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Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process — arXiv:2608.17540

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.17540
License (as archived): CC0-1.0
Archived: 2026-10-18T15:04:58.218Z
SHA-256 of the archived text: 71119d288ecc53baa02a1c6a81491b8b0640700526961c24fe89b8670069409d
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How to cite

Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process — arXiv:2608.17540 — arXiv (en), https://arxiv.org/abs/2608.17540 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_eb3ee2e2ab07 (Metraipolis archive dc_eb3ee2e2ab07, sha256 71119d288ecc53baa02a1c6a81491b8b0640700526961c24fe89b8670069409d)

BibTeX

@misc{dceb3ee2e2ab07,
  title = {Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process — arXiv:2608.17540},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.17540},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_eb3ee2e2ab07; sha256 71119d288ecc53baa02a1c6a81491b8b0640700526961c24fe89b8670069409d},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.17540. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $\alpha$-stable process, where $\alpha\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{\alpha-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/\alpha$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{\alpha-1}$. For stable jump coefficients with Hölder exponent $\wt\eta>1/\alpha$, the estimate gives explicit power rates in $B,A,S$; at $\wt\eta=1/\alpha$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-19 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.17540 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T14:59:27.060Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-19?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).