On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin

TL;DR

This paper extends the expected discounted penalty function (EDPF) to include additional ruin-related variables, providing a new framework for analyzing risk processes with claims modeled by subordinators and Brownian motion.

Contribution

It introduces a generalized EDPF that incorporates new record minima after ruin, enhancing risk assessment models with spectrally negative Lévy processes.

Findings

Extended EDPF characterization for spectrally negative Lévy processes

Application to compute expected discounted capital injections

Framework includes claims modeled by subordinators and Brownian perturbation

Abstract

In this chapter we propose an extended concept of the expected discounted penalty function (EDPF) that takes into account new ruin-related random variables. We add to the EDPF, which was introduced in classical papers [Gerber and Shiu (1997), (1998) and Gerber and Landry (1998)], a sequence of expected discounted functions of new record minima reached by a jump of the risk process after ruin. Inspired by results of Huzak et al. (2004) and developpements in fluctuation theory for spectrally negative L\'evy processes, we provide a characterization for this extended EDPF in a setting involving a cumulative claims modelled by a subordinator, and Brownian perturbation. We illustrate how the extended EDPF can be used to compute the expected discounted value of capital injections (EDVCI) for Brownian perturbed risk model.