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

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 $X_t$ and on its supremum ${\overline{X}}_t:=\underset{0\le s\le t}{sup}{X}_s$. 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.

KEYWORDS:

• First passage
• draw-down process
• spectrally negative process
• scale functions
• dividends
• dividends barrier optimization
• de finetti
• optimal harvesting
• variational problem

Acknowledgments

The authors thank Hongzhong Zhang for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Dan Goreac  http://orcid.org/0000-0002-8693-6672

Notes

1 The equivalence between the two de-Finetti optimality conditions may be checked by differentiating $W_q^{\prime }\left(b^{\ast }\right)=W_q\left(b^{\ast }\right){\nu }_q\left(b^{\ast }\right)$, which yields $0=W_q^{″}\left(b^{\ast }\right)=W_q\left(b^{\ast }\right)\left({\nu }_q^{\prime }\right(b^{\ast })+{\nu }_q^2(b^{\ast }\left)\right)$.

2 ν and δ capture the behavior of excursions of the process away from its running maximum.

3 See Chan et al. (2011) for more information on the smoothness of scale functions for Lévy processes, and note this problem has not yet been studied for spectrally negative Markov processes.

4 We have conducted an analysis for d increasing and with slope $\le 1$ (‘normalized’ in some sense) pointing out that $d^{\prime }\le constant\ne 1$ does not introduce much difficulties.

5 When $\xi =0,d\left(b\right)=b$, we recover the classic de Finetti optimal barrier equation $W_q^{″}\left(b^{\ast }\right)=0\iff {\nu }_q\left(b^{\ast }\right)=\frac{q}{\mu }$

6 The formulas (56(56) $W_q(x,y)=\left[\begin{array}{cc}{\phi }_q^{+}\left(x\right)& {\phi }_q^{+}\left(y\right)\\ {\phi }_q^{-}\left(x\right)& {\phi }_q^{-}\left(y\right)\end{array}\right]={\phi }_q^{+}\left(x\right){\phi }_q^{-}\left(y\right)-{\phi }_q^{+}\left(y\right){\phi }_q^{-}\left(x\right).$(56) ), (59(59) $\begin{array}{r}{W}_q(x,y)=W_q^Z\left(F\right(x)-F(y\left)\right):=w_q\left(F\right(x)-F(y\left)\right),\end{array}$(59) ) differ by the function ${\mathrm{e}}^{(r_{+}+r_{-})F\left(y\right)}$, which does not affect the definition of ${\nu }_q(x,y)$.

7 Equivalently, the function $\frac{r}{F^{\prime }\left(x\right)F^{\prime }\left(y\right)}$ is decreasing in y.

8 As we have already seen for diffusion models, W is computed starting from an expression of type ${\mathrm{e}}^{r_{+}x}-{\mathrm{e}}^{r_{-}x}$. The natural (and immediate) generalization is to replace the linear forms $r_{+}x$ with a function $r_{+}(x,y)$ and, similar, $r_{-}x$ with $r_{-}(x,y)$. The simplest forms are $r_{+}$ a polynomial function and $r_{-}=-\mathrm{\infty }$.