transition in the EM scheme of an SDE driven by $α$-stable noises with $α\in (0,2]$ |
| topic | Probability |
| url | https://arxiv.org/abs/2403.18626 |