Is there an existence and uniqueness theorem for SDEs of the following type:

$dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$,

where $\tilde{W}_{t}$ is say $d$-dimensional Brownian motion and where $\mu$ is a "nice" function of the path of the solution up to time t. (In the case I am interested in, it's a drift that turns on or off depending on whether certain $W_t$-stopping times have happened by time $t$ or not).

I could definitely construct the solution by "gluing together" solutions for intervals of time during which the drift only depends on the position $W_t$, but I was wondering if there is a slicker way.