On a generalization of the expected discounted penalty function to include deficits at and beyond ruin

Abstract

In this paper we propose an extended concept of the expected discounted penalty function (EDPF) that takes into account new ruin-related random variables. In the well-known EDPF introduced in seminal papers by Gerber and Shiu (Insur Math Econ 21:129–137, 1997, N Am Actuar J 2(1):48–78, 1998) and Gerber and Landry (Insur Math Econ 22:263–276, 1998), we consider the expectation of a sequence of discounted penalty functions of new record minima reached by a claim of the risk process after ruin (and before recovery). Inspired by results of Huzak et al. (Ann Appl Probab 14(3):1378–1397, 2004) and developments in fluctuation theory for spectrally negative Levy processes, we provide a characterization for this extended EDPF in a setting involving a cumulative claims modeled by a subordinator, and Brownian perturbation. We illustrate how the extended EDPF can be used to compute the expected discounted value of capital injections for the Brownian perturbed risk model.