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October 2018, 14(4): 1443-1461. doi: 10.3934/jimo.2018015

## Optimal liability ratio and dividend payment strategies under catastrophic risk

 1 School of Statistics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China 2 Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

* Corresponding author: Lyu Chen

Received  December 2016 Revised  August 2017 Published  January 2018

Fund Project: This work was supported by National Natural Science Foundation of China (11571113,11771147), the 111 Project(B14019) and Faculty Research Grant of University of Melbourne

This paper investigates the optimal strategies for liability management and dividend payment in an insurance company. The surplus process is jointly determined by the reinsurance policies, liability levels, future claims and unanticipated shocks. The decision maker aims to maximize the total expected discounted utility of dividend payment in infinite time horizon. To describe the extreme scenarios when catastrophic events occur, a jump-diffusion Cox-Ingersoll-Ross process is adopted to capture the substantial claim rate hikes. Using dynamic programming principle, the value function is the solution of a second-order integro-differential Hamilton-Jacobi-Bellman equation. The subsolution--supersolution method is used to verify the existence of classical solutions of the Hamilton-Jacobi-Bellman equation. The optimal liability ratio and dividend payment strategies are obtained explicitly in the cases where the utility functions are logarithm and power functions. A numerical example is provided to illustrate the methodologies and some interesting economic insights.

Citation: Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015
##### References:
 [1] J. Aharony and I. Swary, Quarterly dividend and earnings announcements and stockholders' returns: An empirical analysis, The Journal of Finance, 35 (1980), 1-12. [2] L. H. R. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732. doi: 10.1137/060659375. [3] S. Asmussen, B. Høgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [4] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709. [5] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302. doi: 10.1214/09-AAP643. [6] L. Bai and J. Paulsen, On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs, Stochastic Processes and Their Applications, 122 (2012), 4005-4027. doi: 10.1016/j.spa.2012.08.009. [7] Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. [8] T. Choulli, M. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596. doi: 10.1088/1469-7688/1/6/301. [9] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rate, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [10] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [11] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053. doi: 10.1214/aoap/1060202833. [12] D. Filipović, M. Mayerhofer and P. Schneider, Density approximations for multivariate affine jump-diffusion processes, J. Econometrics, 176 (2013), 93-111. doi: 10.1016/j.jeconom.2012.12.003. [13] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [14] Z. F. Fu and Z. H. Li, Stochastic equations of non-negative processes with jump, Stochastic Process. Appl., 120 (2010), 306-330. doi: 10.1016/j.spa.2009.11.005. [15] X. Guo, J. Liu and X. Y. Zhou, A constrained non-linear regular-singular stochastic control problem with applications, Stochastic Processes and their Applications, 109 (2004), 167-187. doi: 10.1016/j.spa.2003.09.008. [16] L. He and Z. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance policy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003. [17] Z. Jin, H. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363. doi: 10.1016/j.insmatheco.2015.07.005. [18] Z. F. Li, Y. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203. doi: 10.1016/j.insmatheco.2011.09.002. [19] R. L. Loeffen and J. F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108, Gerber-Shiu Functions / Longevity risk and capital markets. doi: 10.1016/j.insmatheco.2009.09.006. [20] A. Lokka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [21] H. Meng and T.K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279. doi: 10.1137/090773167. [22] M. H. Miller and F. Modigliani, Dividend policy, growth, and the valuation of shares, Journal of Business, 34 (1961), 411-433. [23] M. H. Miller and K. Rock, Dividend policy under asymmetric information, The Journal of Finance, 40 (1985), 1031-1051. [24] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992. [25] J. L. Stein, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012. [26] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle, Economic Modeling, 29 (2012), 198-207. [27] J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494. doi: 10.1017/asb.2014.2. [28] J. Zhu and H. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422. doi: 10.1017/apr.2016.7.

show all references

##### References:
 [1] J. Aharony and I. Swary, Quarterly dividend and earnings announcements and stockholders' returns: An empirical analysis, The Journal of Finance, 35 (1980), 1-12. [2] L. H. R. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732. doi: 10.1137/060659375. [3] S. Asmussen, B. Høgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324. doi: 10.1007/s007800050075. [4] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180. doi: 10.1214/105051606000000709. [5] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302. doi: 10.1214/09-AAP643. [6] L. Bai and J. Paulsen, On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs, Stochastic Processes and Their Applications, 122 (2012), 4005-4027. doi: 10.1016/j.spa.2012.08.009. [7] Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. [8] T. Choulli, M. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596. doi: 10.1088/1469-7688/1/6/301. [9] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rate, Econometrica, 53 (1985), 385-407. doi: 10.2307/1911242. [10] B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [11] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053. doi: 10.1214/aoap/1060202833. [12] D. Filipović, M. Mayerhofer and P. Schneider, Density approximations for multivariate affine jump-diffusion processes, J. Econometrics, 176 (2013), 93-111. doi: 10.1016/j.jeconom.2012.12.003. [13] W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531. doi: 10.1137/S0363012902419060. [14] Z. F. Fu and Z. H. Li, Stochastic equations of non-negative processes with jump, Stochastic Process. Appl., 120 (2010), 306-330. doi: 10.1016/j.spa.2009.11.005. [15] X. Guo, J. Liu and X. Y. Zhou, A constrained non-linear regular-singular stochastic control problem with applications, Stochastic Processes and their Applications, 109 (2004), 167-187. doi: 10.1016/j.spa.2003.09.008. [16] L. He and Z. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance policy, Insurance: Mathematics and Economics, 42 (2008), 976-983. doi: 10.1016/j.insmatheco.2007.11.003. [17] Z. Jin, H. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363. doi: 10.1016/j.insmatheco.2015.07.005. [18] Z. F. Li, Y. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203. doi: 10.1016/j.insmatheco.2011.09.002. [19] R. L. Loeffen and J. F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108, Gerber-Shiu Functions / Longevity risk and capital markets. doi: 10.1016/j.insmatheco.2009.09.006. [20] A. Lokka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [21] H. Meng and T.K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279. doi: 10.1137/090773167. [22] M. H. Miller and F. Modigliani, Dividend policy, growth, and the valuation of shares, Journal of Business, 34 (1961), 411-433. [23] M. H. Miller and K. Rock, Dividend policy under asymmetric information, The Journal of Finance, 40 (1985), 1031-1051. [24] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992. [25] J. L. Stein, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012. [26] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle, Economic Modeling, 29 (2012), 198-207. [27] J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494. doi: 10.1017/asb.2014.2. [28] J. Zhu and H. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422. doi: 10.1017/apr.2016.7.
Optimal liability ratio values versus $c$
Optimal liability ratio values versus $\mu$
Optimal liability ratio values versus $\sigma$
Optimal liability ratio values versus $\gamma$
Optimal liability ratio values versus $\theta$ for logarithmic utility function
Optimal liability ratio values versus $\theta$ for power utility function
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