# Probability Brownian motion lies between $2$ functions

Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \ldots < t_J$ are time points. Let $W_t$ be a standard Brownian motion. Is it possible to further simplify the expression \begin{align*} \mathbb{P}\left(\bigcap_{j=0}^{J-1}\left\{\sup_{t \in [t_j, t_{j+1})}\left\{\left|W_t + a_j\right| - b_j\right\} \le 0\right\}\right) = \mathbb{P}\left(-b(t) - a(t) \le W_t \le b(t) - a(t) \forall t \in [0, t_J]\right) \end{align*} where $a(t) = a_j$ for $t \in [t_j, t_{j+1})$, and similarly for $b(t)$?