The Euler-Maruyama method for SDEs with low-regularity drift
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908489962487808 |
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| author | Wei, Jinlong Hu, Junhao Lv, Guangying Yuan, Chenggui |
| author_facet | Wei, Jinlong Hu, Junhao Lv, Guangying Yuan, Chenggui |
| contents | We study the strong $L^p$-convergence rates of the Euler-Maruyama method for stochastic differential equations driven by Brownian motion with low-regularity drift coefficients. Specifically, the drift is assumed to be in the Lebesgue-Hölder spaces $L^q([0,T]; {\mathcal C}_b^α({\mathbb R}^d))$ with $α\in(0,1)$ and $q\in (2/(1+α),\infty]$. For every $p\geq 2$, by using stochastic sewing and/or the Itô-Tanaka trick, we obtain the $L^p$-convergence rates: $(1+α)/2$ for $q\in [2,\infty]$ and $(1-1/q)$ for $q\in (2/(1+α),2)$. Moreover, we prove that the unique strong solution can be constructed via the Picard iteration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Euler-Maruyama method for SDEs with low-regularity drift Wei, Jinlong Hu, Junhao Lv, Guangying Yuan, Chenggui Probability 65C30, 60H50 We study the strong $L^p$-convergence rates of the Euler-Maruyama method for stochastic differential equations driven by Brownian motion with low-regularity drift coefficients. Specifically, the drift is assumed to be in the Lebesgue-Hölder spaces $L^q([0,T]; {\mathcal C}_b^α({\mathbb R}^d))$ with $α\in(0,1)$ and $q\in (2/(1+α),\infty]$. For every $p\geq 2$, by using stochastic sewing and/or the Itô-Tanaka trick, we obtain the $L^p$-convergence rates: $(1+α)/2$ for $q\in [2,\infty]$ and $(1-1/q)$ for $q\in (2/(1+α),2)$. Moreover, we prove that the unique strong solution can be constructed via the Picard iteration. |
| title | The Euler-Maruyama method for SDEs with low-regularity drift |
| topic | Probability 65C30, 60H50 |
| url | https://arxiv.org/abs/2508.10512 |