I was wondering if anything could be said at all about the well-psedness of the following time-inhomogeneous singular diffusion SDE:


\begin{align}d X\_t&=\sigma(X\_t,t ) d W\_t , \qquad t\geq 0, \, X\_0 \in \mathbb R \\
\sigma(x,t )&=t^{-\alpha} b(x) \qquad \, \alpha>0,
\end{align}


with $b$ as nice as required (say bounded, or Lipschitz continuous and sublinearly growing).


I can imagine for existence one needs at least $\alpha <1/2$ so that $\sigma \in L^2(\mathbb R, [0,T])$. Perhaps something could be achieved by time reversal?


The following paper may be relvant, but I am not entirely sure of its applicability


<https://projecteuclid.org/journals/annals-of-probability/volume-46/issue-3/Strong-solutions-to-stochastic-differential-equations-with-rough-coefficients/10.1214/17-AOP1208.pdf>



Solution of SDE with time power law singular diffusion
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


*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 p. (2005). [ZBL1071.60003](https://zbmath.org/?q=an:1071.60003).


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


*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).


