Smoothness of scale functions for spectrally negative Lévy processes

By Terence Chan, Andreas E. Kyprianou, Mladen Savov. 2009

162 citations recorded by the source corpus.

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.

Subfields: Finance, Management Information Systems, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/82f0aa00a7a8e2c53776ca56ab1004a0e9955330

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Citations in the corpus

Optimal dividends in the dual model under transaction costs (2013, unknown): “, Chan et al. (2011). Hence, (a) Z (q) is C1(0,∞) and C0(R) for the bounded variation case, while it is C2(0,∞) and C1(R) for the unbounded variation case, and (b) Z (q) is C2(0,∞) and C1(R) for the bounded variation case, while it is C3(0,∞) and C2(R) for the unbounded variation case. (3) As in (8.”
Distribution of the Present Value of Dividend Payments in a Lévy Risk Model (2007, methodology): “The scale functions are also differentiable if we impose different assumptions on the spectrally negative Lévy process U ; see Chan and Kyprianou [3] for more details.”
ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS (2013, background): “The following facts about the scale functions are taken from Chan et al. (2011).”
A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “Chan et al. (2009). Otherwise, when X has paths of bounded variation, a necessary and sufficient condition for W (q) to be continuous is that the jump measure Π has no atoms (cf.”
De Finetti's optimal dividends problem with an affine penalty function at ruin (2010, background): “Further, from Theorem 1 in [12], W (q) is twice continuously differentiable on (0,∞) if σ > 0.”
A Lévy Insurance Risk Process with Tax (2008, background): “1 is, for instance, fulfilled if the Lévy measure has a completely monotone density (see [16] for the strict convexity of (W(δ))′ and [5] for infinite differentiability).”
The McKean stochastic game driven by a spectrally negative Lévy process (2008, background): “(9) Consider the exponential change of measure dP(1) dP ∣∣∣∣ Ft = eXt−ψ(1)t.”
Evaluating Scale Functions of Spectrally Negative Lévy Processes (2008, background): “But, nonetheless, it exhibits the important properties of the scale function, as specified in Proposition 1 and those found in, among others, [7], [15], [17], and [21].”
Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes (2017, background): “(B.2) From Chan et al. (2011), we know the following facts about the scale function.”
An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density (2009, unknown): “Note that by Theorem 3, W (q) is infinitely differentiable (this was proved for the first time in [6]) and therefore certainly smooth enough.”