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

Abstract

We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in ${\mathit{W}}_{\mathit{d},\mathrm{loc}}^1$ and the drift in ${\mathit{L}}_{\mathit{d}}$. 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.