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Optimal Dividend Policy when Cash Reserves Follow a Jump-Diffusion Process Under Markov-Regime Switching

Part of: Stochastic systems and control Mathematical economics Stochastic analysis

Published online by Cambridge University Press:  30 January 2018

Zhengjun Jiang
Zhengjun Jiang*
Affiliation:
Beijing Normal University - Hong Kong Baptist University United International College
*
∗ Postal address: Beijing Normal University - Hong Kong Baptist University United International College, 28 Jinfeng Road, Tangjiawan, Zhuhai, 519085, P. R. China. Email address: [email protected]
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Abstract

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In this paper we study the optimal dividend payments for a company of limited liability whose cash reserves in the absence of dividends follow a Markov-modulated jump-diffusion process with positive drifts and negative exponential jumps, where parameters and discount rates are modulated by a finite-state irreducible Markov chain. The main aim is to maximize the expected cumulative discounted dividend payments until bankruptcy time when cash reserves are nonpositive for the first time. We extend the results of Jiang and Pistorius [15] to our setup by proving that it is optimal to adopt a modulated barrier strategy at certain positive regime-dependent levels and that the value function can be explicitly characterized as the fixed point of a contraction.

Keywords

Optimal dividend policyMarkov chainregime switchingjump-diffusion processstochastic control

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 91B70: Stochastic models 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 209 - 223
Copyright
© Applied Probability Trust 

References

Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.CrossRefGoogle 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.CrossRefGoogle Scholar
Avram, F., Palmowski, Z. and Pistorius, M. R. (2014). On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk-process in the presence of a penalty function. Preprint. Available at http://arxiv.org/abs/1110.4965.Google Scholar
Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261–308.CrossRefGoogle Scholar
Belhaj, M. (2010). Optimal dividend payments when cash reserves follow a Jump-diffusion process. Math. Finance 20, 313–325.CrossRefGoogle Scholar
Cadenillas, A., Sarkar, S. and Zapatero, F. (2007). Optimal dividend policy with mean-reverting cash reservoir. Math. Finance 17, 81–109.CrossRefGoogle 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.CrossRefGoogle Scholar
Décamps, J.-P. and Villeneuve, S. (2007). Optimal dividend policy and growth option. Finance Stoch. 11, 3–27.CrossRefGoogle Scholar
De Finetti, B. (1957). Su un'impostazione alternativa della teoria colletiva del rischio. Trans. XV Internat. Congr. Actuaries 2, 433–443.Google Scholar
Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22.CrossRefGoogle Scholar
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.CrossRefGoogle Scholar
Hamilton, J. D. (1990). Analysis of time series subject to changes in regime. J. Econometrics 45, 39–79.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Jeanblanc-Picqué, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257–278.CrossRefGoogle Scholar
Jiang, Z. and Pistorius, M. (2012). Optimal dividend distribution under Markov regime switching. Finance Stoch. 16, 449–476.CrossRefGoogle 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, 428–443.CrossRefGoogle 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.CrossRefGoogle Scholar
Løkka, A. and Zervos, M. (2008). Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance Math. Econom. 42, 954–961.CrossRefGoogle Scholar
Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.CrossRefGoogle Scholar
Sotomayor, L. R. and Cadenillas, A. (2011). Classical and singular stochastic control for the optimal dividend policy when there is regime switching. Insurance Math. Econom. 48, 344–354.CrossRefGoogle Scholar
Wei, J., Wang, R. and Yang, H. (2012). On the optimal dividend strategy in a regime-switching diffusion model. Adv. Appl. Prob. 44, 886–906.CrossRefGoogle Scholar
Wei, J., Yang, H. and Wang, R. (2010). Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. J. Optimization Theory Appl. 147, 358–377.CrossRefGoogle Scholar
Wei, J., Yang, H. and Wang, R. (2010). Optimal reinsurance and dividend strategies under the Markov-modulated insurance risk model. Stoch. Anal. Appl. 28, 1078–1105.CrossRefGoogle Scholar
Zhu, J. (2014). Singular optimal dividend control for the regime-switching Cramér–Lundberg model with credit and debit interest. J. Computational Appl. Math. 257, 212–239.CrossRefGoogle Scholar
Zhu, J. and Chen, F. (2013). Dividend optimization for regime-switching general diffusions. Insurance Math. Econom. 53, 439–456.CrossRefGoogle Scholar