Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift

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 $$\parallel \underset{0\le t\le 1}{\sup }\mathrm{\mid }{X}_t-X_t^n\mathrm{\mid }{\parallel }_{L^p(\mathrm{\Omega })}\le C{n}^{-\frac{1}{2}}\log (n{)}^{\frac{3}{2}},p\ge 2.$$ We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order $1\mathrm{/}2$ cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue–Dini drift framework.