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Open AccessArticle

The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

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Laboratoire de Mathématiques Appliquées, Université de Pau, 64012 Pau, France
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Department of Mathematics, University of Osijek, 31000 Osijek, Croatia
3
Department of Statistics, Middle East Technical University, Ankara 06800, Turkey
*
Author to whom correspondence should be addressed.
Risks 2019, 7(1), 18; https://doi.org/10.3390/risks7010018
Received: 21 November 2018 / Revised: 31 January 2019 / Accepted: 13 February 2019 / Published: 19 February 2019
As is well-known, the benefit of restricting Lévy processes without positive jumps is the “ W , Z scale functions paradigm”, by which the knowledge of the scale functions W , Z extends immediately to other risk control problems. The same is true largely for strong Markov processes X t , with the notable distinctions that (a) it is more convenient to use as “basis” differential exit functions ν , δ , and that (b) it is not yet known how to compute ν , δ or W , Z beyond the Lévy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Lévy cases are very similar (except for the level of work involved in computing the basic functions ν , δ ). We illustrate the potential of the unified framework by introducing a new objective (33) for the optimization of dividends, inspired by the de Finetti problem of maximizing expected discounted cumulative dividends until ruin, where we replace ruin with an optimally chosen Azema-Yor/generalized draw-down/regret/trailing stopping time. This is defined as a hitting time of the “draw-down” process Y t = sup 0 s t X s X t obtained by reflecting X t at its maximum. This new variational problem has been solved in a parallel paper. View Full-Text
Keywords: first passage; drawdown process; spectrally negative process; scale functions; dividends; de Finetti valuation objective; variational problem first passage; drawdown process; spectrally negative process; scale functions; dividends; de Finetti valuation objective; variational problem
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Avram, F.; Grahovac, D.; Vardar-Acar, C. The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps. Risks 2019, 7, 18.

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