On the distribution of cumulative Parisian ruin

TL;DR

This paper introduces the concept of cumulative Parisian ruin, providing explicit formulas for its distribution in certain stochastic models, and explores its relationship with classical and Parisian ruin concepts.

Contribution

It offers the first explicit distribution formulas for cumulative Parisian ruin in compound Poisson and Brownian models, and analyzes its connections with existing ruin concepts.

Findings

Explicit distribution formulas for cumulative Parisian ruin in compound Poisson processes.

Detailed analysis of the Brownian ruin model.

Relationships between cumulative Parisian ruin and classical/Parisian ruin.

Abstract

We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyze for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.