A PONTRYAGHIN MAXIMUM PRINCIPLE APPROACH FOR THE OPTIMIZATION OF DIVIDENDS/CONSUMPTION OF SPECTRALLY NEGATIVE MARKOV PROCESSES, UNTIL A GENERALIZED DRAW-DOWN TIME

Avram Florin 1 Dan Goreac 2
2 PS
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : The first motivation of our paper is to explore further the idea that, in risk control problems, it may be profitable to base decisions both on the position of the underlying process Xt and on its supremum Xt := sup 0≤s≤t Xs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time (see [AY79]), this framework provides a natural unification of draw-down and classic first passage times. We illustrate here the potential of this unified framework by solving a variation of the De Finetti problem of maximizing expected discounted cumulative dividends/consumption gained under a barrier policy, until an optimally chosen Azema-Yor time, with a general spectrally negative Markov model. While previously studied cases of this problem [APP07, SLG84, AS98, AVZ17, AH18, WZ18] assumed either Lévy or diffusion models, and the draw-down function to be fixed, we describe, for a general spectrally negative Markov model, not only the optimal barrier but also the optimal draw-down function. This is achieved by solving a variational problem tackled by Pontryaghins maximum principle. As a by-product we show that in the Lévy case the classic first passage solution is indeed optimal; in the diffusion case, we obtain the optimality equations, but the existence of solutions improving the classic ones is left for future work.
Type de document :
Pré-publication, Document de travail
2018
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https://hal-upec-upem.archives-ouvertes.fr/hal-01961105
Contributeur : Dan Goreac <>
Soumis le : mercredi 19 décembre 2018 - 17:00:43
Dernière modification le : dimanche 23 décembre 2018 - 01:02:46

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  • HAL Id : hal-01961105, version 1
  • ARXIV : 1812.08438

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Avram Florin, Dan Goreac. A PONTRYAGHIN MAXIMUM PRINCIPLE APPROACH FOR THE OPTIMIZATION OF DIVIDENDS/CONSUMPTION OF SPECTRALLY NEGATIVE MARKOV PROCESSES, UNTIL A GENERALIZED DRAW-DOWN TIME. 2018. 〈hal-01961105〉

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