Tanaka's theorem (wikipedia) 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:

- 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 $0$?

- 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 $1$?