Wiener process with drift $W_t+at$

Question

Let $W$ be the standard Wiener process. The task is to establish for which $a$ and $b$ the stopping moment is finite almost surely. $$\sigma_{a,b}=inf\{t \geq 0\space W_t+at=b\}$$ I established that for a=0 and all b it is finite almost surely, but I am not sure on how to approach this drift. Thank you for your hints. Should I use some martingale (in the case of a=0 I used the fact that $W_t$ is a Martingale)?