Tanaka's theorem ([wikipedia](https://en.wikipedia.org/wiki/Tanaka%27s_formula)) implies that $X\_t = |B\_t|$ is a weak solution to the SDE


$dX\_t = dW\_t + dL\_t^0(X\_t)$,


where $W\_t$ is a Brownian motion and $L\_t^0(X\_t)$ is the local time of $X\_t$ at $0$. I have two related questions:


1. Does there exist a (weak) solution to the SDE


$dX\_t = dW\_t + 2 dL\_t^0(X\_t)$


where again $L\_t^0(X\_t)$ is the local time of $X\_t$ at $0$?


2. What about the SDE


$dX\_t = \sqrt{X\_t}dW\_t + 2 dL\_t^1(X\_t)$,


where $L\_t^1(X\_t)$ is the local time of $X\_t$ at $1$?



Weak existence for modified Tanaka SDE
For the first see ch 6 prob 2.24 of Revuz & Yor


