Convexity and smoothness of scale functions and de Finetti's control problem

Andreas E. Kyprianou; Víctor Rivero; Renming Song

doi:10.1007/s10959-009-0220-z

Convexity and smoothness of scale functions and de Finetti's control problem

Andreas E. Kyprianou, Víctor Rivero, Renming Song

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Abstract

We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156-180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.

Original language	English (US)
Pages (from-to)	547-564
Number of pages	18
Journal	Journal of Theoretical Probability
Volume	23
Issue number	2
DOIs	https://doi.org/10.1007/s10959-009-0220-z
State	Published - Jun 2010

Keywords

Control theory
Potential analysis
Scale functions for spectrally negative Lévy processes
Special Bernstein function

ASJC Scopus subject areas

Statistics and Probability
General Mathematics
Statistics, Probability and Uncertainty

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10.1007/s10959-009-0220-z

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title = "Convexity and smoothness of scale functions and de Finetti's control problem",

abstract = "We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156-180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the L{\'e}vy measure of a spectrally negative L{\'e}vy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying L{\'e}vy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.",

keywords = "Control theory, Potential analysis, Scale functions for spectrally negative L{\'e}vy processes, Special Bernstein function",

author = "Kyprianou, {Andreas E.} and V{\'i}ctor Rivero and Renming Song",

note = "Funding Information: The authors gratefully acknowledge funding from EPSRC grant numbers EP/E047025/1 and EP/C500229/1 and Royal Society grant number RE-MA1004. We are also grateful to Ronnie Loeffen for his comments on earlier drafts of this paper, in particular with regard to the computations around (4.4). Thanks are also due to an associate editor and an anonymous referee for their valuable comments on an earlier version of this manuscript.",

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AU - Kyprianou, Andreas E.

AU - Rivero, Víctor

AU - Song, Renming

N1 - Funding Information: The authors gratefully acknowledge funding from EPSRC grant numbers EP/E047025/1 and EP/C500229/1 and Royal Society grant number RE-MA1004. We are also grateful to Ronnie Loeffen for his comments on earlier drafts of this paper, in particular with regard to the computations around (4.4). Thanks are also due to an associate editor and an anonymous referee for their valuable comments on an earlier version of this manuscript.

PY - 2010/6

Y1 - 2010/6

N2 - We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156-180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.

AB - We continue the recent work of Avram et al. (Ann. Appl. Probab. 17:156-180, 2007) and Loeffen (Ann. Appl. Probab., 2007) by showing that whenever the Lévy measure of a spectrally negative Lévy process has a density which is log-convex then the solution of the associated actuarial control problem of de Finetti is solved by a barrier strategy. Moreover, the level of the barrier can be identified in terms of the scale function of the underlying Lévy process. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators and their application to convexity and smoothness properties of the relevant scale functions.

KW - Control theory

KW - Potential analysis

KW - Scale functions for spectrally negative Lévy processes

KW - Special Bernstein function

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JF - Journal of Theoretical Probability

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Convexity and smoothness of scale functions and de Finetti's control problem

Abstract

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