November  2020, 16(6): 2781-2797. doi: 10.3934/jimo.2019080

Optimal investment and reinsurance with premium control

1. 

Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong

2. 

College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350117, China

* Corresponding author: Mi Chen

Received  October 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: The research of Xin Jiang and Kam Chuen Yuen was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU17329216). The research of Mi Chen was supported by National Natural Science Foundation of China (Nos. 11701087 and 11701088), Natural Science Foundation of Fujian Province (Nos. 2018J05003, 2019J01673 and JAT160130), Program for Innovative Research Team in Science and Technology in Fujian Province University, and the grant "Probability and Statistics: Theory and Application (No. IRTL1704)" from Fujian Normal University

This paper studies the optimal investment and reinsurance problem for a risk model with premium control. It is assumed that the insurance safety loading and the time-varying claim arrival rate are connected through a monotone decreasing function, and that the insurance and reinsurance safety loadings have a linear relationship. Applying stochastic control theory, we are able to derive the optimal strategy that maximizes the expected exponential utility of terminal wealth. We also provide a few numerical examples to illustrate the impact of the model parameters on the optimal strategy.

Citation: Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080
References:
[1]

S. AsmussenB. J. Christensen and M. Taksar, Portfolio size as function of the premium: Modelling and optimization, Stochastics, 85 (2013), 575-588.  doi: 10.1080/17442508.2013.797426.

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.

[3]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.

[4]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, vol. 25, Springer-Verlag, New York, 2006.

[5]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.

[6]

B. Højgaard, Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs, Scandinavian Actuarial Journal, 2002,225–245. doi: 10.1080/03461230110106291.

[7]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998,166–180. doi: 10.1016/S0167-6687(98)00007-9.

[8]

S. E. Jabari and H. X. Liu, A stochastic model of traffic flow: Gaussian approximation and estimation, Transportation Research Part B: Methodological, 47 (2013), 15-41.  doi: 10.1016/j.trb.2012.09.004.

[9]

Z. LiangL. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965.

[10]

Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350.  doi: 10.1080/15326340701300894.

[11]

Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016, 18–36. doi: 10.1080/03461238.2014.892899.

[12]

Z. LiangK. C. Yuen and K. C. Cheung, Optimal reinsurance and investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Applied Stochastic Models in Business and Industry, 28 (2012), 585-597.  doi: 10.1002/asmb.934.

[13]

A. Martin-Löf, Premium control in an insurance system, an approach using linear control theory, Scandinavian Actuarial Journal, 1983, 1–27. doi: 10.1080/03461238.1983.10408686.

[14]

X. F. PengL. H. Bai and J. Y. Guo, Optimal control with restrictions for a diffusion risk model under constant interest force, Applied Mathematics & Optimization, 73 (2016), 115-136.  doi: 10.1007/s00245-015-9295-3.

[15]

D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.

[16]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001, 55–68. doi: 10.1080/034612301750077338.

[17]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[18]

J. Thøegersen, Optimal premium as a function of the deductible: Customer analysis and portfolio characteristics, Risks, 4 (2016), 19 pages.

[19]

S. Thonhauser, Optimal investment under transaction costs for an insurer, European Actuarial Journal, 3 (2013), 359-383.  doi: 10.1007/s13385-013-0078-4.

[20]

M. Vandebroek and J. Dhaene, Optimal premium control in a non-life insurance business, Scandinavian Actuarial Journal, 1990, 3–13. doi: 10.1080/03461238.1990.10413869.

[21]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.

[22]

K. C. YuenZ. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009.

[23]

M. ZhouK. C. Yuen and C. C. Yin, Optimal investment and premium control in a nonlinear diffusion model, Acta Mathematicae Applicatae Sinica, 33 (2017), 945-958.  doi: 10.1007/s10255-017-0709-7.

show all references

References:
[1]

S. AsmussenB. J. Christensen and M. Taksar, Portfolio size as function of the premium: Modelling and optimization, Stochastics, 85 (2013), 575-588.  doi: 10.1080/17442508.2013.797426.

[2]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.

[3]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.

[4]

W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2$^{nd}$ edition, Stochastic Modelling and Applied Probability, vol. 25, Springer-Verlag, New York, 2006.

[5]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.

[6]

B. Højgaard, Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs, Scandinavian Actuarial Journal, 2002,225–245. doi: 10.1080/03461230110106291.

[7]

B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models, Scandinavian Actuarial Journal, 1998,166–180. doi: 10.1016/S0167-6687(98)00007-9.

[8]

S. E. Jabari and H. X. Liu, A stochastic model of traffic flow: Gaussian approximation and estimation, Transportation Research Part B: Methodological, 47 (2013), 15-41.  doi: 10.1016/j.trb.2012.09.004.

[9]

Z. LiangL. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965.

[10]

Z. Liang and J. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350.  doi: 10.1080/15326340701300894.

[11]

Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016, 18–36. doi: 10.1080/03461238.2014.892899.

[12]

Z. LiangK. C. Yuen and K. C. Cheung, Optimal reinsurance and investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Applied Stochastic Models in Business and Industry, 28 (2012), 585-597.  doi: 10.1002/asmb.934.

[13]

A. Martin-Löf, Premium control in an insurance system, an approach using linear control theory, Scandinavian Actuarial Journal, 1983, 1–27. doi: 10.1080/03461238.1983.10408686.

[14]

X. F. PengL. H. Bai and J. Y. Guo, Optimal control with restrictions for a diffusion risk model under constant interest force, Applied Mathematics & Optimization, 73 (2016), 115-136.  doi: 10.1007/s00245-015-9295-3.

[15]

D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.

[16]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001, 55–68. doi: 10.1080/034612301750077338.

[17]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[18]

J. Thøegersen, Optimal premium as a function of the deductible: Customer analysis and portfolio characteristics, Risks, 4 (2016), 19 pages.

[19]

S. Thonhauser, Optimal investment under transaction costs for an insurer, European Actuarial Journal, 3 (2013), 359-383.  doi: 10.1007/s13385-013-0078-4.

[20]

M. Vandebroek and J. Dhaene, Optimal premium control in a non-life insurance business, Scandinavian Actuarial Journal, 1990, 3–13. doi: 10.1080/03461238.1990.10413869.

[21]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.

[22]

K. C. YuenZ. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009.

[23]

M. ZhouK. C. Yuen and C. C. Yin, Optimal investment and premium control in a nonlinear diffusion model, Acta Mathematicae Applicatae Sinica, 33 (2017), 945-958.  doi: 10.1007/s10255-017-0709-7.

Figure 1.  Effect of $ \sigma^2 $ on $ p^\star_t $
Figure 2.  Effect of $ \sigma^2 $ on $ u^\star_t $
Figure 3.  Effect of $ \beta $ on $ \pi^\star_t $
Figure 4.  Effect of $ a $ on $ u^\star_t $
Figure 5.  Effect of $ \eta_{min} $ on $ p^\star_t $
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