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 $b(t)$?



Probability Brownian motion lies between $2$ functions
This is the problem of Brownian motion between two *moving* absorbing boundaries. For a linear time dependence some analytical progress can be made, but for arbitrary time dependence no closed-form solution exists. Here are some pointers to the literature:


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moving boundary, Appl. Math. Letters 19, 1399-1405 (2006).


