First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries

The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.