Smoothness of scale functions for spectrally negative Lévy processes. (English) Zbl 1259.60050

Scale functions play an important role in the fluctuation theory of spectrally negative Lévy processes, and appear in various identities for one- and two-sided exit problems. Moreover, they often appear in martingales associated with spectrally negative Lévy processes and, using Itô calculus, it could be verified that they are the solutions to partial integro-differential equations associated with certain boundary value problems. However, the application of Itô calculus requires that functions are sufficiently smooth.
Motivated by these relations, the authors study the differentiability properties of scale functions for Lévy processes with Gaussian component or paths of bounded variation. Their main results can be roughly summarized as follows:
If the process has a Gaussian component then the scale function is, at least, twice differentiable.
If the process has paths of bounded variation and the Lévy measure is sufficiently smooth, then the scale function is, at least, twice differentiable.
The proofs are based on new results regarding the smoothness of renewal measures of subordinators, which are of interest on their own.


60G51 Processes with independent increments; Lévy processes
60J45 Probabilistic potential theory
91B30 Risk theory, insurance (MSC2010)
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