Abstract
This chapter is concerned with the handling of insurance risk. We focus on numerical methods for regime-switching models that are widely used in analyzing optimization problems arising from dividend payment strategy and ruin probability, which are commonly formulated as stochastic control and optimization problems. The regime-switching models better describe the market conditions (such as bull and bear) and other economic conditions. The model is more versatile and realistic, but more complex to deal with. It is virtually impossible to obtain closed-form solutions for the associated optimal control problems that require the solutions of the Hamilton–Jacobi–Bellman (HJB) equations. For the regime-switching models, these HJB equations become HJB systems of equations. Closed-form solutions are virtually impossible to obtain, It is thus necessary to solve the problem numerically. We will use Markov chain approximation techniques to develop numerical solutions, which have the advantages that the regularity of the value function and/or analytic properties of the associated systems of HJB equations and/or quasi-variational inequalities need not be known. A feasible numerical approximation schemes is constructed to find a good approximation to the underlying problems, and the numerical examples are provided to illustrate the performance of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Asmussen, S., Høgaard, B., and Taksar, M. (2000). Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Finance and Stochastics, 4: 299–324.
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-Out. Insurance: Math. and Economics, 20: 1–15.
Brown, J. R., (2001). Private pensions, mortality risk, and the decision to annuitize, Journal of Public Economics, 82: 29–62.
Choulli, T., Taksar, M., and Zhou, X. Y. (2001). Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction. Quant. Finance 1: 573–96.
Davidoff, T., Brown, J. and Diamond, P. (2005). Annuities and individual swelfare, M.I.T. Department of Economics Working Paper Series, The American Economic Review, 95: 1573–1590.
De Finetti, B. (1957). Su unimpostazione alternativa della teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries 2: 433–443.
Gerber, H. and Shiu, E. (2004). Optimal dividends: analysis with Brownian motion. North American Actuarial Journal, 8: 1–20.
Gerber, H. and Shiu, E. (2006). On optimal dividend strategies in the compound Poisson model. North American Actuarial Journal, 10: 76–93.
Hamilton, J. (1989). A new approach to the economic analysis of non-stationary time series. Econometrica, 57: 357–384.
Jin, Z. and Yin, G. (2011). A numerical method for annuity-purchasing decision making to minimize the probability of financial ruin for regime-switching wealth models, Internat. J. Computer Math., 88:1256–1282.
Jin, Z., Yin, G., and Yang, H.L. (2011). Numerical methods for dividend soptimization using regime-switching jump-diffusion models, Math. Control Related Fields, 1: 21–40.
Kapur, S. and Orszag, M., (1999). A portfolio approach to investment and annuitization during retirement. Working Paper. Birkbeck College, London.
Kushner, H. and Dupuis, P. (2001). Numerical Methods for Stochstic Control Problems in Continuous Time, volume 24 of Stochastic Modelling and Applied Probability. Springer, New York, second edition.
Lundberg, F. (1903). I. Approximerad framstallning af sannolikhetsfunktionen: II. Aterforsakring af kollektivrisker, Uppsala.
Milevsky, M., Moore, K., Young, V. R., (2006). Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin, Mathematical Finance, 16: 647–671.
Reddemann, S., Basse, T., von der Schulenburg, and Johann-Matthias G. (2010). On the Impact of the Financial Crisis on the Dividend Policy of the European Insurance Industry Geneva Papers on Risk and Insurance: Issues and Practice, 35(1): 53–62.
Richard, S., (1975). Optimal Consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of Financial Economics, 2: 187–203.
Schmidli, H., (2008). Stochastic Control in Insurance, Springer Verlag.
Sotomayor, L. and Cadenillas, A. (2011). Classical and Singular Stochastic Control for the Optimal Dividend Policy When There Is Regime Switching. Insurance: Mathematics and Economics, 48(3), 344–54.
Vanderhei, J. and Copeland, C., (2003). Can America afford tomorrows retirees: Results from the EBRI-ERF retirement security projection model, Issue Brief of the Employee Benefit Research Institute, www.ebri.org.
Wei, J., Yang, H. and Wang, R. (2010). Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. Journal of Optimization Theory and Applications, 147(2).
Yang, H. and Yin, G. (2004). Ruin probability for a model under Markovian switching regime, In T.L. Lai, H. Yang, and S.P. Yung, editors, Probability, Finance and Insurance, pages 206–217. World Scientific, River Edge, NJ.
Yaari, M. (1965). Uncertain lifetime, life insurance and the theory of the consumer, Review of Economic Studies, 32: 137–150.
Yin, G., Jin, Z., and Yang, H. (2010). Asymptotically optimal dividend policy for regime-switching compound Poisson models Acta Mathematicae Applicatae Sinica, 26:529–542.
Yin, G. and Zhang, Q. (2005). Discrete-time Markov Chains: Two-time-scale Methods and Applications, Springer, New York, NY.
Acknowledgments
The research of Zhuo Jin was supported in part by Faculty Research Grant of the University of Melbourne. The research of George Yin was supported in part by the National Science Foundation under DMS-1207667.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Jin, Z., Yin, G. (2013). Numerical Methods for Optimal Annuity Purchasing and Dividend Optimization Strategies under Regime-Switching Models: Review of Recent Results. In: Zeng, Y., Wu, S. (eds) State-Space Models. Statistics and Econometrics for Finance, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7789-1_10
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7789-1_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7788-4
Online ISBN: 978-1-4614-7789-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)