Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

Abstract

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations. Spectrally negative Lévy processes and scale functionsLet X = {X t : t ≥ 0} be a Lévy process defined on a filtered probability space (Ω, F, F, P), where {F t : t ≥ 0} is the filtration generated by X satisfying the usual conditions. For x ∈ R denote by P x the law of X when it is started at x and write simply P 0 = P. Accordingly we shall write E x and E for the associated expectation operators. In this paper we shall assume throughout that X is spectrally negative meaning here that it has no positive jumps and that it is not the negative of a subordinator. It is well known that the latter allows us to talk about the Laplace exponent ψ(θ) := log E[e θX1 ] for (θ) ≥ 0 where in particular we have the Lévy-Khintchine representationwhere a ∈ R, σ ≥ 0 is the Gaussian coefficient and Π is a measure concentrated on (−∞, 0) satisfying (−∞,0) (1∧x 2 )Π(dx) < ∞. The, so-called, Lévy triple (a, σ, Π) completely characterises the process X. For later reference we also introduce the function Φ : [0, ∞) → [0, ∞) as the right inverse of ψ on (0, ∞) so that for all q ≥ 0 Φ(q) = sup{θ ≥ 0 : ψ(θ) = q}.(2)

The Theory of Scale Functions for Spectrally Negative Lévy Processes

“…The following theorem, taken from Hubalek and Kyprianou [48] (see also Vigon [104]), shows how one may identify a spectrally negative Lévy process X (called the parent process) for a given descending ladder height process H. The proof follows by a straightforward manipulation of the Wiener-Hopf factorization (4.1).…”

“…Theorem 4.1, as stated in its more general form in [48], says that there exists a parent process, say X, that drifts to −∞ such that its Laplace exponent ψ can be factorized as…”

An Insurance Risk Model with Parisian Implementation Delays

“…In nature, this is similar to the use of Erlangian horizons (rather than a deterministic horizon) for the calculation of finite-time ruin probabilities in various risk models (see Asmussen et al (2002) and Ramaswami et al (2008)). As will be shown, all our results are expressed in terms of scale functions for which many explicit examples are known; see, e.g., Hubalek and Kyprianou (2010), Kyprianou and Rivero (2008), as well as the numerical algorithm developed by Surya (2008). Furthermore, mixed Erlang distributions are known to be a very large and flexible class of distributions for modelling purposes (see Willmot and Woo (2007)).…”

Optimal Capital Structure With Scale Effects Under Spectrally Negative Lévy Models

“…However, this can be extended very easily to the phase-type case; see [16]. For other spectrally negative Lévy processes with explicit forms of scale functions, see [23,26,27]. We also remark that the solutions can in principle be computed numerically for any choice of spectrally negative Lévy process by using the approximation algorithms of the scale function such as [16,37].…”

“…The solutions to our generalized model admit semi-explicit expressions written in terms of the scale function, which has analytical forms in certain cases [18,23,26,27] and can be approximated generally using, e.g., [16,37]. In order to illustrate the implementation side, we give an example based on a mixture of Brownian motion and a compound Poisson process with i.i.d.…”