Does there exist a process which, when it is positive behaves like Brownian motion with drift $\mu >0$ and when it is negative like Brownian motion with drift $-\mu<0$? I.e., Brownian motion with drift away from the origin.
It would have an SDE: $$ dX_t = \mu \times \text{sign}(X_t)dt + dB_t $$ but I'm aware that $\text{sign}(x)$is not a Lipschitz function so we can't apply the classic existence result.
Can we say that a process $|X_t|$ exists with drift $\mu$ and behaviour like a reflected Brownian motion? If so, what does the SDE look like? How would I calculate something like $\mathbb{P}(|X_t|\leq r)$?
I don't really know where to begin, but I think it should be true that the probability distribution function of $|X_t|$ is the solution of some partial differential equation (like the pdf of $|B_t|$ is when $B$ is Brownian motion without drift, but as is not the case for $|B_t|$ with the usual omni-directional drift.
Any help is much appreciated.