Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims

The Parisian ruin time, which is the first time the insurer's surplus process has an excursion below level zero that exceeds a prescribed time length, has been extensively analyzed in recent years mainly in the Levy model and its special cases. However, the cumulative Parisian ruin time, which is the first time the total time spent by the surplus process below level zero exceeds a certain time length, has been rarely considered in the literature. In this paper, we study the cumulative Parisian ruin problem in a renewal risk model with general interclaim times and exponential claims. Explicit formulas for the infinite-time cumulative Parisian ruin probability is first derived under a deterministic Parisian clock and then under an Erlang clock, where the latter case can also serve as an approximation of the former. The finite-time cumulative Parisian ruin probability is subsequently analyzed as well when the time horizon is another Erlang random variable. Our formulas are applied in various numerical examples where the interclaim times follow gamma, Weibull, or Pareto distribution. Consequently, we demonstrate that the choice of the interclaim distribution does have a significant impact on the cumulative Parisian ruin probabilities when one deviates from the exponential assumption.