Uniqueness of strong solutions for SDEs with H\"older diffusions

TL;DR

This paper establishes a new uniqueness result for strong solutions of Itô stochastic differential equations with H"older continuous diffusion coefficients of class , using Itô's formula and advanced smoothing techniques.

Contribution

It provides the first known uniqueness result for strong solutions with  H"older diffusions and continuous drifts, expanding the theory of SDEs.

Findings

Uniqueness of strong solutions for  H"older diffusions

Application of Itô's formula with smoothing techniques

Extension of solution theory to less regular coefficients

Abstract

This paper is concerned with the It\^o stochastic differential equations with $\mR^{d\times k}$ diffusions in class of H\"older spaces and continuous $\mR^d$ drifts. We derive a uniqueness result of strong solutions for $\cC^\alpha \ (\alpha\geq \frac{1}{2})$ coefficients and this result is new. Our proof is supported by It\^o's formula and a finer analysis on cut-off and smoothing techniques.