# American Institute of Mathematical Sciences

April  2019, 15(2): 517-535. doi: 10.3934/jimo.2018055

## Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model

 1 College of Economics and Business Administration, Chongqing University, Chongqing 400030, China 2 Department of Mathematics, Wayne State University, MI, USA, 48202

* Corresponding author: Manman Li

Received  August 2017 Revised  October 2017 Published  April 2018

Fund Project: The research of M. Li was supported in part by MOE Project of Humanities and Social Sciences on the west and the border area (No.14XJC910001) and the Fundamental Research Funds for the Central Universities (No.106112016CDJXY100002). The research of G. Yin was supported in part by the National Science Foundation under DMS-1207667

This paper focuses on optimal threshold strategies for a spectrally negative Lévy (SNL) risk process with capital injections and proportional transaction costs. Restricted to solvency constraint, our model requires the shareholders of dividends prevent ruin by injecting capitals. Value function of the firm is assumed to be an expected discounted total [dividends less discounted capital injection]. Under such a setup, we derive certain key identities in connection with value function of the firm of a maximum dividend rate. Under restricted dividend rates and capital injection, we give analytical description of the maximum value function of the firm and the optimal threshold strategy explicitly.

Citation: Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial & Management Optimization, 2019, 15 (2) : 517-535. doi: 10.3934/jimo.2018055
##### References:
 [1] H. Albrecher, J. Hartinger and S. Thonhauser, On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model, ASTIN Bull., 37 (2007), 203-233. doi: 10.1017/S0515036100014847. [2] S. Asmussen, Applied Probability and Queues, World Scientific, 2003, [3] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance Math. Econom., 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0. [4] B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41 (2011), 611-644. [5] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a SNLP, Ann. Appl. Prob., 17 (2007), 156-180. doi: 10.1214/105051606000000709. [6] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [7] T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probab. Theory Relat. Fields, 150 (2011), 691-708. doi: 10.1007/s00440-010-0289-4. [8] H. S. Dai, Z. M. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections, Math. Meth. Oper. Res., 72 (2010), 129-143. doi: 10.1007/s00186-010-0312-7. [9] H. Gerber and E. Shiu, On optimal dividend strategies in the compound poisson model, North American Actuarial J., 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. [10] H. Gerber and E. Shiu, On optimal dividends: from reflection to refraction, J.Comput. Appl. Math., 186 (2006), 4-22. doi: 10.1016/j.cam.2005.03.062. [11] J. M. Harrison and A. J. Taylor, Optimal control of a Brownian storage system, Stoch. Process. Appl., 6 (1978), 179-194. [12] M. Jeanblanc-Picqué and A. N. Shiryaev, Optimization of the flow of dividends, Russian Math. Surveys, 50 (1995), 257-277. [13] N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramer-Lundberg model with capital injections, Insurance Math. Econom., 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] A. E. Kyprianou and R. L. Loeffen, Refracted Lévy processes, Annales de l'Instut Henri Poincaré, 46 (2010), 24-44. doi: 10.1214/08-AIHP307. [15] A. E. Kyprianou, V. Rivero and R. Song, Convexity and smoothness of scale functions and de Finetti's control problem, J. Th. Probab., 23 (2010), 547-564. doi: 10.1007/s10959-009-0220-z. [16] A. E. Kyprianou and F. Hubalek, Old, new examples of scale functions for spectrally negative Lévy processes, Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, 63 (2011), 119-145. [17] A. E. Kyprianou, R. Loeffen and J. Perez, Optimal control with absolutely continuous strategies for spectrally negative Lévy prcoesses, J. Appl. Probab., 49 (2012), 150-166. doi: 10.1239/jap/1331216839. [18] A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed. Springer, 2014. [19] A. Lambert, Completely asymmetric Lévy processes confined in a finite interval, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 251-274. doi: 10.1016/S0246-0203(00)00126-6. [20] M. Li and G. Yin, Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model, preprint, 2014. [21] X. S. Lin and K. P. Pavlova, The compound Poisson risk midel with a threhsold dividend strategy, Insurance Math. Econom., 38 (2006), 57-80. doi: 10.1016/j.insmatheco.2005.08.001. [22] R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680. doi: 10.1214/07-AAP504. [23] R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Probab., 46 (2009), 85-98. doi: 10.1239/jap/1238592118. [24] R. L. Loeffen, J. F. Renaud and X. W. Zhou, Occupation times of intervals until first passage times for spectrally negative Lévy processes, Stoch. Proc. Appl., 124 (2014), 1408-1435. doi: 10.1016/j.spa.2013.11.005. [25] A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance Math. Econom., 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [26] A. C. Y. Ng, On a dual model with a dividend threshold, Insurance Math. Econom., 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011. [27] M. R. Pistorius, On doubly reflected completely asymmetric Lévy processes, Stoch. Proc. Appl., 107 (2003), 131-143. doi: 10.1016/S0304-4149(03)00049-8. [28] M. R. Pistorius, On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum, J. Th. Probab., 17 (2004), 183-220. doi: 10.1023/B:JOTP.0000020481.14371.37. [29] J. L. Pérez and K. Yamazakic, On the refracted-reflected spectrally negative Lévy processes, Stochastic Process. Appl., 128 (2018), 306-331. doi: 10.1016/j.spa.2017.03.024. [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999. [31] S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and refelecting barriers, SIAM J. Control Optim., 22 (1984), 55-75. doi: 10.1137/0322005. [32] D. J. Yao, H. L. Yang and R. M. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64. doi: 10.1016/j.econmod.2013.10.026. [33] C.C. Yin and K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model, Statistics and Probability Letters, 81 (2011), 1841-1846. doi: 10.1016/j.spl.2011.07.022. [34] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007. [35] J. X. Zhu and H. L. 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.

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##### References:
 [1] H. Albrecher, J. Hartinger and S. Thonhauser, On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model, ASTIN Bull., 37 (2007), 203-233. doi: 10.1017/S0515036100014847. [2] S. Asmussen, Applied Probability and Queues, World Scientific, 2003, [3] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance Math. Econom., 20 (1997), 1-15. doi: 10.1016/S0167-6687(96)00017-0. [4] B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41 (2011), 611-644. [5] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a SNLP, Ann. Appl. Prob., 17 (2007), 156-180. doi: 10.1214/105051606000000709. [6] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [7] T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probab. Theory Relat. Fields, 150 (2011), 691-708. doi: 10.1007/s00440-010-0289-4. [8] H. S. Dai, Z. M. Liu and N. Luan, Optimal dividend strategies in a dual model with capital injections, Math. Meth. Oper. Res., 72 (2010), 129-143. doi: 10.1007/s00186-010-0312-7. [9] H. Gerber and E. Shiu, On optimal dividend strategies in the compound poisson model, North American Actuarial J., 10 (2006), 76-93. doi: 10.1080/10920277.2006.10596249. [10] H. Gerber and E. Shiu, On optimal dividends: from reflection to refraction, J.Comput. Appl. Math., 186 (2006), 4-22. doi: 10.1016/j.cam.2005.03.062. [11] J. M. Harrison and A. J. Taylor, Optimal control of a Brownian storage system, Stoch. Process. Appl., 6 (1978), 179-194. [12] M. Jeanblanc-Picqué and A. N. Shiryaev, Optimization of the flow of dividends, Russian Math. Surveys, 50 (1995), 257-277. [13] N. Kulenko and H. Schmidli, Optimal dividend strategies in a Cramer-Lundberg model with capital injections, Insurance Math. Econom., 43 (2008), 270-278. doi: 10.1016/j.insmatheco.2008.05.013. [14] A. E. Kyprianou and R. L. Loeffen, Refracted Lévy processes, Annales de l'Instut Henri Poincaré, 46 (2010), 24-44. doi: 10.1214/08-AIHP307. [15] A. E. Kyprianou, V. Rivero and R. Song, Convexity and smoothness of scale functions and de Finetti's control problem, J. Th. Probab., 23 (2010), 547-564. doi: 10.1007/s10959-009-0220-z. [16] A. E. Kyprianou and F. Hubalek, Old, new examples of scale functions for spectrally negative Lévy processes, Seminar on Stochastic Analysis, Random Fields and Applications VI Progress in Probability, 63 (2011), 119-145. [17] A. E. Kyprianou, R. Loeffen and J. Perez, Optimal control with absolutely continuous strategies for spectrally negative Lévy prcoesses, J. Appl. Probab., 49 (2012), 150-166. doi: 10.1239/jap/1331216839. [18] A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed. Springer, 2014. [19] A. Lambert, Completely asymmetric Lévy processes confined in a finite interval, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 251-274. doi: 10.1016/S0246-0203(00)00126-6. [20] M. Li and G. Yin, Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model, preprint, 2014. [21] X. S. Lin and K. P. Pavlova, The compound Poisson risk midel with a threhsold dividend strategy, Insurance Math. Econom., 38 (2006), 57-80. doi: 10.1016/j.insmatheco.2005.08.001. [22] R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680. doi: 10.1214/07-AAP504. [23] R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Probab., 46 (2009), 85-98. doi: 10.1239/jap/1238592118. [24] R. L. Loeffen, J. F. Renaud and X. W. Zhou, Occupation times of intervals until first passage times for spectrally negative Lévy processes, Stoch. Proc. Appl., 124 (2014), 1408-1435. doi: 10.1016/j.spa.2013.11.005. [25] A. Løkka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance Math. Econom., 42 (2008), 954-961. doi: 10.1016/j.insmatheco.2007.10.013. [26] A. C. Y. Ng, On a dual model with a dividend threshold, Insurance Math. Econom., 44 (2009), 315-324. doi: 10.1016/j.insmatheco.2008.11.011. [27] M. R. Pistorius, On doubly reflected completely asymmetric Lévy processes, Stoch. Proc. Appl., 107 (2003), 131-143. doi: 10.1016/S0304-4149(03)00049-8. [28] M. R. Pistorius, On exit and ergodicity of the completely asymmetric Lévy process reflected at its infimum, J. Th. Probab., 17 (2004), 183-220. doi: 10.1023/B:JOTP.0000020481.14371.37. [29] J. L. Pérez and K. Yamazakic, On the refracted-reflected spectrally negative Lévy processes, Stochastic Process. Appl., 128 (2018), 306-331. doi: 10.1016/j.spa.2017.03.024. [30] K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999. [31] S. E. Shreve, J. P. Lehoczky and D. P. Gaver, Optimal consumption for general diffusions with absorbing and refelecting barriers, SIAM J. Control Optim., 22 (1984), 55-75. doi: 10.1137/0322005. [32] D. J. Yao, H. L. Yang and R. M. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64. doi: 10.1016/j.econmod.2013.10.026. [33] C.C. Yin and K.C. Yuen, Optimality of the threshold dividend strategy for the compound Poisson model, Statistics and Probability Letters, 81 (2011), 1841-1846. doi: 10.1016/j.spl.2011.07.022. [34] M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207. doi: 10.1016/j.econmod.2011.09.007. [35] J. X. Zhu and H. L. 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.
The modified Lévy risk process
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