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

Autori
Avram, Florin ; Grahovac, Danijel ; Vardar-Acar, Ceren

Izvornik
Risks (2227-9091) 7 (2019), 1; 18, 15

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
first passage ; drawdown process ; spectrally negative process ; scale functions ; dividends ; de Finetti valuation objective ; variational problem

Sažetak
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 Xt , 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 Yt=sup0≤s≤tXs−Xt obtained by reflecting Xt at its maximum. This new variational problem has been solved in a parallel paper.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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