The martingale problem for geometric stable-like processes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916542020583424 |
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| author | Iyer, Sarvesh Ravichandran |
| author_facet | Iyer, Sarvesh Ravichandran |
| contents | We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form $$ (\mathcal Lf)(x) = \int_{\mathbb R^d \setminus \{0\}} \left(f(x+h)-f(x) - (\nabla f(x) \cdot h)1_{\|h\| < 1}\right)K(x,h) dh, $$ where $K(x,\cdot)$ is a jump kernel of the form $K(x,h) \sim \frac{l(\|h\|)}{\|h\|^d}$ for each $x \in \mathbb R^d,\|h\|<1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $α$-geometric stable processes, $α\in (0,2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The martingale problem for geometric stable-like processes Iyer, Sarvesh Ravichandran Probability 60J76 (Primary), 60G51 (Secondary) We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form $$ (\mathcal Lf)(x) = \int_{\mathbb R^d \setminus \{0\}} \left(f(x+h)-f(x) - (\nabla f(x) \cdot h)1_{\|h\| < 1}\right)K(x,h) dh, $$ where $K(x,\cdot)$ is a jump kernel of the form $K(x,h) \sim \frac{l(\|h\|)}{\|h\|^d}$ for each $x \in \mathbb R^d,\|h\|<1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $α$-geometric stable processes, $α\in (0,2]$. |
| title | The martingale problem for geometric stable-like processes |
| topic | Probability 60J76 (Primary), 60G51 (Secondary) |
| url | https://arxiv.org/abs/2412.18677 |