[Paper Review] Convexity and smoothness of scale functions and de Finetti's control problem

Andreas E. Kyprianou, Víctor M. Hernández Rivero+1 more|ArXiv.org|Jan 13, 2008

Probability and Risk Models17 references4 citations

TL;DR

This paper establishes smoothness and convexity properties of scale functions for spectrally negative Lévy processes using potential analysis of subordinators. It proves that if the Lévy density is log-convex, then the scale function $ W^{(q)} $ is convex on $ (a^*, ∞) $, where $ a^* $ is the point where $ W^{(q)′} $ attains its global minimum, thereby solving de Finetti’s control problem via a barrier strategy at $ a^* $.

ABSTRACT

Under appropriate conditions, we obtain smoothness and convexity properties of $q$-scale functions for spectrally negative Lévy processes. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators. As an application of the latter results to scale functions, we are able to continue the very recent work of \cite{APP2007} and \cite{Loe}. We strengthen their collective conclusions by showing, amongst other results, that whenever the Lévy measure has a density which is log convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is the largest value at which $W^{(q)\prime}$ attains its global minimum. As a consequence we deduce that de Finetti's classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height $a^*$.

Motivation & Objective

• To establish smoothness and convexity properties of $ q $-scale functions for spectrally negative Lévy processes.
• To extend previous results on de Finetti’s classical actuarial control problem to a broader class of Lévy processes.
• To characterize the optimal barrier level $ a^* $ in terms of the global minimum of $ W^{(q)′} $.
• To link the convexity of scale functions to the log-convexity of the Lévy density via potential analysis of subordinators.
• To provide a rigorous analytical foundation for the optimality of barrier strategies in risk control under general Lévy dynamics.

Proposed method

• Utilizes recent developments in potential analysis of subordinators, particularly the theory of special subordinators.
• Applies the Wiener-Hopf factorization to relate the Laplace exponent of the descending ladder process to the scale function.
• Derives the Lévy density $ \upsilon_q(x) $ of the bivariate descending ladder subordinator $ \widehat{\kappa}(q, \cdot) $, showing it is non-increasing under log-convex Lévy density.
• Employs integration by parts and properties of log-convex functions to prove that $ \upsilon_q(x) = e^{\Phi(q)x} \int_x^\infty e^{-\Phi(q)y} \pi(y) dy $, which is non-increasing when $ \pi $ is log-convex.
• Uses the characterization of $ \widehat{\kappa}(q, \cdot) $ as a Bernstein function with drift $ d = \sigma^2/2 $ and Lévy measure derived from the potential measure of the ascending ladder process.
• Applies the time-space Wiener-Hopf factorization to connect $ \widehat{\kappa}(q, 0) = q / \Phi(q) $, enabling the derivation of the tail behavior of the Lévy measure of $ \widehat{\kappa}(q, \cdot) $.

Experimental results

Research questions

• RQ1 Under what conditions on the Lévy density does the $ q $-scale function $ W^{(q)} $ exhibit convexity on a half-line?
• RQ2 How is the optimal barrier level $ a^* $ in de Finetti’s control problem related to the global minimum of $ W^{(q)′} $?
• RQ3 Can the convexity of $ W^{(q)} $ be established via potential analysis of subordinators when the Lévy density is log-convex?
• RQ4 What is the precise structure of the Lévy measure of the descending ladder subordinator $ \widehat{\kappa}(q, \cdot) $, and how does it relate to the original Lévy measure $ \Pi $?
• RQ5 To what extent can the optimality of barrier strategies in de Finetti’s problem be generalized beyond the diffusion and compound Poisson cases?

Key findings

• If the Lévy density $ \pi $ is log-convex, then the Lévy density $ \upsilon_q $ of the subordinator $ \widehat{\kappa}(q, \cdot) $ is non-increasing.
• The function $ x \mapsto e^{\Phi(q)x} \int_x^\infty e^{-\Phi(q)y} \pi(y) dy $ is log-convex when $ \pi $ is log-convex.
• The scale function $ W^{(q)} $ is convex on $ (a^*, \infty) $, where $ a^* $ is the point where $ W^{(q)′} $ attains its global minimum.
• The optimal control strategy in de Finetti’s problem is a barrier strategy with barrier at $ a^* $, under the log-convexity of the Lévy density.
• The tail of the Lévy measure of $ \widehat{\kappa}(q, \cdot) $ is given by $ \Upsilon_q(z, \infty) = e^{\Phi(q)z} \int_z^\infty e^{-\Phi(q)u} \overline{\Pi}(u) du $, which is consistent with the derived density $ \upsilon_q(x) $.
• The scale function $ W^{(q)} $ is strictly increasing and continuous on $ [0, \infty) $, with Laplace transform $ \int_0^\infty e^{-\theta x} W^{(q)}(x) dx = 1 / (\psi(\theta) - q) $ for $ \theta > \Phi(q) $.