Consider the $d$-dimensional SDE, $d > 1$,

$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$

where $W$ is a standard $d$-dimensional Brownian motion.

I am interested in the case where $\sigma$ is a bounded variation function, and $b$ is assumed as nice as possible.

Question:

Has there been any work done on this case? In one dimension, there are existence results for $\sigma$ of bounded variation, and $b$ moderately irregular (Sobolev/Holder regularity). However I have not been able to find much in the multidimensional case.

The reason I ask is I believe I have a feasible plan to prove existence in the multidimensional case, modulo several (hard) lemmas. However I would like to ensure that the result is new, and also would be of interest.

One potential application I have in mind is bounded variation optimal control in multiple dimensions.