Existence of strong solution to SDEs with non-Lipschitzian drift

Question

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 $b\in C^\alpha$.

Answer 1

This question has recently been addressed in detail in the article "Averaging along irregular curves and regularisation of ODEs" by Catellier and Gubinelli. They give a purely analytical condition on the "irregularity" of $W$ and the regularity of $b$ which guarantees that the fixed point equation $$ x(t) = x_0 + W(t) + \int_0^t b(x(s))\,ds $$ admits a unique solution in the space of continuous functions. In particular, for any fixed $b \in C^\alpha$ with $\alpha>0$, their condition is satisfied for almost every sample path of a Brownian motion. If $W$ is even more irregular (for example a typical trajectory of fractional Brownian motion with small Hurst parameter), then they can even build solutions for distributional drifts $b$. Since the constructions are purely analytic, the solutions are automatically strong.

Answer 2

Another approach for this problem has been developped in "Construction of strong solutions of SDE's via Malliavin calculus" by T. Meyer-Brandis and F. Proske. It has been further developped and extended in "A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s" by O. Menoukeu-Pamen and al. In these papers, the drift is assumed to be only measurable (or bounded and measurable).