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Optimal financing and dividend distribution in a general diffusion model with regime switching

Part of: Mathematical finance Hamilton-Jacobi theories, including dynamic programming

Published online by Cambridge University Press:  10 June 2016

Jinxia Zhu  and
Hailiang Yang
Jinxia Zhu*
Affiliation:
The University of New South Wales
Hailiang Yang*
Affiliation:
The University of Hong Kong
*
* Postal address: School of Risk and Actuarial Studies, The University of New South Wales, Kensington Campus, NSW 2052, Australia. Email address: [email protected]
** Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: [email protected]
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Abstract

We study the optimal financing and dividend distribution problem with restricted dividend rates in a diffusion type surplus model, where the drift and volatility coefficients are general functions of the level of surplus and the external environment regime. The environment regime is modeled by a Markov process. Both capital injection and dividend payments incur expenses. The objective is to maximize the expectation of the total discounted dividends minus the total cost of the capital injection. We prove that it is optimal to inject capital only when the surplus tends to fall below 0 and to pay out dividends at the maximal rate when the surplus is at or above the threshold, dependent on the environment regime.

Keywords

Dividendgeneral diffusionoptimizationoptimal financingregime-switching

MSC classification

Primary: 49L20: Dynamic programming method
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 406 - 422
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Bäuerle, N. (2004).Approximation of optimal reinsurance and dividend payout policies.Math. Finance 14,99–113.CrossRefGoogle Scholar
[2]Cadenillas, A.,Sarkar, S. and Zapatero, F. (2007).Optimal dividend policy with mean-reverting cash reservoir.Math. Finance 17,81–109.CrossRefGoogle Scholar
[3]Dickson, D. C. M. and Waters, H. R. (2004).Some optimal dividends problems.ASTIN Bull. 34,49–74.CrossRefGoogle Scholar
[4]Fleming, W. H. and Soner, H. M. (1993).Controlled Markov Processes and Viscosity Solutions (Appl. Math. (New York) 25).Springer,New York.Google Scholar
[5]He, L. and Liang, Z. (2008).Optimal financing and dividend control of the insurance company with proportional reinsurance policy.Insurance Math. Econom. 42,976–983.Google Scholar
[6]Højgaard, B. and Taksar, M. (2001).Optimal risk control for a large corporation in the presence of returns on investments.Finance Stoch. 5,527–547.Google Scholar
[7]Ikeda, N. and Watanabe, S. (1977).A comparison theorem for solutions of stochastic differential equations and its applications.Osaka J. Math. 14,619–633.Google Scholar
[8]Jiang, Z. and Pistorius, M. (2012).Optimal dividend distribution under Markov regime switching.Finance Stoch. 16,449–476.CrossRefGoogle Scholar
[9]Krylov, N. V. (1996).Lectures on Elliptic and Parabolic Equations in Hölder Spaces.American Mathematical Society,Providence, RI.CrossRefGoogle Scholar
[10]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.Google Scholar
[11]Paulsen, J. (2008).Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs.SIAM J. Control Optimization 47,2201–2226.CrossRefGoogle Scholar
[12]Scheer, N. and Schmidli, H. (2011).Optimal dividend strategies in a Cramér–Lundberg model with capital injections and administration costs.Europ. Actuarial J. 1,57–92.Google Scholar
[13]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.CrossRefGoogle Scholar
[14]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
[15]Taksar, M. I. (2000).Optimal risk and dividend distribution control models for an insurance company.Math. Meth. Operat. Res. 51,1–42.Google Scholar
[16]Yao, D.,Yang, H. and Wang, R. (2011).Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs.Europ. J. Operat. Res. 211,568–576.Google Scholar
[17]Zhu, J. (2014).Dividend optimization for a regime-switching diffusion model with restricted dividend rates.ASTIN Bull. 44,459–494.Google Scholar
[18]Zhu, J. (2015).Dividend optimization for general diffusions with restricted dividend payment rates.Scand. Actuarial J. 2015,592–615.Google Scholar
[19]Zhu, J. and Yang, H. (2015).Optimal financing and dividend distribution in a general diffusion model with regime switching. Preprint. Available at http://arxiv.org/abs/1506.08360.Google Scholar