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 $1$?