We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy 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.

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

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the n-th moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are

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Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes. In this paper we work out this extension to a generalized risk model driven by a nondecreasing Levy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, in...

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On The Expected Discounted Penalty function for Lévy Risk Processes

A serious gap in the Proof of Pakes’s paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev’s theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Levy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin’s theorem is proved.

Convolution equivalence and distributions of random sums

The structure of various Gerber–Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg’s fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.

Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

We obtain a new fluctuation identity for a general Levy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kluppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of Levy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Levy process is spectrally one sided.

R. A. Doney

Papers

Overshoots and undershoots of L\'{e}vy processes