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 $b(t)$?
 A: 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:


*

*C. Park and F.J. Schuurmann, Evaluation of barrier-crossing probabilities of Wiener paths, J. Appl. Prob. 13, 267-275 (1976).

*C. Jennen and H.R. Lerche, First exit densities of Brownian motion
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*J. Durbin, The first-passage density of a continuous Gaussian process to a general boundary, J. Appl. Prob. 22, 99-122 (1985).

*P. Salminen, On the first hitting time and last exit time for a
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*T.H. Sheike, A boundary crossing result for Brownian motion, J. Appl. Prob. (1992), 29, 448-453

*G.O. Roberts and C.F. Shortland, Pricing barrier options with
time-dependent coefficients, Mathematical Finance 7, 83-93 (1997).

*C.F. Lo and C.H. Hui, Computing the first passage
time density of a time- dependent Ornstein-Uhlenbeck process to a
moving boundary, Appl. Math. Letters 19, 1399-1405 (2006).

