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 It\^{o}--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 $$\Big\|\sup_{0\le t\le1}|X_t-X_t^n|\Big\|_{L^p(\Omega)} \le C n^{-\frac12}\log(n)^{\frac32}, \quad p\ge2.$$ 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.