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

TL;DR

This paper proves the unique strong solvability of certain Itô equations with coefficients in Morrey classes, extending known results even for constant diffusion, under minimal regularity assumptions.

Contribution

It establishes the existence and uniqueness of strong solutions for Itô equations with diffusion and drift in Morrey classes, broadening the class of equations with well-posed solutions.

Findings

Unique strong solutions exist under minimal regularity.

Results hold even for constant diffusion coefficients.

The work extends classical solvability results to Morrey class coefficients.

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.