Computing the finite-time expected discounted penalty function for a family of Lévy risk processes

Abstract

Ever since the first introduction of the expected discounted penalty function (EDPF), it has been widely acknowledged that it contains information that is relevant from a risk management perspective. Expressions for the EDPF are now available for a wide range of models, in particular for a general class of Lévy risk processes. Yet, in order to capitalize on this potential for applications, these expressions must be computationally tractable enough as to allow for the evaluation of associated risk measures such as Value at Risk (VaR) or Conditional Value at Risk (CVaR). Most of the models studied so far offer few interesting examples for which computation of the associated EDPF can be carried out to the last instances where evaluation of risk measures is possible. Another drawback of existing examples is that the expressions are available for an infinite-time horizon EDPF only. Yet, realistic applications would require the computation of an EDPF over a finite-time horizon. In this paper we address these two issues by studying examples of risk processes for which numerical evaluation of the EDPF can be readily implemented. These examples are based on the recently introduced meromorphic processes, including the beta and theta families of Lévy processes, whose construction is tailor-made for computational ease. We provide expressions for the EDPF associated with these processes and we discuss in detail how a finite-time horizon EDPF can be computed for these families. We also provide numerical examples for different choices of parameters in order to illustrate how ruin-based risk measures can be computed for these families of Lévy risk processes.

The Theory of Scale Functions for Spectrally Negative Lévy Processes

“…See for example Corollary 2 and Remark 2 in [65] as well as [66]. The second statement can be established using the same technique as in the proof of Theorem 1 in [64].…”

Extended Gerber–Shiu functions in a risk model with interest

“…In principle, the above expression is the finite horizon GerberShiu function of Kuznetsov and Morales (2014) with the time horizon integrated with respect to the first entrance time to (a, ∞).…”

“…The first expected value in the definition of f (x) looks similar as the finite-time Gerber-Shiu function of Kuznetsov and Morales (2014). But we replace here the deterministic horizon by a stopping time.…”

High order expansions for renewal functions and applications to ruin theory

“…Remark 12. It is worth comparing Theorem 10 with the results of Kuznetsov and Morales [14]. They consider a so called meromorphic risk process {R t , t ≥ 0}, which amounts to assume that the claims Z k 's have density…”

“…for some positive coefficients (b m ) m≥1 and increasing sequence (ρ m ) m≥1 satisfying ρ m → +∞. Corollary 1 of [14] states that the Laplace exponent Λ(z) := log E e zY1 of the Lévy process {R t , t ≥ 0} admits a meromorphic extension on z ∈ and that all the solutions of the (extended) Lundberg equation (25) are real, negative and simple. Furthermore, denoting these solutions by (−ζ n ) n≥1 , the ruin probability (21) has expansion…”