Let $m: \mathbb R_+\to [0,1]$ be continuous and decreasing. Consider

$$X_t=1+bt+\int_0^t\frac{\sigma}{1+m(s)}dW_s,\quad \forall t\ge 0,$$

where $b, \sigma>0$ are given and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Do we have an explicit (or semi-explicit) formula for the probability

$$\mathbb P[\inf_{0\le s\le t}X_s>0]?$$

Any answer and comments are highly appreciated.