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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.22020 |
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| _version_ | 1866918072798937088 |
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| author | Kyprianou, Andreas E. Mantelos, Harry S. Rivero, Victor |
| author_facet | Kyprianou, Andreas E. Mantelos, Harry S. Rivero, Victor |
| contents | We start by remarking a one-to-one correspondence between self-similar Markov processes (ssMps) on a Banach space and Markov additive processes (MAPs) that is analogous to the well-known one between positive ssMps and Lévy processes through the renowned Lamperti-transform, with the main difference that ours is norm-dependent. We then consider multidimensional self-similar Markov processes obtained by killing or by reflecting a stable process or Brownian motion in the orthant and we then fully describe the MAPs associated to them using the $L_1$-norm. Namely, we describe the MAP underlying the ssMp obtained by killing a $d$-dimensional $α$-stable process when it leaves the orthant and the one obtained by reflecting it back in the orthant continuously (or by a jump); finally, we also describe the MAP underlying $d$-dimensional Brownian motion reflected in the orthant. The first three of the aforementioned examples are pure-jump, and the last is a diffusion, so their characterization is given through their Lévy system, generator and/or through the modulated SDE that defines them, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Norm-dependent Lamperti-type MAP representations of stable processes and Brownian motions in the orthant Kyprianou, Andreas E. Mantelos, Harry S. Rivero, Victor Probability We start by remarking a one-to-one correspondence between self-similar Markov processes (ssMps) on a Banach space and Markov additive processes (MAPs) that is analogous to the well-known one between positive ssMps and Lévy processes through the renowned Lamperti-transform, with the main difference that ours is norm-dependent. We then consider multidimensional self-similar Markov processes obtained by killing or by reflecting a stable process or Brownian motion in the orthant and we then fully describe the MAPs associated to them using the $L_1$-norm. Namely, we describe the MAP underlying the ssMp obtained by killing a $d$-dimensional $α$-stable process when it leaves the orthant and the one obtained by reflecting it back in the orthant continuously (or by a jump); finally, we also describe the MAP underlying $d$-dimensional Brownian motion reflected in the orthant. The first three of the aforementioned examples are pure-jump, and the last is a diffusion, so their characterization is given through their Lévy system, generator and/or through the modulated SDE that defines them, respectively. |
| title | Norm-dependent Lamperti-type MAP representations of stable processes and Brownian motions in the orthant |
| topic | Probability |
| url | https://arxiv.org/abs/2506.22020 |