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Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs

Part of: Optimality conditions Stochastic systems and control Markov processes Existence theories

Published online by Cambridge University Press:  01 July 2016

Jostein Paulsen
Jostein Paulsen*
Affiliation:
University of Bergen
*
∗ Postal address: Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, 5008 Bergen, Norway. Email address: [email protected]
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Abstract

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The problem of optimal dividends paid until absorbtion at zero is considered for a rather general diffusion model. With each dividend payment there is a proportional cost and a fixed cost. It is shown that there can be essentially three different solutions depending on the model parameters and the costs. (i) Whenever assets reach a barrier y*, they are reduced to y* - δ* through a dividend payment, and the process continues. (ii) Whenever assets reach a barrier y*, everything is paid out as dividends and the process terminates. (iii) There is no optimal policy, but the value function is approximated by policies of one of the two above forms for increasing barriers. A method to numerically find the optimal policy (if it exists) is presented and numerical examples are given.

Keywords

Optimal dividenddiffusion modelimpulse controlbarrier strategy

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 49J15: Optimal control problems involving ordinary differential equations 49K15: Problems involving ordinary differential equations 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 669 - 689
Copyright
Copyright © Applied Probability Trust 2007 

References

Alvarez, L. H. R. (2004a). A class of solvable impulse control problems. Appl. Math. Optimization 49, 265–295.CrossRefGoogle Scholar
Alvarez, L. H. R. (2004b). Stochastic forest stand value and optimal timber harvesting. SIAM J. Control Optimization 42, 1972–1972.CrossRefGoogle Scholar
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.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
Cadenillas, A., Choulli, T., Taksar, M. and Zhang, L. (2006). Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm. Math. Finance 16, 181–202.Google Scholar
Cai, J., Gerber, H. U. and Yang, H. (2006). Optimal dividends in an Ornstein–Uhlenbeck type model with credit and debit interest. N. Amer. Actuarial J.. 10, 94–119.Google Scholar
Choulli, T., Taksar, M. and Zhou, X. Y. (2003). A diffusion model for optimal dividend distribution for a company with constraints on risk control. SIAM J. Control Optimization 46, 1946–1946.Google Scholar
Décamps, J. P. and Villeneuve, S. (2007). Optimal dividend policy and growth option. Finance Stoch. 11, 3–27.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (2004). Optimal dividends: analysis with Brownian motion. N. Amer. Actuarial J.. 8, 1–20.Google Scholar
Grandits, P., Hubalek, F., Schachermayer, W. and Zigo, M. (2007). Optimal expected exponential utility of dividend payments in a Brownian risk model. Scand. Actuarial J., 73–107 Google Scholar
Højgaard, B. and Taksar, M. (2004). Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quantitative Finance 4, 315–327.Google Scholar
Jeanblanc-Picqué, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Russian Math. Surveys 50, 257–277.CrossRefGoogle Scholar
Krylov, N. V. (1996). Lectures on Elliptic and Parabolic Equations in Hölder Spaces (Graduate Studies Math. 12). American Mathematical Society, Providence, RI.Google Scholar
Lungu, E. M. and Øksendal, B. (1997). Optimal harvesting from a population in a stochastic crowded environment. Math. Biosci. 145, 47–75.CrossRefGoogle Scholar
Paulsen, J. (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance Stoch. 7, 457–473.Google Scholar
Paulsen, J. and Gjessing, H. K. (1997). Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance Math. Econom. 20, 215–223.Google Scholar
Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optimization 22, 55–75.Google Scholar
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar