It seems natural to view the Itô process on a different time scale where it becomes a Itô diffusion, and in turn, invoke existence/uniqueness theory for Itô diffusion. 


In particular, under the (deterministic) time change given by $\tau(t) = (t-2 \alpha t )^{\frac{1}{1-2 \alpha}}$ (so that  $\tau'(t) = \tau(t)^{2 \alpha}$ and $\tau(0)=0$),  the time-changed process is (in law) an Itô diffusion that satisfies $$
d \tilde{X}_t = b(\tilde{X}_t) d B_t
$$ where $\tilde{X}_t = X_{\tau(t)}$ and $B_t=\int_0^{\tau(t)} s^{-\alpha} dW_s$ is again a standard Brownian motion.  Note that this time change only makes sense for $\alpha \in [0,1/2)$. 


Sufficient conditions for existence and uniqueness of the time-changed process can be found in, e.g., Section 1.2 of the monograph 

<cite authors="Cherny, Alexander S.; Engelbert, Hans-Jürgen">_Cherny, Alexander S.; Engelbert, Hans-Jürgen_, [**Singular stochastic differential equations.**](http://dx.doi.org/10.1007/b104187), Lecture Notes in Mathematics 1858. Berlin: Springer (ISBN 3-540-24007-1/pbk). viii, 128&nbsp;p. (2005). [ZBL1071.60003](https://zbmath.org/?q=an:1071.60003).</cite>

For more about the correspondence between an Itô diffusion and Itô process via time change, see

<cite authors="Øksendal, Bernt">_Øksendal, Bernt_, [**When is a stochastic integral a time change of a diffusion?**](http://dx.doi.org/10.1007/BF01045159), J. Theor. Probab. 3, No. 2, 207-226 (1990). [ZBL0698.60046](https://zbmath.org/?q=an:0698.60046).</cite>