Level Crossings and Absorption of an Insurance Model

This chapter discusses a particular piecewise‐deterministic Markov process (PDMP) to catastrophic events occurring at random times and with random intensities. It considers the insurance model by Kovacevic and Pflug describing the evolution of a capital subject to random heavy loss events. The chapter presents a local‐time crossing relation for the PDMP. This local‐time crossing relation allows for the proof of the so‐called Kac‐Rice formula, giving an explicit form for the average number of continuous crossings by the process of a given level. The chapter provides the results on the estimation of the absorption probability and hitting time for the PDMP. The motion of the process depends on an easily estimable quantity in a parametric, semi‐parametric or non‐parametric setting. The chapter focuses on a procedure for estimating the Markov kernel R of the post‐jump locations, formula leads to estimate the transition density by the plug‐in estimator