On strong solutions of Itô’s equations with σ∈Wd1 and b∈Ld

We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in Wd,loc1 and the drift in Ld. We prove the unique strong solvability for any starting point and prove that, as a function of the starting point, the solutions are Hölder continuous with any exponent <1. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.