Probability Brownian motion lies between $2$ functions

Question

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 $b(t)$?

Answer 1

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:

1. C. Park and F.J. Schuurmann, Evaluation of barrier-crossing probabilities of Wiener paths, J. Appl. Prob. 13, 267-275 (1976).
2. C. Jennen and H.R. Lerche, First exit densities of Brownian motion through one- sided moving boundaries, Z. Wahr. Verw. Gebiete 55, 133-148 (1981).
3. J. Durbin, The first-passage density of a continuous Gaussian process to a general boundary, J. Appl. Prob. 22, 99-122 (1985).
4. P. Salminen, On the first hitting time and last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Prob. 20, 411-426 (1988).
5. T.H. Sheike, A boundary crossing result for Brownian motion, J. Appl. Prob. (1992), 29, 448-453
6. G.O. Roberts and C.F. Shortland, Pricing barrier options with time-dependent coefficients, Mathematical Finance 7, 83-93 (1997).
7. 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).