On strong solutions of Itô's equations with a$\,\in W^{1}_{d}$ and b$\,\in L_{d}$
Nicolai V. Krylov · arXiv (Cornell University) · 2020
We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{d,loc}$, and the drift in $L_{d}$. 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.