Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915131310473216 |
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| author | Pavlyukevich, Ilya Thipyarat, Sooppawat |
| author_facet | Pavlyukevich, Ilya Thipyarat, Sooppawat |
| contents | For a solution $X$ of a Lévy-driven $d$-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme $X^h$ has a strong convergence of order $\frac12$: for each $T\in [0,\infty)$ and all $x\in\mathbb R^d$ we have $$ \mathbf E \sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1}{2}}(1+|x|),\quad h\to 0. $$ We also determine the rate of the locally uniform strong convergence: for each $N\in(0,\infty)$ and $\varepsilon\in (0,1)$ we have $$ \mathbf E\sup_{|x|\leq N}\sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1-\varepsilon}{4d}},\quad h\to 0. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_19175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs Pavlyukevich, Ilya Thipyarat, Sooppawat Probability 65C30, 60H10, 60G51, 60H35 For a solution $X$ of a Lévy-driven $d$-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme $X^h$ has a strong convergence of order $\frac12$: for each $T\in [0,\infty)$ and all $x\in\mathbb R^d$ we have $$ \mathbf E \sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1}{2}}(1+|x|),\quad h\to 0. $$ We also determine the rate of the locally uniform strong convergence: for each $N\in(0,\infty)$ and $\varepsilon\in (0,1)$ we have $$ \mathbf E\sup_{|x|\leq N}\sup_{kh\leq T}|X_{kh}(x)-X^h_{kh}(x)|\leq C h^{\frac{1-\varepsilon}{4d}},\quad h\to 0. $$ |
| title | Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs |
| topic | Probability 65C30, 60H10, 60G51, 60H35 |
| url | https://arxiv.org/abs/2501.19175 |