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

Avram F., Grahovac D., Vardar Acar C.

RISKS, vol.7, no.1, 2019 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 7 Issue: 1
  • Publication Date: 2019
  • Doi Number: 10.3390/risks7010018
  • Journal Name: RISKS
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Keywords: first passage, drawdown process, spectrally negative process, scale functions, dividends, de Finetti valuation objective, variational problem, NEGATIVE LEVY PROCESSES, PARISIAN REFLECTION, EXIT PROBLEMS, DRAWDOWN, TIMES, ERGODICITY
  • Middle East Technical University Affiliated: Yes


As is well-known, the benefit of restricting Levy 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 nu,delta, and that (b) it is not yet known how to compute nu,delta or W,Z beyond the Levy, diffusion, and a few other cases. The unifying framework outlined in this paper suggests, however, via an example that the spectrally negative Markov and Levy cases are very similar (except for the level of work involved in computing the basic functions nu,delta). 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.