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Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market
doi: 10.3934/jimo.2022020
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## Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

 1 School of Mathematical Sciences, Nankai University, Tianjin 300071, China 2 School of Finance, Nankai University, Tianjin 300071, China

*Corresponding author: Huayue Zhang

Received  September 2021 Revised  December 2021 Early access February 2022

This paper studies the optimal proportional reinsurance and investment strategy for an insurer who invests one paired assets, where their price spread is described by Ornstein-Uhlenbeck (O-U) processes. The insurer's objective is to maximize the expected exponential utility of the terminal wealth in a finite time horizon under two risk models: a classical risk model and a diffusion model. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we characterize the optimal strategies and provide a verification result for the value function via the exponential integrability of the square of an O-U process. Finally, numerical examples are performed to obtain sensitivity analysis.

Citation: Pengxu Xie, Lihua Bai, Huayue Zhang. Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022020
##### References:
 [1] L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] F. E. Benth and K. H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl., 23 (2005), 687-704.  doi: 10.1081/SAP-200064457. [3] W. K. Bertram, Analytic solutions for optimal statistical arbitrage trading, Physica A: Statistical Mechanics and its Applications, Issue 11, 389 (2010), 2234-2243. [4] M. Boguslavsky and E. Boguslavskaya, Arbitrage under power, Risk, 17 (2004), 69-73. [5] M. Brachetta and H. Schmidli, Optimal reinsurance and investment in a diffusion model, Decis. Econ. Finance, 43 (2020), 341-361.  doi: 10.1007/s10203-019-00265-8. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [7] R. J. Elliott, J. van der Hoek and W. P. Malcolm, Pairs trading, Quant. Finance, 5 (2005), 271-276.  doi: 10.1080/14697680500149370. [8] E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative value arbitrage rule, Social Science Electronic Publishing, 19 (2006), 797-827. [9] H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph Series, 8., Distributed by Richard D. Irwin, Inc., Homewood, Ill., 1979. [10] J. Grandell, Aspects of Risk Theory, Springer, New York, 1991. doi: 10.1007/978-1-4613-9058-9. [11] B. H$\phi$jgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007. [12] Z. B. Liang, L. H. Bai and J. Y. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Appl. Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965. [13] Y. X. Lin, M. McCrae and C. Gulati, Loss protection in pairs trading through minimum profit bounds: A cointegration approach, J. Appl. Math. Decis. Sci., 2006 (2006), 1-14.  doi: 10.1155/JAMDS/2006/73803. [14] R. C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [15] S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal Pairs Trading: A Stochastic Control Approach, American Control Conference IEEE, 2008. doi: 10.1109/ACC.2008.4586628. [16] D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214. [17] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 1 (2001), 55-68.  doi: 10.1080/034612301750077338. [18] G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis, John Wiley Sons, 2004. [19] K. Vladislav, Optimal Convergence Trading, Technical report, EconWPA, 2004. [20] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.

show all references

##### References:
 [1] L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] F. E. Benth and K. H. Karlsen, A note on Merton's portfolio selection problem for the Schwartz mean-reversion model, Stoch. Anal. Appl., 23 (2005), 687-704.  doi: 10.1081/SAP-200064457. [3] W. K. Bertram, Analytic solutions for optimal statistical arbitrage trading, Physica A: Statistical Mechanics and its Applications, Issue 11, 389 (2010), 2234-2243. [4] M. Boguslavsky and E. Boguslavskaya, Arbitrage under power, Risk, 17 (2004), 69-73. [5] M. Brachetta and H. Schmidli, Optimal reinsurance and investment in a diffusion model, Decis. Econ. Finance, 43 (2020), 341-361.  doi: 10.1007/s10203-019-00265-8. [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [7] R. J. Elliott, J. van der Hoek and W. P. Malcolm, Pairs trading, Quant. Finance, 5 (2005), 271-276.  doi: 10.1080/14697680500149370. [8] E. Gatev, W. N. Goetzmann and K. G. Rouwenhorst, Pairs trading: Performance of a relative value arbitrage rule, Social Science Electronic Publishing, 19 (2006), 797-827. [9] H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph Series, 8., Distributed by Richard D. Irwin, Inc., Homewood, Ill., 1979. [10] J. Grandell, Aspects of Risk Theory, Springer, New York, 1991. doi: 10.1007/978-1-4613-9058-9. [11] B. H$\phi$jgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007. [12] Z. B. Liang, L. H. Bai and J. Y. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Appl. Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965. [13] Y. X. Lin, M. McCrae and C. Gulati, Loss protection in pairs trading through minimum profit bounds: A cointegration approach, J. Appl. Math. Decis. Sci., 2006 (2006), 1-14.  doi: 10.1155/JAMDS/2006/73803. [14] R. C. Merton, Option pricing when the underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [15] S. Mudchanatongsuk, J. A. Primbs and W. Wong, Optimal Pairs Trading: A Stochastic Control Approach, American Control Conference IEEE, 2008. doi: 10.1109/ACC.2008.4586628. [16] D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214. [17] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 1 (2001), 55-68.  doi: 10.1080/034612301750077338. [18] G. Vidyamurthy, Pairs Trading: Quantitative Methods and Analysis, John Wiley Sons, 2004. [19] K. Vladislav, Optimal Convergence Trading, Technical report, EconWPA, 2004. [20] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.
The sample path of $u^*$
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The influence of $z$ on $u^*$
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