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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
[1]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[2]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[3]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[4]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[5]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[6]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[7]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[8]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[9]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[10]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[11]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[12]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[13]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[14]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[15]

Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302

[16]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[17]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[18]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[19]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[20]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (157)
  • HTML views (526)
  • Cited by (0)

Other articles
by authors

[Back to Top]