On strong solutions of It\^o's equations with $D\sigma $ and $b $ in Morrey classes containing $L_{d}$

Abstract

We consider It\^o uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{2+\varepsilon,loc}$, and the drift in a Morrey class containing $L_{d}$. We prove the unique strong solvability in the class of admissible solutions for any starting point. The result is new even if the diffusion is constant.