Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift
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| Format: | Preprint |
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2026
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| _version_ | 1866910020992499712 |
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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 investigate the strong approximation of stochastic differential equations whose drift is square-integrable in time and Dini continuous in space, while the diffusion coefficient is non-constant and uniformly elliptic. Using a refined Itô--Tanaka trick combined with parabolic regularity estimates, we first establish strong well-posedness and the stochastic flow property. Under additional Lipschitz regularity of the diffusion matrix, we then analyze a polygonal-type Euler--Maruyama scheme and prove the strong error estimate \[ \Big\|\sup_{0\le t\le1}|X_t-X_t^n|\Big\|_{L^p(Ω)} \le C n^{-\frac12}\log(n)^{\frac32}, \quad p\ge2. \] We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order $1/2$ cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue--Dini drift framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12456 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift Wei, Jinlong Hu, Junhao Lv, Guangying Yuan, Chenggui Probability Primary 65C30, Secondary 60H10, 35K10 We investigate the strong approximation of stochastic differential equations whose drift is square-integrable in time and Dini continuous in space, while the diffusion coefficient is non-constant and uniformly elliptic. Using a refined Itô--Tanaka trick combined with parabolic regularity estimates, we first establish strong well-posedness and the stochastic flow property. Under additional Lipschitz regularity of the diffusion matrix, we then analyze a polygonal-type Euler--Maruyama scheme and prove the strong error estimate \[ \Big\|\sup_{0\le t\le1}|X_t-X_t^n|\Big\|_{L^p(Ω)} \le C n^{-\frac12}\log(n)^{\frac32}, \quad p\ge2. \] We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order $1/2$ cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue--Dini drift framework. |
| title | Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift |
| topic | Probability Primary 65C30, Secondary 60H10, 35K10 |
| url | https://arxiv.org/abs/2602.12456 |