A pontryaghin maximum principle approach for the optimization of dividends/consumption of spectrally negative markov processes, until a generalized draw-down time

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 $X_t $Xt and on its supremum $\overline X_t:=\sup _{0\leq s\leq t} X_s $X¯t:=sup0≤s≤tXs. Strongly connected to Azema-Yor/generalized draw-down/trailing stop time 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 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 Pontryaghin's 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 behavior of associated solutions for further explicit models and the question of whether they do better than the classic solution is left for future work. Instead, we illustrate the novelty by a toy example, with a conveniently chosen scale-like function.