Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution.

I want to know if we assume $b$ satisfies some better condition but non-Lipschitz, like $b\in C^\alpha$, can we get the existence of strong solution to this equation without using Zvonkin's transform.

I tried to consider the Eular approximation: $$X^n_t=X^n_{t_n}+b(X_{t_n})(t-t_n)+(W_t-W_{t_n})\quad X^n_0=x$$ where $t_n=[t2^n]/2^n$. But I can prove the relatively compact property of $\{X^n_{.}\}$ in the metric space $$d_F(X^n,X^m)=E\frac{\sup_{t\leq 1}|X_t^n-X_t^m|}{1+\sup_{t\leq 1}|X_t^n-X_t^m|}$$ when $b\in C^\alpha$.