  Consider a multi-dimensional Brownian motion which models the surplus
processes of multiple lines of business of an insurance company. Our main
result gives exact asymptotics for the cumulative Parisian ruin probability as
the initial capital tends to infinity. An asymptotic distribution for the
conditional cumulative Parisian ruin time is also derived. The obtained results
on the cumulative Parisian ruin can be seen as generalizations of some of the
results derived in Debicki et al (2018, Stochastic Processes and Their
Applications). As a particular interesting case, the two-dimensional Brownian
motion risk model is discussed in detail.
