Abstract
Given a spectrally-negative Lévy process and independent Poisson observation times, we consider a periodic barrier strategy that pushes the process down to a certain level whenever the observed value is above it. We also consider the versions with additional classical reflection above and/or below. Using scale functions and excursion theory, various fluctuation identities are computed in terms of the scale functions. Applications in de Finetti’s dividend problems are also discussed.
Highlights
In actuarial risk theory, the surplus of an insurance company is typically modeled by a compoundPoisson process with a positive drift and negative jumps (Cramér–Lundberg model) or more generally by a spectrally-negative Lévy process
There exists a variety of tools available to compute various quantities that are useful in insurance mathematics
The work in Avram et al (2007) obtained the expected net present value (NPV) of dividends until ruin; a sufficient condition for the optimality of a barrier strategy is given in Loeffen (2008)
Summary
The surplus of an insurance company is typically modeled by a compound. Obtained the expected NPV of dividends and capital injections under a double barrier strategy They showed that it is optimal to reflect the process at zero and at some upper boundary, with the resulting surplus process being a doubly-reflected Lévy process. This process models the controlled surplus process under a combination of the classical and periodic barrier dividend strategies This is a generalization of the Brownian motion case as studied in Avanzi et al (2016). By shifting the process (by − a), it models the controlled surplus process under a combination of the classical and periodic barrier dividend strategies as in (2) with additional classical capital injections For these four processes, we compute various fluctuation identities that include:. Throughout the paper, for any function f of two variables, let f 0 (·, ·) be the partial derivative with respect to the first argument
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