Invariant Sublinear Expectations
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912128639696896 |
|---|---|
| author | Song, Yongsheng |
| author_facet | Song, Yongsheng |
| contents | We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\inΘ}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\inΘ^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., \[\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.\] Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $Θ^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities.
As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14177 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariant Sublinear Expectations Song, Yongsheng Probability 28A12, 28D05, 37A30 We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\inΘ}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\inΘ^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., \[\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.\] Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $Θ^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities. As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation. |
| title | Invariant Sublinear Expectations |
| topic | Probability 28A12, 28D05, 37A30 |
| url | https://arxiv.org/abs/2411.14177 |