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The Optimal Dividend Problem in the Dual Model

Part of: Stochastic processes Mathematical finance Stochastic systems and control

Published online by Cambridge University Press:  22 February 2016

Erik Ekström  and
Bing Lu
Erik Ekström*
Affiliation:
Uppsala University
Bing Lu*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden.
∗ Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden.
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Abstract

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We study de Finetti's optimal dividend problem, also known as the optimal harvesting problem, in the dual model. In this model, the firm value is affected both by continuous fluctuations and by upward directed jumps. We use a fixed point method to show that the solution of the optimal dividend problem with jumps can be obtained as the limit of a sequence of stochastic control problems for a diffusion. In each problem, the optimal dividend strategy is of barrier type, and the rate of convergence of the barrier and the corresponding value function is exponential.

Keywords

Optimal distribution of dividendsde Finetti's dividend problemoptimal harvestingsingular stochastic controljump diffusion model

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60G51: Processes with independent increments; L'evy processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 746 - 765
Copyright
© Applied Probability Trust 

References

Alvarez, L. H. R. and Rakkolainen, T. A. (2009). Optimal payout policy in presence of downside risk. Math. Meth. Operat. Res. 69, 27–58.CrossRefGoogle Scholar
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.CrossRefGoogle Scholar
Avanzi, B., Gerber, H. U. and Shiu, E. S. W. (2007). Optimal dividends in the dual model. Insurance Math. Econom. 41, 111–123.CrossRefGoogle Scholar
Avanzi, B., Shen, J. and Wong, B. (2011). Optimal dividends and capital injections in the dual model with diffusion. ASTIN Bull. 41, 611–644.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.CrossRefGoogle Scholar
Bayraktar, E. and Egami, M. (2008). Optimizing venture capital investments in a Jump diffusion model. Math. Meth. Operat. Res. 67, 21–42.CrossRefGoogle Scholar
Bayraktar, E. and Xing, H. (2009). Pricing American options for Jump diffusions by iterating optimal stopping problems for diffusions. Math. Meth. Operat. Res. 70, 505–525.CrossRefGoogle Scholar
Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359–372.CrossRefGoogle Scholar
Dai, H., Liu, Z. and Luan, N. (2010). Optimal dividend strategies in a dual model with capital injections. Math. Meth. Operat. Res. 72, 129–143.CrossRefGoogle Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman & Hall, London.CrossRefGoogle Scholar
Dayanik, S., Poor, H. V. and Sezer, S. O. (2008). Multisource Bayesian sequential change detection. Ann. Appl. Prob. 18, 552–590.CrossRefGoogle Scholar
Gugerli, U. S. (1986). Optimal stopping of a piecewise-deterministic Markov process. Stochastics 19, 221–236.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
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
Zhanblan-Pike, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257–277.Google Scholar