TheW,Zscale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems


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

ESAIM-PROBABILITY AND STATISTICS, vol.24, pp.454-525, 2020 (Journal Indexed in SCI) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 24
  • Publication Date: 2020
  • Doi Number: 10.1051/ps/2019022
  • Title of Journal : ESAIM-PROBABILITY AND STATISTICS
  • Page Numbers: pp.454-525
  • Keywords: Spectrally negative processes, scale functions, Gerber-Shiu functions, Skorokhod regulation, dividend optimization, capital injections, processes with Poissonian, Parisian observations, generalized drawdown stopping, OPTIMAL DIVIDEND PROBLEM, OPTIMAL PERIODIC DIVIDEND, MARKOV ADDITIVE PROCESSES, COMPOUND POISSON-PROCESS, OCCUPATION TIMES, EXIT PROBLEMS, RUIN PROBABILITIES, RISK THEORY, PARISIAN REFLECTION, BRANCHING-PROCESSES

Abstract

In the last years there appeared a great variety of identities for first passage problems of spectrally negative Levy processes, which can all be expressed in terms of two "q-harmonic functions" (or scale functions)WandZ. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental "two-sided exit" problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the "W,Zalphabet". It is important to note - as is currently being shown - that these identities apply equally to other spectrally negative Markov processes, where however theW,Zfunctions are typically much harder to compute. We collect below our favorite recipes from the "W,Zkit", drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.