The boundary crossing problem in one-dimensional jump-diffusion processes has significant applications in finance and insurance, making it a topic of great interest. The computation of the boundary crossing distribution for diffusion processes, however, has long been a difficult problem due to the absence of explicit formulas for general boundaries.

In this study, new formulas for piecewise linear boundary crossing probabilities and densities of Brownian motion with a compound Poisson process are derived. These formulas can be employed to approximate the first passage time distributions for general nonlinear boundaries, broadening the scope of applicability. The method can also be extended to other diffusion processes, such as geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps. The numerical computation of these formulas can be performed using Monte Carlo integration, a method that is both straightforward and easy to implement. Numerical examples are provided to illustrate the utility of this approach.

Furthermore, a conclusion has been reached regarding the existence of the first passage time density. If the first passage time density of a Brownian motion crossing a boundary exists, then the first passage time density of a Brownian motion with a compound Poisson process crossing the same boundary also exists. This finding is significant as it offers insights into the behavior of these complex processes when they encounter boundaries, furthering our understanding of their dynamics and potential applications in various fields.