I am having an SDE for which I would be in trouble if there were no strong solution.
The SDE is -
$ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$
where $b_1$ and $b_2$ are two independent brownian motions. If the usual Lipschitz conditions were satisfied then I'd have a unique strong solution. Unfortunately, there is a point $x_1$ where all $\mu, \sigma_1$ and $\sigma_2$ have a jump.
All $\mu, \sigma_1, \sigma_2$ are bounded from below by $c_1 > 0$ and bounded from above by $c_2 > 0$. Moreover $\sigma_1, \sigma_2$ have bounded variation.
I know of a paper by Nakao that tells that if I had just one brownian motion instead of 2 then boundedness from below by a strictly positive number along with bounded variation of $\sigma$ will give me a strong solution.
I was wondering if something similar can be said about my SDE with two brownian motions instead of one. Note that $x$ is still a scalar
