  We derive explicit formulas for probabilities of Brownian motion with jumps
crossing linear or piecewise linear boundaries in any finite interval. We then
use these formulas to approximate the boundary crossing probabilities for
general nonlinear boundaries. The jump process can be any integer-valued
process and jump sizes can have general distributions. Moreover, the jump sizes
can be even correlated and/or non-identically distributed. The numerical
algorithm is straightforward and easy to implement. Some numerical examples are
presented.

Probability