On the cumulative Parisian ruin of multi-dimensional Brownian motion models

Consider a multi-dimensional Brownian motion which models different lines of business of an insurance company. Our main result gives an approximation for the cumulative Parisian ruin probability as the initial capital becomes large. An approximation for the conditional cumulative Parisian ruin time is also derived. As a particular interesting case, the two-dimensional Brownian motion models are discussed in detail. Our results suggest that the company should not merge its two lines of business if we consider the cumulative Parisian ruin probability as a measure of risk. This provides an evidence in supporting the principle of portfolio diversification.