Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem

By Andreas E. Kyprianou, V. Rivero, Renming Song. 2008

65 citations recorded by the source corpus.

We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156–180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.

Subfields: Applied Mathematics, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6732356adae20818719c0bd6c8985569469b6e2a

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In Applied Mathematics

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Citations in the corpus

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory (2011, background): “Spectrally negative Lévy processes have been considered recently in Avram et al. (2007), Biffis and Kyprianou (2010), Erder and Klüppelberg (2009), Kyprianou et al. (2010), Loeffen (2009), Renaud and Zhou (2007), among others, in the context of insurance risk models.”
On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function (2011, background): “[26], and Loeffen and Renaud [29], who showed, respectively, that log-convexity of the density and of the survival functions suffice (the second condition is more general).”
A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “Other advantages of scale functions can be seen in recent literature that looks at dividend payments (for example in the form of reflection or refraction strategies) and have proved to be key in understanding optimal barrier strategies (see: Zhou, 2005; Renaud and Zhou, 2007; Kyprianou and Palmowski”
Smoothness of scale functions for spectrally negative Lévy processes (2009, unknown): “More recently, it has been shown in [11] that, in the case of bounded variation paths, W is continuously differentiable if and only if (cid:5) has no atoms.”
The Theory of Scale Functions for Spectrally Negative Lévy Processes (2011, result): “Following additional contributions in [74], the final word on concavity-convexity currently stands with the following theorem, taken from [81], which overlaps all of the aforementioned results.”
Optimal dividends problem with a terminal value for spectrally positive Levy processes (2013, background): “Avram et al (2007), Loeffen (2008) and Kyprianou et al. (2010) studied the case of S = 0 for spectrally negative Lévy processes.”