On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes

By Ronnie Loeffen. 2008

224 citations recorded by the source corpus.

We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433-443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156-180] studied the case when the risk process is modeled by a general spectrally negative Levy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3846420aa4eb8ffb27ecbc64773854ee1597244

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Citations in the corpus

An optimal dividends problem with transaction costs for spectrally negative Lévy processes (2009, unknown): “To be more specific, the q-scale function of X, W (q) : R → [0, ∞) where q ≥ 0, is the unique function such that W (q) (x) = 0 for x 0 and on [0, ∞) is a strictly increasing and continuous function characterized by its Laplace transform which is given by ∞ 0 e −θx W (q) (x)dx = 1 ψ(θ) − q for θ > ”
ON OPTIMAL DIVIDENDS IN THE DUAL MODEL (2012, methodology): “Regarding the spectrally negative Lévy model, we refer the reader to [6] for examples where barrier strategies are suboptimal and to [15] for a sufficient condition for optimality.”
A Lévy Insurance Risk Process with Tax (2008, background): “1 are the gamma process and the inverse Gaussian process (for more examples, see [16]).”
Special, conjugate and complete scale functions for spectrally negative Lévy processes (2007, background): “For example, in the theory of optimal stopping [1; 2] and [38], in the theory of optimal control [3], [40; 51] and [44], in the theory of queuing and storage models [21] and [4], in the theory of branching processes [10], [29] and [43], in the theory of insurance risk and ruin [15], [34], [33], [20]”
On magnitude, asymptotics and duration of drawdowns for L\'{e}vy models (2015, background): “To do so, we use the perturbation approach which has been developed by many researchers in similar contexts (e.g., Dassios and Wu [12], Landriault et al. [23], Li and Zhou [25], Loeffen et al. [26], and Zhang [37]).”
An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density (2009, background): “It was then shown in [20] that when X has a Lévy measure which has a completely monotone density, these conditions on the scale function are satisfied and, in particular, that W(q)′ is strictly convex on (0,∞).”
Optimal dividends problem with a terminal value for spectrally positive Levy processes (2013, background): “Avram et al (2007), Loeffen (2008) and Kyprianou et al. (2010) studied the case of S = 0 for spectrally negative Lévy processes.”
The Theory of Scale Functions for Spectrally Negative Lévy Processes (2011, methodology): “Loeffen’s solution is intricately connected to certain analytical properties of the associated scale functionW (q).”
Old and New Examples of Scale Functions for Spectrally Negative Levy Processes (2007, background): “In the very recent work of Loeffen (2007), the classical dividend control problem of De Finetti’s dividend problem has been considered in the context of the driving risk process being a general spectrally negative Lévy process denoted as usual X .”
Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem (2008, methodology): “As an application of the latter results to scale functions, we are able to continue the very recent work of [2] and [26].”
A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “…and have proved to be key in understanding optimal barrier strategies (see: Zhou, 2005; Renaud and Zhou, 2007; Kyprianou and Palmowski, 2007; Albrecher, Renaud and Zhou, 2008; Kyprianou and Zhou, 2009; Loeffen, 2009; Kyprianou, Rivero and Song, 2008b; Kyprianou and Loeffen, 2008; Loeffen, 2008a,b).”
On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function (2011, methodology): “10) is both necessary and sufficient, as follows by examining the variational inequality characterizing the problem; see Loeffen [27], Lemmas 1, 2.”