  The first passage time (FPT) problem is ubiquitous in many applications. In
finance, we often have to deal with stochastic processes with jump-diffusion,
so that the FTP problem is reducible to a stochastic differential equation with
jump-diffusion. While the application of the conventional Monte-Carlo procedure
is possible for the solution of the resulting model, it becomes computationally
inefficient which severely restricts its applicability in many practically
interesting cases. In this contribution, we focus on the development of
efficient Monte-Carlo-based computational procedures for solving the FPT
problem under the multivariate (and correlated) jump-diffusion processes. We
also discuss the implementation of the developed Monte-Carlo-based technique
for multivariate jump-diffusion processes driving by several compound Poisson
shocks. Finally, we demonstrate the application of the developed methodologies
for analyzing the default rates and default correlations of differently rated
firms via historical data.
