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Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes

Part of: Markov processes Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Andreas E. Kyprianou ,
Ronnie Loeffen  and
José-Luis Pérez
Andreas E. Kyprianou*
Affiliation:
University of Bath
Ronnie Loeffen*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
José-Luis Pérez*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
∗∗∗ Current address: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. R. L. gratefully acknowledges support from the AXA Research Fund.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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Abstract

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In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

Keywords

Scale functionruin problemde Finetti dividend problemcomplete monotonicity

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 150 - 166
Copyright
© Applied Probability Trust 

Footnotes

J.-L. P. acknowledges financial support from CONACyT, grant number 000000000129326.

References

Albrecher, H. and Thonhauser, S. (2009). Optimality results for dividend problems in insurance. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 103, 295–320.Google Scholar
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.Google Scholar
Avanzi, B. (2009). Strategies for dividend distribution: a review. N. Amer. Actuarial J.. 13, 217–251.Google Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156–180.Google Scholar
Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261–308.Google Scholar
Bayraktar, E. and Egami, M. (2008). Optimizing venture capital investments in a Jump diffusion model. Math. Methods Operat. Res. 67, 21–42.Google Scholar
Belhaj, M. (2010). Optimal dividend payments when cash reserves follow a Jump-diffusion process. Math. Finance 20, 313–325.Google Scholar
Biffis, E. and Kyprianou, A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46, 85–91.Google Scholar
Boguslavaskaya, E. (2006). Optimization problems in financial mathematics: explicit solutions for diffusion models. , University of Amsterdam.Google Scholar
Chan, T., Kyprianou, A. E. and Savov, M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Relat. Fields 150, 691–708.Google Scholar
De Finetti, B. (1957). Su un'impostazione alternativa della teoria collettiva del rischio. In Transactions of the 15th International Congress of Actuaries, Vol. II, pp. 433–443.Google Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Mit. Verein. Schweiz. Versicherungsmath. 69, 185–227.Google Scholar
Gerber, H. (1974). The dilemma between dividends and safety and a generalization of the Lundberg–Cramér formulas. Scand. Actuarial J.. 1974, 46–57.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (2006). On optimal dividend strategies in the compound Poisson model. N. Amer. Actuarial J.. 10, 76–93.Google Scholar
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities for competing claim processes. J. Appl. Prob. 41, 679–690.Google Scholar
Huzak, M., Perman, M., Šikić, H. and Vondraček, Z. (2004). Ruin probabilities and decompositions for general perturbed risk processes. Ann. Appl. Prob. 14, 1378–1397.Google Scholar
Jeanblanc-Picqué, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257–277.Google Scholar
Klüppelberg, C., Kyprianou, A.E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, {24–44}.Google Scholar
Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 349–365.Google Scholar
Kyprianou, A. E., Rivero, V. and Song, R. (2010). Convexity and smoothness of scale functions and de Finetti's control problem. J. Theoret. Prob. 23, 547–564.Google Scholar
Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 1669–1680.Google Scholar
Loeffen, R. L. (2009). An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone Jump density. J. Appl. Prob. 46, {85–98}.CrossRefGoogle Scholar
Loeffen, R. L. (2009). An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insurance Math. Econom. 45, 41–48.Google Scholar
Loeffen, R. L. and Renaud, J.-F. (2010). De Finetti's optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46, 98–108.CrossRefGoogle Scholar
Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Radner, R. and Shepp, L. (1996). Risk vs. profit potential: a model for corporate strategy. J. Econom. Dynamics Control 20, 1373–1393.Google Scholar
Schmidli, H. (2008). Stochastic Control in Insurance. Springer, London.Google Scholar
Stroock, D. W. (1999). A Concise Introduction to the Theory of Integration, 3rd edn. Birkhäuser, Boston, MA.Google Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.Google Scholar